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STUDY OF ELECTRICAL PROPERTIES AND STRUCTURE OF NASICON-TYPE SOLID ELECTROLYTES (FAST IONIC CONDUCTORS, MICROWAVE MEASUREMENTS, IMPEDANCE SPECTROSCOPY)

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JU L VJ — L International
8610521
D ygas, J o z e f R o m an
STUDY OF ELECTRICAL PROPERTIES AND STRUCTURE OF NASICON-TYPE
SOLID ELECTROLYTES
Northwestern University
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NORTHWESTERN UNIVERSITY
STUDY OF ELECTRICAL PROPERTIES AND STRUCTURE
OF NASICON-TYPE SOLID ELECTROLYTES
A DISSERTATION
SUBMITTED TO THE GRADUATE SCHOOL
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
f o r t h e degree
DOCTOR OF PHILOSOPHY
F i e l d of M a t e r i a l s Science
and Engineering
By
J o z e f R. Dygas j "3 Vs ^
EVANSTON, ILLINOIS
June, 1986
,
To my w i f e , Malgorzata
a s a small c o n s o l a t i o n f o r a l l t h a t we missed
ii
ABSTRACT
STUDY OF ELECTRICAL PROPERTIES AND STRUCTURE
OF NASICON-TYPE SOLID ELECTROLYTES
J o ze f R. Dygas
Dense, p o l y c r y s t a l l i n e samples of N a ^ r j S i P j O ^ an*3 Na3Zr2Si2P012
were pre pared by s o l i d s t a t e r e a c t i o n s between m ech an ic al ly mixed r e a ­
g e n t s . The e l e c t r i c a l p r o p e r t i e s were i n v e s t i g a t e d in t h e te mp er at ur e
range 20 to 450°C by t h e a . c . impedance s pe c tro s c op y a t fr e q u e n c i e s 1
t o 7x10 Hz and usi ng waveguide te c h n i q u e s a t microwave f r e q u e n c i e s .
A computer program was developed fo r t h e n o n l i n e a r l e a s t - s q u a r e s
a n a l y s i s of t h e complex impedance s p e c t r a which made use of s t a t i s t i c a l
weights r e f l e c t i n g r e s o l u t i o n of t h e measuring equipment. The a . c .
impedance s p e c t r a of both compounds (samples with io n - b l o c k in g e l e c ­
t r o d e s ) were modelled usi ng e q u i v a l e n t c i r c u i t s which co n ta i n ed con­
s t a n t phase elem ents .
For Na2Zr2SiP20j2> t h e g ra in -b ou nda ry p o l a r i z a ­
t i o n was i d e n t i f i e d over t h e te mp er at ur e range 50 to 410°C.
For
Na3Zr2S i 2 P0 ^ 2 > two d i s t i n c t p o l a r i z a t i o n p r o c e s s e s were observed in
h ig h - and lo w-temp erature re g i o n s . The io n - b l o c k in g e l e c t r o d e s were
r e p r e s e n t e d by a c o n s t a n t phase element. Onset of d i s p e r s i o n observed
in th e hig h-f re quen c y p a r t of th e ad m itt a nc e s p e c t r a a t room tempera­
t u r e was a s s o c i a t e d with frequency-dependent hopping c o n d u c t i v i t y .
New te c h n i q u e s f o r measuring the complex p e r m i t t i v i t y a t microwave
fr e q u e n c i e s were developed in v olv in g: (1) use of n o n l i n e a r l e a s t -
sq ua re s e s t i m a t i o n f o r v a r i a b l e te r m i n a t i o n measurements in a wave­
guide ; (2) a r i g o r o u s s o l u t i o n of t h e boundary va lu e problem fo r d e t e r ­
mination of the p e r m i t t i v i t y of a c e n te r e d E-plan s l a b in a r e c t a n g u l a r
waveguide. The microwave c o n d u c t i v i t i e s of both compounds a r e highe r
than t h e i r low-frequency bulk c o n d u c t i v i t i e s and in c r e a s e with fr eq u e n­
cy acc ord ing to a power-law dependence in agreement with th e p r e d i c ­
t i o n s of hopping models. At room te m p e ra t u re , th e l o c a l motion of ions
dominates the microwave c o n d u c t i v i t y .
C r y s t a l s t r u c t u r e s of th e two compounds were determined a t 20 and
300°C us in g neutron powder d i f f r a c t i o n and p r o f i l e re fin em en t. The high
i o n i c c o n d u c t i v i t y of Na2Zr2Si2P0^2 a t te m p e ra t ur es around 300°C was
c o r r e l a t e d with the wide openings between th e N a(l) and Na(2) s i t e s and
th e high r e l a t i v e occupancy f a c t o r of th e Na(2) p o s i t i o n s .
ACKNOWLEDGEMENTS
With g r e a t a p p r e c i a t i o n , I e x p re ss my g r a t i t u t e to my a d v i s o r ,
Pr o f e s s o r D. H. Whitmore, f o r h i s h e l p f u l guidance, c o n ti n u in g encour­
agement and s u p p o r t , which have made th e completion of t h i s work
possible.
I am p a r t i c u l a r l y indebted t o P r o fe s s o r M. E. Brodwin f o r h i s
v a l u a b l e counsel in t h e m a t t e r s of microwave te ch ni ques and e l e c t r i c a l
p r o p e r t i e s of m a t e r i a l s and fo r th e o p p o r t u n i t y of using th e microwave
ch arac te riz atio n laboratory.
I wish to thank P r o f e s s o r W. H. Baur fo r major c o n t r i b u t i o n to the
powder neu tro n d i f f r a c t i o n study and fo r i n s t r u c t i v e d i s c u s s i o n s r e ­
g a rd in g s t r u c t u r e d e t e r m i n a t i o n .
The a s s i s t n c e of Dr. J. Faber c o n t r i ­
buted t o t h e accomplishment of th e neu tron d i f f r a c t i o n experiment.
S p e c ia l thanks a r e due t o Dr. W. Jakubowski who i n i t i a t e d my i n t ­
e r e s t in f a s t i o n i c cond ucto rs and su ggested gra du a te s t u d i e s a t
Northwestern U n i v e r s i t y .
I would l i k e t o thank th e P r o f e s s o r s of th e M a t e r i a l s Science
and Engineering Department fo r t h e i r t u t o r s h i p .
The p a r t i c i p a t i o n in th e Ion ic T ra ns po rt Thrust Group gave oppor­
t u n i t y fo r many s t i m u l a t i n g i n t e r a c t i o n s with fell ow workers.
Thanks
a r e due t o Dr. F. Wong, Dr. M. S. Ansari and S. Anderson fo r d i s c u s ­
s i o n s re g a rd in g th e microwave measurements.
The e n l i g h t e n i n g de bat e s
with Dr. A. Pechenik and h i s f r i e n d s h i p have been e s p e c i a l l y rewarding.
I a p p r e c i a t e th e f r i e n d s h i p and he lp I r e c ei ve d from my c o l l e a g u e s ; Dr.
v
J . B. Phipps, Dr. C. D. Chaney, Dr. Y.-T. T s a i , S. B h a t t a c h a r j a , S. W.
Smoot, J. Olson, S. Lindsey and J . Westwood.
P r o f e s s i o n a l a s s i s t a n c e from J . Baker, P. Weiss and M. Greenley in
coping with e l e c t r o n i c equipment and t e c h n i c a l su pp or t of t h e M a t e r i a l s
Science Shop a r e g r a t e f u l l y acknowledged.
Mrs. L. Williams helped to
type t h i s t h e s i s .
L a s t , but not l e a s t , I would l i k e t o thank my p a r e n t s and my wife
fo r t h e i r p a t i e n c e and love .
The c o n t r i b u t i o n of my w if e , Malgorzata,
t o s o l v i n g th e s c a t t e r i n g problem f o r th e E-plane s l a b in a r e c t a n g u l a r
waveguide, her he lp in p r e p a r a t i o n of t h i s manuscript and, most of a l l ,
her c o n ti n u in g i n s p i r a t i o n a r e s p e c i a l l y c h e r is h e d .
This work has been su pport by t h e N a tio n al Science Foundation
(Grant DMR-8015803), t h e Department of Energy (Grant 0E-AC02-76ER02564)
and t h e Cabell Fellow ship .
TABLE OF CONTENTS
PAGE
A b s t r a c t .........................................................................................................................
iii
L i s t of t a b l e s .............................................................................................................
x
L i s t of f i g u r e s ..........................................................................................................
xi
I.
I n t r o d u c t i o n ...................................................................................................
II.
Review of t h e l i t e r a t u r e
2 .1 .
NASI CON..............................................................................................................
2 .2 .
Frequency dependent c o n d u c t i v i t y of f a s t i o n i c cond uctors
1
5
16
2 . 2 . 1 . Experimental d a t a ........................................................................................
16
2 . 2 . 2 . Theor ie s of t h e frequency-dependent i o n i c c o n d u c t i v i t y . . . .
21
III.
Measurement and a n a l y s i s of t h e complex impedance/admit­
ta nc e s p e c t r a
3 .1 .
I n t r o d u c t i o n ...................................................................................................
34
3 .2 .
C om pute r-c ontro lle d impedance measuring s yst em .........................
30
3 .3 .
Nonlinear l e a s t - s q u a r e s a n a l y s i s of complex s p e c t r a usi ng
weights based on i n s t r u m e n t a l r e s o l u t i o n .....................................
44
3.4
Comparison of d i f f e r e n t we ightin g schemes....................................
53
3 .5 .
Models of t h e a . c . response of s o l i d e l e c t r o l y t e / e l e c t r o d e s
c e l l s .................................................................................................................
63
IV.
Measurement of th e complex p e r m i t t i v i t y a t microwave
frequencies.
4 .1 .
I n t r o d u c t i o n ...................................................................................................
79
4 .2 .
The s t a n d a r d v a r i a b l e t e r m i n a t i o n method......................................
82
4 .3 .
A p p l i c a t i o n of n o n l i n e a r l e a s t - s q u a r e s f i t t i n g fo r e v a l u a ­
t i o n of t h e v a r i a b l e t e r m i n a t i o n measurements..........................
100
4 . 3 . 1 . E st im at ion of s c a t t e r i n g para mete rs f o r a r e c i p r o c a l d i s ­
c o n t i n u i t y in a single-mode waveguide...........................................
100
vii
PAGE
4.3.2.
D i r e c t e s t i m a t i o n of th e complex pro p a g a ti o n c o n s t a n t fo r
hom ogen eou sl y-f ill ed waveguide...........................................................
Ill
4.3.3.
Scheme f o r d a t a re d u c t i o n and examples........................................
117
4.4
E f f e c t of e l e c t r i c a l c o n t a c t between t h e waveguide and the
sample on t h e measured p e r m i t t i v i t y ................................................
131
The c e n t r a l l y - l o c a t e d E-plane s l a b in a r e c t a n g u l a r wave­
g u i d e .................................................................................................................
134
4.5.1.
Experimental s e t u p ....................................................................................
134
4.5.2.
C a l c u l a t i o n of th e s c a t t e r i n g m a t r i x .............................................
137
4.5.3.
Ev al uat io n of the complex p e r m i t t i v i t y from th e v a r i a b l e
t e r m i n a t i o n measurements........................................................................
145
4.5.4.
Numerical and exper iment al t e s t s .....................................................
148
V.
P r e p a r a t i o n of two NASICON compounds
5 .1 .
P r e p a r a t i o n of NASICON c e r a m i c s ........................................................
5 .2 .
X-ray powder d i f f r a c t i o n , m i c r o s t r u c t u r e c h a r a c t e r i z a t i o n
and chemical a n a l y s i s ..............................................................................
T i m e - o f - f l i g h t neutron powder d i f f r a c t i o n and s t r u c t u r e
refi ne ment by th e R i e t v e l d method
4 .5 .
VI.
153
155
6 .1 .
Experimental p r o c e d u r e ...........................................................................
164
6 .2 .
R e s u l t s and d i s c u s s i o n ...........................................................................
166
VII.
E l e c t r i c a l p r o p e r t i e s of NASICON s o l i d e l e c t r o l y t e s
7 .1 .
Complex impedance measurements and a n a l y s i s ................................
183
7 .2 .
R e s u l t s of microwave measurements.....................................................
206
7 .3 .
D i s c u s s i o n .......................................................................................................
214
V II I .
Summary..............................................................................................................
221
Notes and r e f e r e n c e s ..................................................................................................
224
viii
PAGE
Appendices:
A.
D e t a i l s of impedance matching and c a l i b r a t i o n fo r th e
computer c o n t r o l l e d impedance measuring s y s te m ............................
228
B.
Algorithm f o r n o n l i n e a r r e g r e s s i o n .....................................................
243
C.
Program PIRDAC f o r n o n l i n e a r l e a s t - s q u a r e s a n a l y s i s of th e
complex impedance/admittance s p e c t r a ................................................
247
The l e a s t - s q u a r e s e s t i m a t i o n of pa ram ete rs in n o n l i n e a r
i m p l i c i t model................................................................................................
274
P a r t i a l d e r i v a t i v e s and o t h e r formulas f o r t h e v a r i a b l e
t e r m i n a t i o n method.......................................................................................
278
Computer p r i n t o u t s f o r examples o f d a t a re d u c t i o n in th e
v a r i a b l e t e r m i n a t i o n method...................................................................
279
D e t a i l s of c a l c u l a t i o n f o r c e n t r a l l y l o c a t e d E-plane s l a b
in a r e c t a n g u l a r waveguide.....................................................................
289
V i t a ....................................................................................................................................
294
D.
E.
F.
G.
ix
LIST OF TABLES
PAGE
TABLE
3.1
R e s u l t s of n o n l i n e a r l e a s t - s q u a r e s f i t t i n g with d i f f e r e n t
weights f o r t h e s y n t h e t i c impedance sp ect rum ................................
58
R e s u l t s of n o n l i n e a r l e a s t - s q u a r e s f i t t i n g with d i f f e r e n t
weights f o r t h e exper iment al impedance spectrum of th e RC
c i r c u i t of F i g . 3 . 4 ........................................................................................
60
R e s u l t s of d a t a re d u c t i o n f o r measurement of p e r m i t t i v i t y of
NASICON (x=2) a t 12 3 .6°C, 8.37 GHz......................................................
130
Examples of t h e s c a t t e r i n g parameters c a l c u l a t e d fo r a cen­
t e r e d E-plan s l a b in X-band waveguide and e r r o r s of the
p e r m i t t i v i t y e s t i m a t e d using s y n t h e t i c d a t a ..................................
151
Cell c o n s t a n t s , c e l l volumes and r e s i d u a l s fo r Na7Zr7SiP70 , 9
and Na3Zr2S i 2P01 2 ................................................................... . . . . . . . r . : .
169
6.2
S t r u c t u r a l pa rame te rs of th e rhombohedral p h a s e s ........................
170
6.3
S t r u c t u r a l para mete rs f o r th e monoclinic Na3Zr2S i 2POj2 phase
171
6.4
I n t e r a t o m i c d i s t a n c e s fo r th e rhombohedral s t r u c t u r e s .............
172
6.5
In t e r a t o m i c d i s t a n c e s f o r th e monoclinic room tempe rature
s t r u c t u r e of Na3Zr2S i 2P012.......................................................................
173
Observed and s c a l e d up occupancy f a c t o r s and r e l a t i v e oc­
cu pancies fo r Na s i t e s ................................................................................
180
Apparent a c t i v a t i o n e n e r g i e s and p r e e x p o n e n ti a l f a c t o r s for
th e bulk i o n i c c o n d u c t i v i t y .....................................................................
187
Estimated a c t i v a t i o n e n e r g i e s and p r e e x p o n e t i a l f a c t o r s fo r
th e microwave c o n d u c t i v i t y .......................................................................
210
3.2
4.1
4.2
6.1
6 .6
7.1
7.2
x
LIST OF FIGURES
FIGURE
3.1
PAGE
Schematic diagram o f th e c o m p u te r - c o n tr o ll e d impedance
measuring s yst e m .............................................................................................
40
Equi valen t c i r c u i t r e p r e s e n t i n g th e a . c . s i g n a l path in
th e impedance measuring s yst em ...............................................................
42
Eq uivalent c i r c u i t used fo r computer g e n e r a t i o n of s y n t h e t i c
impedance d a t a . ...............................................................................................
56
3.4
Schematic arrangement of t h e t e s t RC c i r c u i t ...............................
56
3.5
Residual e r r o r s of th e a b s o l u t e va lu e and th e phase of
impedance f o r t h e t e s t R.C. c i r c u i t of F i g . 3 . 4 ............................
62
3.6
Complex plane p l o t s of adm ittanc e f o r th e c i r c u i t of F i g . 3.4
64
3.7
C i r c u i t i n c l u d i n g c o n s t a n t phase elements which i s equ iva ­
l e n t to th e sum of two Cole-Cole l i k e e x p r e s s io n s for th e
impedance............................................................................................................
69
3.8
Generalized fo ur element e q u i v a l e n t c i r c u i t 6CFC.........................
69
3.9
G en eralized e q u i v a l e n t c i r c u i t 13C0M...................................................
69
3.10 G en eralized e q u i v a l e n t c i r c u i t 17T0T...................................................
73
3.11 G en eralized e q u i v a l e n t c i r c u i t 12C0R...................................................
73
3.12 Eq uivalent c i r c u i t 18MAC with g e n e r a l i z e d f i n i t e le ng th
d i f f u s i o n impedance......................................................................................
75
3.2
3.3
4.1
4.2
4.3
4.4
I l l u s t r a t i o n of t h e v a r i a b l e t e r m i n a t i o n method for measure­
ment of the complex p e r m i t t i v i t y ..........................................................
83
Voltage s t a n d i n g wave r a t i o and phase of th e v o l t a g e r e f l e c ­
t i o n c o e f f i c i e n t as f u c t i o n of d i s t a n c e t o s h o r t c i r c u i t for
t h r e e samples of th e complex p e r m i t t i v i t y e = 1 6 - j 8 ......................
89
R e l a t i v e e r r o r s of t h e c o n d u c t i v i t y and of th e r e a l p a r t of
the p e r m i t t i v i t y caused by e r r o r of VSWR, AW=0.2dB+0.03W.. . .
90
R e l a t i v e e r r o r s of th e c o n d u c t i v i t y and of th e r e a l p a r t of
the p e r m i t t i v i t y caused by e r r o r of p o s i t i o n of v o l t a g e
minimum................................................................................................................
91
xi
PAGE
FIGURE
4.5
4.6
4.7
R e l a t i v e e r r o r s of th e c o n d u c t i v i t y and of t h e r e a l p a r t of
th e p e r m i t t i v i t y caused by e r r o r of r e f l e c t i o n c o e f f i c i e n t
of s h o r t c i r c u i t , ARs = 0.0 1..........................................................................
R e l a t i v e e r r o r s of th e c o n d u c t i v i t y and of th e r e a l p a r t of
th e p e r m i t t i v i t y caused by e r r o r of p o s i t i o n of s h o r t c i r c u i t
92
93
R e l a t i v e e r r o r s of the c o n d u c t i v i t y and of th e r e a l p a r t of
th e p e r m i t t i v i t y caused by e r r o r of sample l e n g t h .........................
94
To ta l r e l a t i v e e r r o r s of t h e c o n d u c t i v i t y and of th e r e a l
p a r t of p e r m i t t i v i t y f o r t h e worst combination of e r r o r s
95
R e l a t i v e e r r o r s of th e c o n d u c t i v i t y and of t h e r e a l p a r t of
th e p e r m i t t i v i t y caused by e r r o r of p o s i t i o n of v o l t a g e m in i­
mum - l i n e a r approximation e q u i v a l e n t of F i g . 4 . 4 ..........................
96
4.10 Voltage s t a n d i n g wave r a t i o and phase of the v o l t a g e r e f l e c ­
t i o n c o e f f i c i e n t as f u n c t i o n of reduced d i s t a n c e to s h o r t
c i r c u i t , f o r t h r e e samples of complex p e r m i t t i v i t y e = 2 0 -j 4 0 ..
97
4.8
4. 9
4.11 To ta l r e l a t i v e e r r o r s of t h e c o n d u c t i v i t y and of the r e a l p a r t
of p e r m i t t i v i t y f o r th e worst combination of e r r o r s .................... 98
4.12 S c a t t e r i n g m a tr ix r e p r e s e n t a t i o n for a d i s c o n t i n u i t y in
a s i n g l e mode waveguide................................................................................
101
4.13 Cross s e c t i o n o f th e waveguide sample ho ld e r f o r measurement
of th e p e r m i t t i v i t y using a c e n te r e d E-plane s l a b ........................
136
4.14 C o n fi gura tio n of a f i n i t e - l e n g t h , c e n t e r e d E-plane s l a b in a
r e c t a n g u l a r waveguide....................................................................................
138
5.1
O p ti c a l micrographs of th e sample of Na3Zr2S i 2P012 .......................
158
5.2
O p ti c a l micrographs of the sample of Na2Zr2SiP20 12.......................
159
6.1
R i e t v e l d re fin em en t p r o f i l e f o r the room-temperature phase
of Na3Zr2S i 2P012 ...............................................................................................
168
View, in t h e d i r e c t i o n approximately p a r a l l e l t o [0001], of
the h ig h - t e m p e r a tu r e phase of Na3ZR2S i 2P012 (300°C)....................
175
View, in th e d i r e c t i o n [1010], of t h e h ig h - t e m p e r a tu r e phase
of Na3Zr2S i 2P01 2 ...............................................................................................
176
6.2
6.3
7.1
The bulk i o n i c c o n d u c t i v i t y of Na3Zr2S i 2P012 and
xii
FIGURE
Na2Zr2S iP2 ° 1 2 ...........................................................................................................
7.2
Equi val en t c i r c u i t used t o model t h e complex ad mi tta nce s pe c ­
trum of the sample of Na9Zr9SiP90 19 with i o n -b l o c k in g gold
e l e c t r o d e s ............................ .. .............................................................................
7.3
Complex ad m itt a nc e of th e Na2Zr2SiP20 ,2 sample with gold e l e c
t r o d e s a t 19.5°C...............................................................................................
7.4
Complex ad m itt a nce of th e Na9Zr~SiP90 19 sample with gold
e l e c t r o d e s a t 54.5°C.......................................................................................
7.5
Complex ad mi tta nce of the Na9Zr9SiP90 , 9 sample with gold
e l e c t r o d e s a t 195°C.........................................................................................
7.6
Complex ad m itt a nc e o f th e Na9Zr9SiP90 19 sample with gold
e l e c t r o d e s a t 4 0 8 . 5°C....................................................................................
7.7
Temperature dependence of th e grain-b oun da ry c o n d u c t i v i t y of
Na2Zr2S iP 2 ° 1 2 ................................................................................................................
7.8
Eq uiv ale nt c i r c u i t used to model th e complex ad m itt a nce s p ec ­
trum of th e sample of Na3Zr2S i 2P0 1 2 with io n- b l o ck in g
platinum e l e c t r o d e s .........................................................................................
7 .9
Complex ad m itt a nc e of th e Na-Zr^Si-PO,9 sample with platinum
e l e c t r o d e s a t 4 0 8 .5°C....................................................................................
7.10 Complex ad m itt a nce of th e Na..Zr9S i 9P 0 ,9 sample with platinu m
e l e c t r o d e s a t 203°C.........................................................................................
7.11 Complex ad m itt a n ce of t h e Na,Zr9S i 9P 0 , 9 sample with platinu m
e l e c t r o d e s a t 1 3 4 .5°C....................................................................................
7.12 Complex ad mi tta nce of th e Na,Zr9S i 9P 0 ,9 sample with platinum
e l e c t r o d e s a t 50°C...........................................................................................
7.13 Temperature dependence of th e c o n d u c t i v i t y of N a^ Zr-SiP-0,9
a t d i f f e r e n t f r e q u e n c i e s .................................................. ..
7.14 Temperature dependence of the c o n d u c t i v i t y of Na,Zr9S i 9P 0 ,9
a t d i f f e r e n t f r e q u e n c i e s ...................................................
.r.
7.15 Temperature dependence of th e r e a l p a r t of p e r m i t t i v i t y of
Na2 Zr 2 SiP20 j 2 a t microwave f r e q u e n c i e s ...........................................
7.16 Temperature dependence of the r e a l p a r t of p e r m i t t i v i t y of
Na3Zr2S i 2 POj2 a t microwave f r e q u e n c i e s ...........................................
FIGURE
PAGE
7.17 Frequency dependence of t h e r e a l p a r t of microwave c o n d u c ti v ­
i t y of Na2Zr2SiP 20 j 2 a t s e v e r a l t e m p e r a t u r e s
7.18
Frequency dependence of t h e r e a l p a r t of microwave c o n d u c ti v ­
i t y o f Na2Zr2S i 2P0^2 a t s e v e r a l t e m p e r a t u r e s
xiv
215
216
I. INTRODUCTION
High io n i c c o n d u c t i v i t y (of th e or der 10
-4
-10
-1
-1 -1
Si cm ) in the
s o l i d s t a t e , a t te m p e ra t u re s f a r below the m e lti ng p o i n t , i s th e unique
pro p e rt y of a group of m a t e r i a l s c a l l e d f a s t i o n i c c on duc tor s. This
pro p e rt y per mi ts t e c h n o l o g i c a l a p p l i c a t i o n s in v a r i o u s e le c tr o c h e m ic a l
de vic e s [1-5] such as primary and secondary b a t t e r i e s , f u e l c e l l s , gas
s e n s o r s , i o n - s e l e c t i v e e l e c t r o d e s , e l e c tr o c h r o m i c d i s p l a y s , water vapor
e l e c t r o l y z e r s , e t c . In r e c e n t y e a r s , as t h e p o t e n t i a l for important
a p p l i c a t i o n s of f a s t conductors in th e f i e l d s of energy conversion and
s t o r a g e has become e v i d e n t , the r e s e a r c h e f f o r t on th os e m a t e r i a l s has
in c re a s e d s u b s t a n t i a l l y [ 1 , 4 ] . Fa st i o n i c conductors pose a c ha ll e n g e
fo r both exp erim ent al and t h e o r e t i c a l r e s e a r c h aimed a t improving our
unde rs tand in g of v a r i o u s p h y s i c a l phenomena in such m a t e r i a l s [6 - 8 ].
Among th e a l k a l i ion c o nduct or s, the NASICON family of compounds
has been i n t e n s i v e l y s t u d i e d s i n c e the d is c o v e r y of high sodium ion
m o b i l i t y in th e Nai +xZr2S i xP3_x° i 2 s ystem [ 9 . 1 0 ] . Named for i t s sodium
s u p e r i o n i c c o n d u c t i v i t y , th e Na.jZr 2 S i 2 P0 j 2 compound was proposed as an
a l t e r n a t i v e s o l i d e l e c t r o l y t e to the Na B" or 8 aluminas in the sodiumsu lphur c e l l . Although the a p p l i c a b i l i t y of NASICON in p r a c t i c a l Na/S
c e l l s i s s t i l l u n c e r t a i n , a b a s i c un de rs ta n d in g of th e c o n d u c t i v i t y
mechanism in t h i s m a t e r i a l i s of g r e a t s c i e n t i f i c i n t e r e s t because the
NASICON compounds t y p i f y a l a r g e c l a s s of s o l i d m a t e r i a l s e x h i b i t i n g
f a s t i o n i c t r a n s p o r t through a network of i n t e r s e c t i n g channels in a
r i g i d framework c r y s t a l s t r u c t u r e . The f a c t t h a t s i n g l e c r y s t a l s of the
1
2
composition x x 2 , i . e . in the range of high io n i c c o n d u c t i v i t y , were not
grown u n t i l l r e c e n t l y [1 1] , had s e r i o u s l y l i m i t e d th e number of e x p e r i ­
mental probes a v a i l a b l e to study Na+ ion t r a n s p o r t in t h i s m a t e r i a l .
In th e p r e s e n t work, dense p o l y c r y s t a l l i n e NASICON ceramics of two
compositions, Na3Zr2S i 2P012 (x=2) and Na2Zr2SiP2012 (x=l) were s y n th e ­
s i z e d . The x=2 compound e x h i b i t s high io n i c c o n d u c t i v i t y (cr=0.215fl- 1 cm-1
a t 300°C) and undergoes a phase t r a n s i t i o n around 150°C from the roomtem pe rat ure monoclinic symmetry C2/c to th e rhombohedral symmetry R32/c.
The x=l compound i s rhombohedral a t room temperat ure and has io n i c
c o n d u c t i v i t y an order of magnitude lower than the x=2 compound. The
e l e c t r i c a l p r o p e r t i e s of th e two compounds were s t u d i e d between 20 and
450°C and over a wide range of f r e q u e n c i e s . The c r y s t a l s t r u c t u r e was
determined by neu tron powder d i f f r a c t i o n a t room te mp erat ure and a t
300°C. A comparison of the r e s u l t s f o r th e s e two compounds pe rm its the
c o r r e l a t i o n of the s t r u c t u r a l and e l e c t r i c a l p r o p e r t i e s with th e high
io n i c m o b i l i t y .
A review of the l i t e r a t u r e on NASICON and on the fr equ ency-dependence of the c o n d u c t i v i t y of f a s t i o n i c conductors i s p r e s e n t e d in
Chapter I I .
The phase of t h i s i n v e s t i g a t i o n which d e a l t with a . c . impedance
7
sp ect ro sc op y in th e frequency range 1 - 10 Hz was aimed a t th e c h a r a c ­
t e r i z a t i o n of the e l e c t r i c a l response of a c e l l c o n s i s t i n g of th e poly­
c r y s t a l l i n e NASICON compound with s p u t t e r e d , io n - b l o c k in g e l e c t r o d e s .
A new and g r e a t l y improved method for a n a l y s i s of the complex impedance
s p e c t r a was developed, based on n o n l i n e a r , l e a s t - s q u a r e s f i t t i n g of
3
model response f u n c t i o n s to the d a t a . This scheme involved weighting the
complex impedance d a t a acc ording to the experimental u n c e r t a i n t i e s i n ­
h e re n t in the measuring equipment. The method and i t s computer implemen­
t a t i o n a re d e s c r ib e d in Chapter I I I and Appendices B and C.
Measurements of th e c o n d u c t i v i t y and th e p e r m i t t i v i t y in the micro­
wave reg ion were undertaken in order to probe th e e l e c t r i c a l response of
the NASICON compounds on a time s c a l e c h a r a c t e r i s t i c of the t r a n s p o r t
pro c e ss es in the f a s t io n i c conductor. The frequency dependence of the
c o n d u c t i v i t y can provide infor mat ion about the mechanism of io n i c t r a n s ­
p o r t in th e m a t e r i a l [ 6 , 7 ] . The microwave measurements of the high io n ic
c o n d u c t i v i t y of NASICON a t e l e v a t e d te mp erat ures turne d out to be more
d i f f i c u l t than expected because of problems with e l e c t r i c a l c o n t a c t
between a sample and a microwave waveguide. New experimental te chn iq ues
had to be developed f o r measurement of the complex p e r m i t t i v i t y a t
microwave f r e q u e n c i e s a f t e r a p p l i c a t i o n of the s ta nd a rd methods f a i l e d
to y i e l d r e l i a b l e r e s u l t s . Chapter IV p r e s e n t s a new method, developed
in t h i s i n v e s t i g a t i o n , for d a t a re d u c ti o n in the v a r i a b l e t e r m in a ti o n
measurements in waveguides and a new, rig o r o u s method for the de term in a ­
t i o n of the complex p e r m i t t i v i t y using a c en te re d E-plane s l a b in a
r e c t a n g u l a r waveguide. The in v e rs e problem of fi n d i n g the c o n s t i t u t i v e
parameters of a s l a b from the observed s c a t t e r i n g of the fundamental
waveguide mode was s o lv e d , based on a complete s o l u t i o n of the boundary
value problem.
P r e p a r a t i o n of NASICON ceramics and m a t e r i a l c h a r a c t e r i z a t i o n are
de sc ri b e d in Chapter V. The r e s u l t s of the t i m e - o f - f l i g h t neutron powder
d i f f r a c t i o n study a r e p re s e n te d in Chapter VI. Using the R ie tv e ld method
of s t r u c t u r e re fi n e m e n t, i t was p o s s i b l e to o b t a i n a c c u r a t e atomic
p o s i t i o n s , temp er ature f a c t o r s for the atoms and s i t e occupancies for
p o l y c r y s t a l l i n e samples. Comparison of th e r e s u l t s for the two composi­
t i o n s s t u d i e d led to c oncl us io n t h a t th e high i o n ic c o n d u c t i v i t y of
Na2 Zr 2 S i 2 P0 ^2 > e v i d e n t a t high te m p e ra t u re s , i s r e l a t e d to change of
sodium atom occupancies and to widening of the pathways between the Na
sites.
The r e s u l t s of e l e c t r i c a l measurements and t h e i r a n a l y s i s are
pr e se n te d in Chapter VII. Although a comprehensive u nd er st and ing of the
phenomena r e s p o n s i b l e fo r d i s p e r s i o n of the c o n d u c t i v i t y observed over a
wide range of f r e q u e n c i e s and te m p e ra t ur es i s s t i l l l a c k i n g , an i n t e r ­
p r e t a t i o n in terms of hopping mechanism has been proposed for the f r e ­
quency-dependent c o n d u c t i v i t y observed in the microwave region.
II. REVIEW OF THE LITERATURE
2.1 NASICON
Skeleton S t r u c t u r e s
The dis co ve ry of h ig h, sodium-ion c o n d u c t i v i t y in t h e compounds
Nal+xZr2s i xp3-x°12 <wlth 1 - 8<x<2.4 ) was t h e r e s u l t of a s y s t e m a t i c
s ea rc h a t M.I.T. Lincoln L a b o r a t o r i e s f o r f a s t a l k a l i ion cond uctors
with " s k e l e t o n " s t r u c t u r e s [12].
A skeleton s tr u c tu r e c o n sis ts of a
r i g i d oxide framework s t a b i l i z e d by e l e c t r o n s donated by a l k a l i ions
which occupy t h r e e - d i m e n s i o n a l l y l i n k e d i n t e r s t i t i a l space.
Strong
chemical bonding w i t h i n t h e r i g i d framework makes t h e bonding between
t h e a l k a l i ions and t h e network more i o n i c and l e s s s i t e s p e c i f i c .
If
t h e c r o s s s e c t i o n of an i n t e r s t i t i a l passageway i s l a r g e enough to
accommodate th e a l k a l i i o n , t h e high i o n i c m o b i l i t y may be expected.
Besides NASICON, s e v e r a l o t h e r f a s t i o n i c conductors with s k e l e t o n
s t r u c t u r e s and th r e e - d i m e n s i o n a l c o n d u c t i v i t y p a t h s , have been d is c o v ­
ered [ 9 , 1 2 ] .
Some o f them a r e s t r u c t u r a l l y s i m i l a r to NASICON, fo r
example »a3Sc2P3o 12 (131. N a ^ Y ^ r - ^ P j O ^ [1 4 ] . H a ^ l ^ Z r ^ P j O ^
(L=Cr, In , Yb) [ 1 5 ] , Na3Zr2 - xYbxs i 2-x Pl+x012
In case of o t h e r s »
p a r t of the a l k a l i ions c o n t e n t i s mobile, while the balanc e of the
a l k a l i ions p a r t i c i p a t e in forming th e r i g i d framework, as exem pl ifi ed
by th e compounds Na5YSi40 12 [ 1 7 ] , Na5GdSi40 12 [1 8 ] , L i14ZnGe40 16
(LISICON) [19].
There e x i s t a l s o s k e l e t o n s t r u c t u r e s having an i n t e r s t i t i a l space
c o n s i s t i n g of p a r a l l e l , one-dimensional t u n n e l s , as ex em pl ifi ed by K-
6
h o l l a n d i t e , KxM9x/ 2 T*8-x/20 12 ^ 0 ] , and 0_eucry p t i t e . LiAlSiO^ [21].
S t r u c t u r e and Phase T r a n s i t i o n s
The compounds in the system Nai +xZr2S i xP3 -x ° 1 2 ’ ° ^ 3 * belong to
rhombohedral space group r 32c , except in th e range 1.8<x<2.2 where a
d i s t o r t i o n to monoclinic C2/c space group t a k e s p la ce a t room tempera­
t u r e [10].
The c r y s t a l s t r u c t u r e of the monoclinic phase of
Na3Zr2S i 2P012 was o r i g i n a l l y proposed by Hong [10] on the b a s i s of a
s i n g l e - c r y s t a l , X-ray s t r u c t u r e refi ne me nt of the rhombohedral endmember NaZr2P30^2> of t h e s o l i d s o l u t i o n range.
According to Hong's
pro p o s a l, the s k e l e t o n ( Zr2S i xP3_x°i2 ^ ^ +X^_ i s formed by ZrOg o c t a hedra s h a r in g i t s s i x c o r n e r s with SiO^ and P04 t e t r a h e d r a , each t e t r a hedron in tu r n s h a r i n g i t s four c o r n e r s with o c ta h e d ra .
Each 0
2
-
anion
bonds s t r o n g l y to t e t r a h e d r a l l y and o c t a h e d r a l l y c o o r d in a te d c a t i o n s .
There a r e four Na p o s i t i o n s per formula u n i t .
In th e rhombohedral
symmetry, t h e r e a re two d i s t i n g u i s h a b l e s i t e s : N a(l) in p o s i t i o n s 6(b)
of th e space group R32c and Na(2) in p o s i t i o n s 18 (e ). In t h e monoclinic
symmetry the t h r e e Na(2) s i t e s a r e d i s t o r t e d i n t o one Na(2) s i t e [ in
p o s i t i o n s 4(e) of t h e space group C2/c] and two Na(3) s i t e s [ in p o s i t i o n s 8 ( f ) ] . S i t e N a( l) i s o c t a h e d r a l l y - c o o r d i n a t e d by s i x 0
2_
ions of
the two ZrOg o c ta he dr a and i s th e only occupied Na atom p o s i t i o n in
NaZr2P30 , 2 .
The Na(2) s i t e s a r e i r r e g u l a r l y , e i g h t - c o o r d i n a t e d and a re
arra nged o c t a h e d r a l l y around a N a(l) s i t e a t a d i s t a n c e of approximately
3.5 8.
Hong concluded t h a t a l l Na+ ions a r e mobile and t h a t the conduction
path c o n s i s t s of N a ( l ) , Na(2) and Na(3) s i t e s .
The path fo r th e Na+
ions between two neighbor N a(l) s i t e s le a d s through an Na(2) or Na(3)
site.
Each Na(2) or Na(3) s i t e i s connected through a " b o t t l e n e c k " to
two N a(l) s i t e s , but t h e r e i s no passageway for th e Na+ ions between
Na(2) s i t e s or Na(2) and Na(3) s i t e s .
The t h r e e - d i m e n s i o n a l conduction
network i s in t e r c o n n e c t e d a t each Na(l) s i t e by b o t t l e n e c k s to two Na(2)
and four Na(3) s i t e s .
The jump d i s t a n c e between the s i t e s i s 3 . 5 - 3 . 8 8.
The ion exchange experiments i n d i c a t e d t h a t a l l Na+ ions in
Na.jZr2Si2P0j2 a r e mobile and can be exchanged fo r Li+ , Ag+ and K+ ions
in molten s a l t s c o n t a i n i n g t h e s e c a t i o n s .
Tranqui e t . a l .
[22] s t u d i e d th e s i n g l e - c r y s t a l s t r u c t u r e of
N a ^ Z ^ S i j O ^ a t 20, 300 and 620°C.
Through an a n a l y s i s of th e compo­
n e n ts of thermal v i b r a t i o n s and d i f f e r e n c e - F o u r i e r maps, they concluded
t h a t the main d i f f u s i o n path le ad s through pathways between th e Na(2)
s i t e s , while ion exchange between N a(l) and Na(2) s i t e s i s only moder­
ate.
They s t a t e d , in c o n t r a s t to th e f i n d i n g s of Hong [ 10] , t h a t the
opening of b o t t l e n e c k between Na(l) and Na(2) i s s m a l le r than th e Na+
ion and they found t h a t a d i r e c t pathway between Na(2) s i t e s can accom­
modate Na+ io n s .
Four such pathways l e a d i n g to each Na(2) s i t e form a
th r e e - d im e n s i o n a l network.
The jump d i s t a n c e i s 4.72 8.
The opening of
the proposed pathway i s l i m i t e d on one s i d e by th e Na+ ion a t th e Na(l)
s i t e , which v i b r a t e s with high a mp lit ude, thu s changing th e pathway
cross section.
The mo noc lin ic , room-temperature phase of Na.jZr2Si2P0^2 undergoes a
phase t r a n s i t i o n to a rhombohedral phase which i s common t o the e n t i r e
8
Nal+xZr2S i xP3-x°12
system a t high te m pe rat ure s .
The t r a n s i t i o n
te mp er atu re around 150°C has been r e p o r t e d by von Alpen e t . a l .
th e b a s i s of s p e c i f i c heat measurements and by B oil ot e t . a l .
th e b a s i s of X-ray d i f f r a c t i o n .
[23] on
[24] on
The change of c r y s t a l symmetry i s
accompanied by anomalies in the s p e c i f i c he at [23] and in th e d i l a t o m e t r i c curve [24].
Attempts to grow s i n g l e c r y s t a l s of Na3Zr 2 S i 2 POj 2 y i e l d e d c r y s t a l s
with compositions ly i n g o u t s i d e the Nai +xZr2S*xP3-x°12
range.
B o il o t e t . a l .
s° l ution
[25] prepared s i n g l e c r y s t a l s of composition
Na5Zr(P04 ) 3 usi ng a f l u x te ch n i q u e.
They found t h a t , d e s p i t e having a
d i f f e r e n t space group, th e c r y s t a l s t r u c t u r e of Na5Zr(P04 ) 3 r e t a i n e d the
main f e a t u r e s of the NASICON-type s t r u c t u r e , but h a l f of the zirconium
atoms were re p la c e d by sodium atoms.
T he re fo re , B o ilo t e t . a l . proposed
t h a t NASICON compounds can be prepa red with a la r g e zirconium d e f i c i e n ­
cy.
Kohler e t . a l . [11] prepared s i n g l e c r y s t a l s of th e composition
Na3 . 1 Zr1. 78S i 1.24Pl . 76°12 wh*cfl were ntonoclinic a t room te mp era tu re.
S t r u c t u r a l refi ne me nt was p o s s i b l e only in th e h i g h - t e m p e r a t u r e , rhombo­
he d ra l phase. With th e a i d of a j o i n t p r o b a b i l i t y - d e n s i t y fu n c t i o n f o r
th e sodium atoms, which was based on anharmonic tempe rature f a c t o r s ,
Kohler e t . a l . showed t h a t the conduction path invol ve s both sodium
s i t e s , as has been o r i g i n a l l y proposed by Hong [10].
R e l a ti n g the
p r o b a b i l i t y d e n s i t y f u n c t i o n to the e f f e c t i v e , o n e - p a r t i c l e p o t e n t i a l of
the sodium io n s , they c a l c u l a t e d the h e ig h t of the p o t e n t i a l b a r r i e r to
be 0.22 eV, in good agreement with t h e observed a c t i v a t i o n energy fo r
conductivity.
The d e t a i l e d s t r u c t u r e of the monoclinic phase has not been f u l l y
d e sc ri b e d y e t , altho ugh some p r e li m in a r y f i n d i n g s from neu tron powder
d i f f r a c t i o n re fi n e m e n ts have been pre se n te d [2 6] , which confirmed the
s t r u c t u r e proposed by Hong [10].
P r e p a r a t i o n of NASICON and Phase S t a b i l i t y
Nal+xZr2S*xP3-x°12 comP°unds were f i r s t obta ine d by a s o l i d s t a t e
r e a c t i o n between Si02 , ZrO,,, NH4 H2
P 0 4
an<^ Na2C03 £9,10].
Appropriate
mi xtu re s of th e r e a g e n t s were heated a t 190°C for 16 hours and c a l c i n e d
a t 900°C fo r 5 hours.
Use of Na3P04 »12H20, Si 0 2 and Zr02 as raw mate­
r i a l s allowed the use of a s i n g l e s t e p c a l c i n a t i o n a t 1150°C [27],
Zircon (ZrSiO^) and Na^PO^ were a l s o used as s t a r t i n g m a t e r i a l s [28].
In or de r t o i n c r e a s e th e r e a c t i v i t y and th e homogeneity of the
powders, wet chemical p ro c e ss e s have been developed [ 2 4 , 2 9 , 3 0 , 3 1 ] .
B o il o t e t . a l . [24] prepa red a ge l by mixing n e a r l y - s a t u r a t e d , water
s o l u t i o n s of th e sodium s i l i c a t e s , (NH4 ) 2HP04 and Zr0(N03 ) 2 «2H20.
Higher r e a c t i v i t y of an amorphous powder prepared from the d r i e d gel
allowed th e formation of NASICON a t 1000°C.
Quon e t . a l .
[29] mixed a
S i0 2 s o l and a f i n e Zr(0H)4 s l u r r y with a s o l u t i o n of Na2C03 , NH4H2P04
and formic a c i d (HCOOH).
The mixture was d i s p e r s e d with anhydrous i s o ­
propanol and d r i e d a t 110°C and th e powder was c a l c i n e d a t 750°C.
McEntire e t . a l .
[30] used a proc ess inv ol vi ng g e l l a t i o n of a c o l l o i d a l
suspension of s i l i c a in a s o l u t i o n c o n t a i n i n g ammonium-zirconylc ar bon ate (NH4 ) 3Zr0H(C03 ) 3.2H20, NH4H2P04 , NaN03 and c i t r i c a c i d .
A
g l a s s y g e l , formed a t 100°C, was pyrolyzed a t 400°C i n t o the component
oxides and c a l c i n e d a t 900°C.
Engell e t . a l . [31] prepared NASICON from
10
metal al ko xid e d e ri v e d g e l s .
Clear g e l s were prepared from dry H3po4 ,
t e t r a e t h y l o r t h o s i l i c a t e S i ( 0 E t ) 4 , zirconium propoxide Zr(0Prn ) 4 and
sodium methoxide NaOMe d i s s o l v e d in propanol.
The e f f e c t s of d i f f e r e n t
f a c t o r s on th e formation of NASICON were s t u d i e d .
C l e a r f i e l d e t . a l . [32, 33] developed hydrothermal procedures to
s y n t h e s i z e NASICON.
Hydrothermal r e a c t i o n s of a c r y s t a l l i n e ot-zirconium
phosphate (a-Zr(HP04 )2*H20) and sodium s i l i c a t e (Na4S i 0 4 ) were c a r r i e d
out in t e f l o n - l i n e d Pa rr bombs a t 300°C.
This produced a hydrated phase
which upon h e a t i n g t o 1100°C y i e l d e d NASICON of composition
Na3 . 3 Zr1.65S i 1 . 9 Pl . l ° 1 1 . 5 Extensive s t u d i e s of th e s i n t e r i n g pro c e ss fo r Na3Zr2Si2P02 were
conducted by a group from Ceramatec, Inc. [ 2 7 ,2 8 ,3 0 ] .
NASICON was
s i n t e r e d to 98% d e n s i t y in 2 hours a t 1260°C and in 16 hours a t 1230°C.
At te mp era tu res above 1260°C, b l o a t i n g of the ceramic occurred which was
accompanied by e x s o l u t i o n of Z ^
and appearance of s p h e r i c a l voids.
Chemical a n a l y s i s of samples f i r e d a t d i f f e r e n t te m pe ra tu res led to the
con clusi on t h a t decomposition i n t o th e compound oxides and d e n s i f i c a t i o n
oc curred s im ult a n eo us ly duri ng s i n t e r i n g .
Th e re fo re , NASICON should be
s i n t e r e d a t an optimum te m pe ra tu re which r e s u l t s in good mechanical
p r o p e r t i e s and a minimum of decomposition. For the s ta n d a rd x=2 NASICON,
th e optimum s i n t e r i n g tempe rat ure was found to be 1225°C.
The appearance of f r e e Zr02 in Na3Zr2Si2P012 ceramics s i n t e r e d above
1175°C [30] r a i s e s th e q u e s ti o n of th e phase s t a b i l i t y in the
Nal+xZr2S*xP3 x°12 s ys te m -
was observed t h a t , usi ng a
zirconium
d e f i c i e n c y in th e s t a r t i n g m a t e r i a l s , e l i m i n a t e s t r a c e s of f r e e z i r c o n i a
11
in the f i r e d ceramic [24].
The s o d a - r i c h co mpositions (x*2.3) were a l s o
prepared as a s i n g l e phase [28]. Von Alpen e t . a l .
[34] mixed the
s i n g l e - p h a s e end compounds of the Nai +xZr2S*xP3-x°12 s e r *es and s t * 11
observed s e g r e g a t i o n of ZrC> 2 from compositions in th e range of
1.8<x<2.4.
They proposed a noth er system of s o l i d s o l u t i o n s with the
composition Nai +zZr2 - z/ 3 S i z P3_z° i 2 - 2 z / 3 f o r
In th e composition
range 0<z<1.6 a rhombohedral phase e x i s t s ; whereas, for 1.6<z<3, a
monoclinic phase with high i o n i c c o n d u c t i v i t y was found, and s e g r e g a ti o n
of Zr ( > 2 did not occur. The s i l i c a t e end-member of th e s e r i e s N a^ZrS ijO ^
was prepared [35] and found to be a f a i r i o n i c conductor.
The re p o rt e d
compositions of the mixed phases did n o t , however, agree with the pro­
posed g e ner al formula.
Based on chemical a n a l y s i s of the h y d ro th e rm a ll y - p re p a re d NASICONl i k e phases, C l e a r f i e l d e t . a l . [32] proposed a n o n s t o i c h i o m e t r i c f o r ­
mula Nai +z+4yZr2_x_yS^z P3_z° i 2 - 2 x ’ w*“ ch r e q u i r e s both zirconium and
oxygen d e f i c i e n c y .
Another g e n e r a l formula Nai +zZr2_yS*xP3-x°12 ( z_x“
4y=0), was proposed by Kohler e t a l [1 1] , based on a s i n g l e - c r y s t a l ,
X-ray s t r u c t u r e re fin em en t.
This l a s t formula i n c o r p o r a t e s , as a spe­
c i a l c a s e , the o r i g i n a l Hong s o l i d s o l u t i o n s e r i e s [10] but expands the
NASICON phase to a comp osi tio nal plane in th e q u a r t e r n a r y system (Zr02S i 0 2 ~Na2 0 -P 2 0 ^) which in c lu d e s th e z i r c o n i u m - d e f i c i e n t compositions.
However, i t appears t h a t the compound Na^Zr^ 25^P04^3 Pr e d i c t e d
this
formula does not e x i s t , as has been demonstrated by the phase s t u d i e s of
the orthophosphate j o i n , Na3P04-NaZr2 (P04 ) 3 , in the Na2 0 -Z r 0 2 -P 2 0 5 s y s ­
tem [36, 37],
12
On the o th e r hand, th e s i n g l e - p h a s e Na3Zr2S i 2P012 ceramic was ob­
t a i n e d by c a r e f u l p r o c e s s i n g , e i t h e r from mecha nic al ly mixed r e a g e n t s
[10,26] or from p r e c i p i t a t e d g e l s [ 2 9 ,3 1 ] .
I t may be concluded t h a t the
o r i g i n a l l y proposed [10] s e r i e s of s o l i d s o l u t i o n s Nai +xZr2S i xP3_x° i 2
does e x i s t , but t h a t th e phases in th e range i.5<x<2.5 a r e u n s t a b l e a t
high te m p e ra t u re s .
The exac t s t o i c h i o m e t r y of NASICON may depend upon
th e p r e p a r a t i o n procedure and NASICON-like phases can be o bt a in e d with
compositions o u t s i d e the simple s o l i d s o l u t i o n s e r i e s .
Io nic C onduct ivi ty
The room-temperature value of th e i o n i c c o n d u c t i v i t y of
Na3Zr2S i 2P012 i s about 6xlO-4 (flcm)-1 [ 9 , 3 8 ] ,
At 300°C, th e c o n d u c t i ­
v i t y ranges from 0.1 to 0.35 (flcm)- * and depends on t h e p r e p a r a t i o n
te chn iq ue [ 9 , 2 8 , 3 9 , 4 0 , 4 1 ] .
The h i g h e s t va lu e <,30o'»r = 0,35
has
been r e p o r t e d f o r samples prepared by th e ge l pro c e ss [28].
Electronic
c o n d u c t i v i t y i s n e g l i g i b l e in t h i s m a t e r i a l , as i s e v i d e n t from complex
impedance measurements in which blo ck in g e l e c t r o d e s had been used.
The Arrhenius p l o t of the c o n d u c t i v i t y fo r Na3Zr2S i 2P0^2 over a
te mp erat ur e range from 20 to 450°C e x h i b i t s a t h r e e - s l o p e behavior
[2 3 , 3 9 , 4 1 ] .
The a p p ar en t a c t i v a t i o n e n e r g i e s r e p o r t e d by Bogusz e t a l
[39] a re Ea ^=0.19 eV, Ea2=0.43 eV, and Ea3=0.33 eV, fo r th e h i g h - ,
i n t e r m e d i a t e - , and lo w-temp erature r e g io n s .
The d a t a fo r the bulk
c o n d u c t i v i t y were e x t r a c t e d from impedance measurements, us in g the com­
plex ad mittance a n a l y s i s .
The i n t e r m e d i a t e - t e m p e r a t u r e reg io n (120 -
180°C) i s t r e a t e d by some i n v e s t i g a t o r s [2 3, 24 ,4 1 ] as a t r a n s i t i o n
region over which a change in a c t i v a t i o n energy for i o n i c conduction
13
o ccu rs.
They r e p o r t two v a l u e s of a c t i v a t i o n energy: (1) a t tempera­
t u r e s above about 190°C (Ea l =0.21 eV [2 3] , 0. 20 eV [2 4 ] , 0.27 eV [ 4 1 ] ) ;
(2) a t te m p e ra t u re s below about 120°C (Ea 2=0.36 eV [2 3 ] , 0.40 eV [24] ,
0.37 eV [ 4 1 ] ) .
The change in the a c t i v a t i o n energy for i o n i c c o n d u c ti v ­
i t y i s a s s o c i a t e d with the s t r u c t u r a l phase t r a n s i t i o n [23,24] which has
been d e s c r i b e d above.
The broad te mp erat ure range over which t h i s
change occurs might i n d i c a t e a p r o g r e s s i v e d e cr ea se of monoclinic d i s ­
t o r t i o n , as well as an i n c r e a s i n g d i s o r d e r among the mobile io ns .
For
compositions with x>2.2, Arrhenius p l o t s , c h a r a c t e r i z e d by two s t r a i g h t l i n e segments, were r e p o r t e d [42],
The i o n i c c o n d u c t i v i t y in th e system Nai +xZr 2 S i >5 P3 _}{0 i 2 depends on
th e composition.
C on d u c t iv it y of the end-compounds, N a Z ^ P j O ^ and
Na4Zr2S i 3012, i s low; o2oo°c = 8x10
(flcm)- * [ 22 ] , r e s p e c t i v e l y .
-6
(ftcm)
-1
[14] and <*300oc = 3.5x10
-4
According to Kafal as and Cava [43] t h e r e i s
a broad maximum in c o n d u c t i v i t y over the composition range 1.8<x<2.5,
with th e h i g h e s t va lu e o c c u r r in g a t th e composition x=2.3.
Goodenough
e t a l [9] and B o i l o t e t a l [24] re p o r t e d a maximum in t h e c o n d u c t i v i t y
f o r t h e composition x=2, whereas Bogusz e t a l [42] observed the h i g h e s t
c o n d u c t i v i t y fo r x=2.2.
Only two c o n d u c t i v i t y measurements have been r e p o r t e d fo r the x=l
composition. B o il o t e t a l [24] measured a ° 300oC = 0.018 (12cm)-1 and the
a c t i v a t i o n energy ECl = 0.29 eV fo r th e te mp erat ur e range from 120 to
400°C.
Goodenough e t a l [12] measured a Ojqqoq = 0.026 (ficm)- '1', Eg =
0.27 eV for x=1.2; and <*300oC = 0.0029 (ftcm)"1 , Ea = 0.31 eV fo r x=0.8.
The v a r i a t i o n of the c o n d u c t i v i t y of t h e s e compounds with composi-
14
t i o n i s r e l a t e d t o t h e s t r u c t u r a l changes.
The c o n d u c t i v i t y maximum
o c c u r r in g around x=2.2, c o i n c i d e s with t h e maximum l e n g th of the rhombo­
h e d ra l c a x i s , which may a l s o be c o r r e l a t e d with the s i z e of th e pathway
between the Na s i t e s [ 1 0 , 1 1 , 2 2 ] . The importance of th e pathway s i z e i s
supported by th e o b s e r v a t i o n of a p o s i t i v e a c t i v a t i o n volume fo r the
c o n d u c t i v i t y proc ess [43],
A more d e t a i l e d d e s c r i p t i o n of s t r u c t u r a l
changes a s s o c i a t e d w it h S i / P r a t i o was given by S c h i o l e r e t a l [26] in
terms of a c o o p e r a t i v e r o t a t i o n of th e Si / P t e t r a h e d r a which causes an
opening-up of th e b o t t l e n e c k in the Na+ ion pathway for t h e composition
x=2.
Another mechanism i n f l u e n c i n g th e composition dependence of the con­
d u c t i v i t y has been proposed by Richards [44] who c a l c u l a t e d th e hopping
c o n d u c t i v i t y in a t w o - s u b l a t t i c e s t r u c t u r e in which e xcl us io n of double­
occupancy of s i t e s was invoked.
Using a c r y s t a l s t r u c t u r e corresponding
t o the aforementioned conduction network proposed by Hong [10] and
assuming deeper p o t e n t i a l w e l l s fo r th e N a(l) s i t e s , Richards obt ained a
c o n d u c t i v i t y maximum c orr esp ond in g to x=1.5 and found v a lu es approaching
zero for x=0 and x=3.
This behavior a r i s e s from c o r r e l a t i o n s between
jumps of Na+ ions .
Another t h e o r e t i c a l i n t e r p r e t a t i o n has been proposed by Jacobson e t
a l [45] who c a l c u l a t e d , by means of s t o c h a s t i c Langevin dynamics s im ula ­
t i o n , th e c o n d u c t i v i t y of i n t e r a c t i n g p a r t i c l e s in a p e r i o d i c one­
dimensional p o t e n t i a l .
The i n t e r i o n i c f o r c e s s t r o n g l y reduce the con­
d u c t i v i t y in c as e s when the p a r t i c l e c o n c e n t r a t i o n i s commensurate with
the p e r i o d i c p o t e n t i a l and s t r o n g l y enhance i t f o r incommensurate d e n s i ­
15
ties.
With the proper ch oice of pa ra m et e rs , th e c o n d u c t i v i t y vs. s t o i ­
chiometry curve given by Kafalas [43] was reproduced around th e maximum.
These model c a l c u l a t i o n s p r e d i c t a deep minimum in c o n d u c t i v i t y fo r the
composition x=l and a noth er small maximum fo r x*0.6, the l a t t e r not
having been observed to d a t e .
The major weakness of t h i s model, be s id e s
i t s one-dimensional c h a r a c t e r , i s th e assumption t h a t a l l the s i t e s are
e q u i v a l e n t which i s not the case fo r N a(l) and Na(2) s i t e s in the
NASICON system.
P r a c t i c a l C o n s id e r a ti o n s
The sodium ion c o n d u c t i v i t y of the x=2 NASICON a t 300°C, the op er a­
t i n g te mp er at ur e of t h e sodium-sulphur c e l l , i s comparable to t h a t of
the b e s t p o l y c r y s t a l l i n e Na 8"-alumina [27],
NASICON can be f a b r i c a t e d
i n t o dense p o l y c r y s t a l l i n e ceramic and has been proposed as an a l t e r n a ­
t i v e s o l i d e l e c t r o l y t e f o r sodium-sulphur c e l l s .
advantages over Na 8"-al umi n a, v i z . :
NASICON has some
(1) th e i o n ic motion i s t h r e e -
di me nsi on al, and i s not conf ine d to two-dimensional conduction l a y e r s as
in 8”-alumina; (2) lo w e r- te m p e ra tu re ceramic p ro c e ss in g (around 1200°C
compared with 1550°C and highe r needed fo r B"-alumina) can be used,
which s u b s t a n t i a l l y reduces sodium l o s s e s ; (3) NASICON appea rs to be
i n s e n s i t i v e to m oi s tu re [ 27] , whereas 8"-alumina undergoes de g ra d a ti o n
upon exposure to m o is tu re .
For s u c c e s s f u l a p p l i c a t i o n s in sodium-sulphur b a t t e r i e s , NASICON
ceramic must be s t a b l e in c o n t a c t with molten sodium and p o l y s u l f i d e s a t
te m pe ra tu re s 300 - 400°C as well as to be a b le to pass i o n i c c u r r e n t a t
high d e n s i t y w ith out d e g r a d a t i o n .
T e s ts of the chemical s t a b i l i t y in
16
molten sodium performed by s e v e r a l groups have led to d i f f e r e n t con clu ­
sions.
The Ceramatec group r e p o rt e d m o d e r a t e - t o - s e v e r e c o r r o s i o n of the
x=2 NASICON when i t was immersed in l i q u i d sodium a t 400°C [28], Dynamic
t e s t s of Na+ i o n - t r a n s f e r in a sodium c e l l r e s u l t e d in the f a i l u r e of
2
the NASICON e l e c t r o l y t e tu be s a f t e r pa ss in g l e s s than 5 Ah/cm“ a t c u r 2
r e n t d e n s i t i e s of 50 mA/cm
[28].
S od a-r ich compositions (x=2.4) were
found to be more d u ra b le in sodium c e l l s .
Su cc ess ful t e s t of the x=2
compound in th e sodium c e l l s , which r e s u l t e d in pa ssi ng 2200 Ah/cm
2
at a
c u r r e n t d e n s i t y of 0. 84 A/cm^ a t 320°C, were r e p o r t e d from China [46],
A z i r c o n i a - d e f i c i e n t composition, Na^ ■^Zr1 ^^Si 2 jPq
7
° ; q . which can be
s y n th e s iz e d without any f r e e Z ^ , was only moderately a f f e c t e d by
molten sodium a t 300°C [34].
Yde-Anderson e t a l [40] observed good
s t a b i l i t y in molten sodium fo r the z i r c o n i a - d e f i c i e n t composition
N a 2
9 4
Zrl
4 9 P 0
8 S * 2
2 ° 1 0
8 5
‘
0n the o t h e r hand *-he de g ra d a ti o n o f
NASICON due to chemical r e a c t i o n with l i q u i d sodium a t 300°C was ob­
served by Schmid e t a l [4 7] , both for the s t o i c h i o m e t r i c and the
z i r c o n i a - d e f i c i e n t compositions.
The r e s i s t a n c e of NASICON to co rr o s io n
in molten sodium appea rs to depend on both th e composition and p r e p a r a ­
t i o n technique and can probably be improved.
A p p li c a t io n of NASICON in
th e sodium-sulphur b a t t e r y i s not ye t imminent.
2.2 Frequency-dependent C ond uct ivi ty of Fast I on ic Conductors
2 . 2 . 1 Experimental Data
In c o n t r a s t with the l a r g e number of o b s e r v a t i o n s t h a t have been
made of the d . c . or the low-frequency a . c . c o n d u c t i v i t y of v a r io u s io n ic
c on ducto rs , t h e r e a r e only few f a s t ion cond ucto rs fo r which th e conduc­
17
t i v i t y has been measured w it h in the microwave re g io n .
The i n t e r e s t in
the frequency-dependent c o n d u c t i v i t y of i o n i c conduction s t a r t e d , when
Funke and coworkers r e p o r t e d microwave measurements of the c o n d u c t i v i t y
of ot-Agl [4 8] , a - c u l [49] and s-CuBr [50],
The measurements a t fr eq ue n­
c i e s ranging from 10 to 40 GHz were made usi ng a r e c t a n g u l a r waveguide
with the two o p p o s it e narrow w a ll s re p la c e d by th e samples.
The conduc­
t i v i t y of a l l t h r e e compounds was found to d e cr ea se with frequency to a
minimum in the microwave region ( e . g . a t 20 GHz fo r ot-Agl) and then
i n c r e a s e a t highe r frequency to small maximum ( a t 30 GHz f o r ot-Agl).
The c o n d u c t i v i t y a t th e minimum was about two times lower than the d . c .
va lu e .
These ex perimental f i n d i n g s led to development of numerous
t h e o r e t i c a l models aimed a t e x p l a i n i n g the frequency-dependence of con­
d u c t i v i t y [ e . g . 6-8] .
Gebhardt e t a l [51] used an open c a v i t y p e r t u r b a t i o n method with
s p h e r i c a l samples of a-Agl to cover th e frequency range 4-40 GHz.
Their
r e s u l t s were not c o n s i s t e n t with the e a r l i e r f i n d i n g s of Funke and J o s t
[48]. No frequency-dependence for the a-Agl c o n d u c t i v i t y was observed.
Roemer and Luther [ 5 2 ,5 3 ] , usi ng waveguide t r a n s m i s s i o n and r e f l e c t i o n
t e c h n i q u e s , a l s o did not observe any frequency dependence of the conduc­
t i v i t y of ot-Agl in th e microwave re gi on and th e r e p o r t e d c o n d u c t i v i t y
was equal to th e d . c . v a lu e .
R ece ntl y, the dis cr ep a nc y was s e t t l e d by
a t t r i b u t i n g the frequency-dependence of th e c o n d u c t i v i t y re p o r t e d by
Funke and J o s t [48] to c ur re nt- im pe di ng s t r u c t u r e s on the s u r f a c e of the
sample which r e s u l t e d from machining the e -p hase .
Samples of a-Agl
formed d i r e c t l y from the melt e x h i b i t , a t microwave f r e q u e n c i e s , a
18
c o n d u c t i v i t y equal to th e d . c . v a lu e .
The c o n d u c t i v i t i e s of a-CuI and
8-CuBr a r e a l s o frequency independent, a c c or di ng to new microwave meas­
urements [55].
The i o n ic c o n d u c t i v i t y of a-RbAg4l(. was measured up to f a r - i n f r a r e d
f r e q u e n c i e s by Funke and Schneider [56].
An i n c r e a s e in t h e c o n d u c ti v ­
i t y with frequency was observed in th e microwave reg ion and the n a t u r e
of th e frequency-dependence v a r i e d with te m p e ra t u re .
At 373 K, the
c o n d u c t i v i t y i s equal to th e d . c . value up to * 33 GHz and then i n ­
c r e a s e s to a subsequent p l a t e a u .
At te m p e ra t u re s below 300 K, a slow
i n c r e a s e of the c o n d u c t i v i t y with frequency s t a r t s a t ^ 20 GHz.
The
Arrhenius p l o t s of th e microwave c o n d u c t i v i t y of <*-RbAg4I 5 e x h i b i t e d
two-slope behavior with a l a r g e r sl ope a t high t e m p e ra t u re s .
An i n t e r ­
p r e t a t i o n of t h e s e ex per im ent al f i n d i n g s was given in terms of momentary
l o c a l p o t e n t i a l s exper ien ce d by the mobile io ns .
A d e t a i l e d study of t h e complex c o n d u c t i v i t y of Ag2 HgI4 in the r a dio
and microwave frequency range has been r e p o r t e d by Wong e t a l [ 5 7 ,5 8 ] ,
At fr e q u e n c i e s from 0 .3 to 70 GHz, s t a n d i n g wave measurements were made
with the sample f i l l i n g c r o s s - s e c t i o n of a c o a x i a l or a r e c t a n g u l a r
waveguide and backed by a v a r i a b l e s h o r t - c i r c u i t . Below 300 MHz, an R-X
meter was used to measure th e impedance of a c o a x i a l c a p a c i t o r f i l l e d
with the sample.
For th e io n i c - c o n d u c t i n g ot-phase, th e c o n d u c t i v i t y
e x h i b i t i n g power-law frequency dependence (o
un with n = 0 .6 ) was
observed, t o g e t h e r with a p o s i t i v e d i e l e c t r i c c o n s t a n t which d e c r e a s e s
with i n c r e a s i n g frequency.
For the e l e c t r o n i c - c o n d u c t i n g B-phase, th e
s lo p e of a c o n d u c t i v i t y v e rs u s frequency p l o t i s g r e a t e r than u n i t y on a
19
l o g - l o g s c a l e and th e d i e l e c t r i c c o n s t a n t v a r i e s very l i t t l e with f r e ­
quency.
In th e oc-phase, t h e c o n d u c t i v i t y i s t h e r m a l l y - a c t i v a t e d a t a l l
f r e q u e n c i e s and th e a c t i v a t i o n energy d e c r e a s e s with i n c r e a s i n g fr e q u e n ­
cy.
The 8-phase c o n d u c t i v i t y i s th e r m a ll y a c t i v a t e d only below 1 GHz.
Microwave and f a r - i n f r a r e d measurements on 8-alumina have been p e r ­
formed by Strom e t a l [5 9] , and by Barker e t a l [60].
Strom e t a l [59]
used a c a v i t y - p e r t u r b a t i o n method a t t h r e e microwave f r e q u e n c i e s and
l a s e r spectro photo m etr y a t f r e q u e n c i e s between 240 and 3600 GHz.
Meas­
urements were made a t te m pe ra t ure s 4.2 to 300°K on s i n g l e - c r y s t a l s of Na
B-alumina and th e c o n d u c t i v i t y was found to be independent of the f r e ­
quency in the microwave reg ion but i t s v a l u e s were l a r g e r than the d . c .
conductivity.
The te m p e ratu re dependence of the microwave c o n d u c t i v i t y
was i n i t i a l l y d e s c r i b e d to be q u a d r a t i c (o ^ T^) between 30° and 300°K.
In a l a t e r paper [6 1 ] , the same te mp era tu re dependence was p r e s e n t e d as
a tw o-slope Arrhenius p l o t with the high te m p e ra t u re a c t i v a t i o n energy
equal to the a c t i v a t i o n energy fo r d . c . c o n d u c t i v i t y .
Barker e t a l [60]
used a c a v i t y p e r t u r b a t i o n method a t f r e q u e n c i e s 0.38 to 2.4 GHz, wave­
guide t r a n s m i s s i o n measurements a t 24 and 85 GHz and f a r - i n f r a r e d i n t e r ­
ferom ete r fo r f r e q u e n c i e s between 3 x 1 0 ^ and 1.2 x 1 0 ^ Hz.
Measure­
ments were made on Na, K and Ag 8-aluminas a t t e m p e ra t u re s 100 to 500 K.
The room-temperature c o n d u c t i v i t y spectrum of Na 8-alumina e x h i b i t s a
power-law frequency dependence above 1 GHz and up to t h e f a r - i n f r a r e d
peak (o ^ wn with n in the range 0.6 to 0 . 8 ) .
This be havior has been
a s s o c i a t e d with ion-hopping in a d i s o r d e r e d system.
The r e a l p a r t of
the d i e l e c t r i c c o n s t a n t i n c r e a s e s above th e i n f r a r e d va lu e as the f r e -
20
quency i s de cr ea se d.
The f a r - i n f r a r e d c o n d u c t i v i t y peak a t 1.8 x 10
Hz was a s s o c i a t e d with v i b r a t i o n s of th e Na+ io ns .
12
An Arrhe nius p l o t
c o n d u c t i v i t y vs. te m pe ratu re a t 1 GHz d i s p l a y s approx ima te ly a two
s t r a i g h t l i n e be ha vio r.
The l a r g e r a c t i v a t i o n energy o c c u r r i n g a t
hig he r t e m p e ra t u re s corresponded to th e a c t i v a t i o n energy for th e d . c .
conductivity.
Microwave measurements on K - h o l l a n d i t e (K^ 54Mg0 77T i 7 23°12^’ a
f a s t i o n i c conductor with one-dimensional channels in a s k e l e t o n s t r u c ­
t u r e , have been performed by Khanna e t a l [62] and by Yoshikado e t a l
[63].
Khanna e t a l [62] used a c a v i t y - p e r t u r b a t i o n method with long
ne edle- sh ap ed c r y s t a l s .
At 9 GHz, the c o n d u c t i v i t y was t h e r m a l l y -
a c t i v a t e d with an a c t i v a t i o n energy equal to 0.034 eV over the tempera­
t u r e range 77 - 300 K.
Yoshikado e t a l [63] used a t h i n , c e n t r a l l y -
l o c a t e d d i e l e c t r i c s l a b in a r e c t a n g u l a r waveguide.
Their tr e a tm e n t of
the d a ta i s q u e s t i o n a b l e because the sample was ne edl e- sh ap ed r a t h e r
than being a s l a b ext end in g along the waveguide.
The c o n d u c t i v i t y a t
32.8 GHz and 9.54 GHz showed an approximate two-slope be havior in the
Arrhenius p l o t , over th e tem pe rat ure range 83 - 616 K.
Below 300 K, the
a c t i v a t i o n energy was 0.034 eV; above 300 K, i t was 0.075 eV.
The
d i e l e c t r i c c o n s t a n t was found to i n c r e a s e with i n c r e a s i n g te mp era tu re
and was approximately 80 a t room tem pe ratu re.
The measurements were
l a t e r extended to lower fr e q u e n c i e s and four kinds of K - p r i d e r i t e s were
i n v e s t i g a t e d [64],
A f te r s u b t r a c t i o n of the e f f e c t s of e l e c t r o d e s , the
c o n d u c t i v i t y i n c re a s e d with frequency from 100 Hz to microwave fr e q u e n ­
c i e s , where s a t u r a t i o n was observed.
F r a c t i o n a l power law frequency
dependence of the c o n d u c t i v i t y (o * «n ) was observed over a broad range
of f r e q u e n c i e s and t h e e n t i r e c o n d u c t i v i t y spectrum from 100 Hz to 37
GHz was modeled by an e q u i v a l e n t c i r c u i t based on th e "moving box" model
[65].
The frequency-dependent c o n d u c t i v i t y of h o l l a n d i t e was e a r l i e r
i n v e s t i g a t e d a t f r e q u e n c i e s below 10
6
Hz and a model r e l a t i n g th e anoma­
lous low frequency c o n d u c t i v i t y o - wn to a random d i s t r i b u t i o n of
p o t e n t i a l b a r r i e r s was developed [66].
At low t e m p e r a t u r e s , where the d . c . c o n d u c t i v i t y i s low, the f r e ­
quency dependence of the c o n d u c t i v i t y can be observed a t f r e q u e n c i e s f a r
below the microwave re g io n .
A power-law, frequency-dependence of the
c o n d u c t i v i t y of Na 8-alumina with an exponent •''0.6 was observed a t
fr e q u e n c i e s from 10
5
to 10
7
Hz and te m p e ra t u re s around 100 K [67] .
A
phenomenogical r e l a t i o n between the d . c . c o n d u c t i v i t y and the freque ncy dependent c o n d u c t i v i t y was e s t a b l i s h e d and a simple method fo r ex­
t r a c t i n g the r a t e of hopping and the c a r r i e r c o n c e n t r a t i o n was proposed
[ 6 8 ,6 9 ] .
S im ila r be havior of c o n d u c t i v i t y was observed fo r p o l y c r y s t a l ­
l i n e L i4S i 0 4 , Ag7I 4As04 g l a s s and CACNO^/KNO^ g l a s s [69].
t h e s e c as e s the d . c . c o n d u c t i v i t y was lower than 10
_4 _ i
fl
_i
cm
In a l l of
.
2 . 2 . 2 Theor ies of Frequency-Dependent Ioni c C ondu ctivi ty
In g e n e r a l , a f a s t i o n ic conductor c o n t a i n s two kinds of p a r t i c l e s :
mobile ions and ions which a re bound to t h e i r e q u i l i b r i u m s i t e s .
The
mobile ions can move in a p o t e n t i a l provided by the l a t t i c e c o n s i s t i n g
of the bound io ns .
The coupled motion of the l a t t i c e cage ions and the
d i f f u s i n g ions r e p r e s e n t s a very complex p h y s ic a l s i t u a t i o n .
Specific
22
t h e o r e t i c a l models g r e a t l y s i m p l i f y the r e a l s i t u a t i o n and emphasize
only c e r t a i n a s p e c t s of th e problem.
The hopping models, based on a master e q u a ti o n , d e s c r i b e the motion
of th e mobile ions in terms of a s e r i e s of d i s c r e t e jumps between
definite la ttic e site s.
All of th e i n t e r a c t i o n s a re re p r e s e n t e d only by
t h e i r i n f l u e n c e on th e hopping r a t e s of the mobile s p e c i e s .
No d e s c r i p ­
t i o n of th e dynamics d ur in g the jump i s provid ed , beyond d e f i n i n g the
hopping r a t e .
This kind of d e s c r i p t i o n i s re a so na ble only i f th e a v e r ­
age re s id e n c e time i s c o n s id e ra b ly longer than the hopping time, t h a t
i s , th e time i t t a k e s to reach q u a s i - e q u i l i b r i u m a f t e r a jump of mobile
ion. The a b s o l u t e r a t e th e o ry which i s th e b a s i s fo r t h e r m a l l y - a c t i v a t e d
hopping r a t e s , i s a p p l i c a b l e only fo r thermal e n e r g i e s , kT, much lower
than th e h e i g h t of th e p o t e n t i a l b a r r i e r s s e p a r a t i n g the e q u i l i b r i u m
s i t e s of t h e mobile io n s .
The l i m i t a t i o n of such d i s c r e t e ion motion i s r e la xed by s t o c h a s t i c
models fo r continu ous d i f f u s i o n .
The continuous motion of a p a r t i c l e in
a p o t e n t i a l f i e l d can be de sc ri b e d in a s t o c h a s t i c manner i f th e i n t e r ­
a c t i o n s with th e l a t t i c e v i b r a t i o n s a r e re p r e s e n te d by random f o r c e s and
damping of the ion motion.
The model becomes i n c r e a s i n g l y complicated
and i n t r a c t a b l e when more a s p e c t s of the r e a l system a re excluded from
r e p r e s e n t a t i o n by simple random p ro c e ss es and a re t r e a t e d e x p l i c i t l y .
In p r a c t i c e , i f one i s to t r e a t the dynamics of continuous motion, one
has to s a c r i f i c e the d e t a i l s of g e om et ric al c o n f i g u r a t i o n and l i m i t the
s o l u t i o n e i t h e r to very small systems or to c e r t a i n ranges of parame­
t e r s , such as a h i g h - f r i c t i o n or a hi g h -t em p e ra tu re l i m i t .
23
An e x c e l l e n t review of the t h e o r i e s developed fo r f a s t i o n i c t r a n s ­
po r t was given by D i e t e r i c h , Fulde and Peschel [ 7 ] .
In the fo ll ow in g,
th e r e s u l t s of th o s e t h e o r i e s of the dynamic c o n d u c t i v i t y a re b r i e f l y
reviewed, which may be r e l e v a n t to the case of f a s t ion t r a n s p o r t in
framework s t r u c t u r e s such as found in NASICON.
Hopping Models
Several a u t h o r s [7 0 ,7 1 ,7 2 ] have shown for mally t h a t , for a g e ne r al
model of ion hopping from one m e t a s t a b l e c o n f i g u r a t i o n of the system to
a n o t h e r , th e r e a l p a r t of th e c o n d u c t i v i t y i n c r e a s e s monotonically with
frequency or i s frequency independent.
d u c t i v i t y i s always p o s i t i v e .
The imaginary p a r t of the con­
This i s e q u i v a l e n t to a p o s i t i v e r e a l
p a r t of the d i e l e c t r i c c o n s t a n t by v i r t u e of th e r e l a t i o n
e(u) = - j o ( w ) / u e o +
(2 .1 )
This p r o p o s i t i o n a p p l i e d to a r e a l system should hold up to f r e q u e n c i e s
comparable with t h e v i b r a t i o n a l f r e q u e n c i e s of th e mobile ions.
The b a s i c assumptions which lead to th e aforementioned p r o p e r t i e s of
the hopping c o n d u c t i v i t y a r e as foll ow s. The p r o b a b i l i t y per u n i t time
of a t r a n s i t i o n r gb from c o n f i g u r a t i o n a to c o n f i g u r a t i o n b depends only
on th e i n i t i a l and th e f i n a l c o n f i g u r a t i o n s and on the tem pe rat ure .
The
t r a n s i t i o n r a t e s must obey the d e t a i l e d - b a l a n c e c o n d it io n :
Pa r ab = PbFba
( f o r a11 a and b)
( 2 ‘ 2)
where P° i s t h e p r o b a b i l i t y of having c o n f i g u r a t i o n a in the e q u il ib r iu m
Cl
state.
The c o n d u c t i v i t y i s frequency-dependent when the s t a t i c e l e c t r i c
f i e l d d i s t o r t s the e q u i l i b r i u m d i s t r i b u t i o n fu nc tio n [7 ,7 1 ] ,
24
A simple phenomenological e x pr e ss io n fo r th e c o n d u c t i v i t y in a
hopping model, proposed by Kimball and Adams [7 0] , i s :
R
o(w) = oQ
O(oo)-O(0)
= 0 ( o o ) -----------------
1+j(l)T
(2 .3 )
1 + jut
where cjq = o(oo) a r i s e s from th e change in the hopping r a t e induced by
the e l e c t r i c f i e l d and corre spo nds to the r e s u l t fo r s i n g l e - p a r t i c l e ,
u n c o r r e l a t e d hopping or the s o - c a l l e d m e a n - fi e ld approximation [73].
The second term a r i s e s from th e change of p r o b a b i l i t y d i s t r i b u t i o n which
r e s u l t s from th e presence of the f i e l d .
The f a c t o r R i s a measure of
the "bounce-back" e f f e c t : an in c r e a s e d p r o b a b i l i t y fo r an ion of the
jumping back t o i t s o r i g i n a l s i t e due e i t h e r to i n t e r a c t i o n s with ot he r
ions or to th e pr es enc e of i n e q u i v a l e n t s i t e s .
The c h a r a c t e r i s t i c time
, t , can be much s h o r t e r than the i n v e rs e of the hopping r a t e , as deduced
from the d . c . c o n d u c t i v i t y [ 7 3 ,7 4 ] .
Note t h a t R = 1 - f C where f C =
o ( 0 )/a(«>) i s th e c u r r e n t c o r r e l a t i o n f a c t o r in tr oduce d by Sato and
Kikuchi [74].
Wong and Brodwin [75] s t u d i e d independent p a r t i c l e hopping in a
system with n - s u b l a t t i c e s and p e r i o d i c boundary c o n d i t i o n s .
They used
an e x p l i c i t formula fo r th e f l u x of ions between th e s i t e s and were ab le
to t r e a t both th e disp la cem en t and th e d r i f t p o r t i o n s of the c u r r e n t .
For a t w o - s u b l a t t i c e system, the frequency-dependence of c o n d u c t i v i t y
was expressed by a formula e q u i v a l e n t to Eq. ( 2 . 3 ) .
The a c t i v a t i o n
energy for the low-frequency c o n d u c t i v i t y co rresponds to th e highe r
potential b a rrie r.
At high f r e q u e n c i e s , motion i s l i m i t e d to hopping
over a low p o t e n t i a l b a r r i e r which det ermi ne s the a c t i v a t i o n energy.
For a g e n e r a l n - s u b l a t t i c e system, the complex c o n d u c t i v i t y was given in
the form:
n-1
(2 .4 )
where 1 / Tpj i s an ei g en v a lu e of the e q u i l i b r i u m t r a n s f e r m a tr ix .
Independent p a r t i c l e hopping in a one-dimensional d i s o r d e r e d l a t t i c e
has been i n v e s t i g a t e d by Bernasconi e t a l [76, 77]. The d i s o r d e r was
modeled by a n e a r e s t - n e i g h b o r hopping r a t e r which was taken as an
independent random v a r i a b l e d i s t r i b u t e d acc ord ing to a c e r t a i n p ro b a b i­
l i t y density p (r).
Using a c o n t i n u e d - f r a c t i o n expansion, s o lv in g asymp­
t o t i c a l l y an i n t e g r a l eq u at i o n and app ly in g a s c a l i n g h y p o t h e s i s , they
were a b l e t o c a l c u l a t e the frequency-dependent c o n d u c t i v i t y in the lowfrequency l i m i t (<■> -» 0) fo r two c l a s s e s of p r o b a b i l i t y d e n s i t i e s fo r the
hopping r a t e :
(i)
p ( r ) such t h a t
«
m= I
0
p(r)
— dr < *
r
(1 - a) r"a
(ii)
(2 .5 a )
0
1, 0<a< 1
(2.5b)
P( r) =
0
oth e rw is e
The asy mp to tic be hav io r of the c o n d u c t i v i t y in the «
r e s p e c t i v e l y as [77].-
0 l i m i t was given
where nQ i s th e d e n s i t y , e th e charge of th e c a r r i e r s , P0 ^
a re c o n s t a n t s .
and
D is ordered systems of c l a s s ( i ) a s y m p t o t i c a l l y behave as
ordered systems with f i n i t e d . c . c o n d u c t i v i t y , whereas systems of c l a s s
(ii)
have anomalous power-law frequency-dependent c o n d u c t i v i t y which
va n is h e s in the d . c . l i m i t .
The above r e s u l t s were a p p l i e d t o r a t i o n a l i z e th e exper iment al lowfrequency c o n d u c t i v i t y of K - h o l l a n d i t e [66].
I t was assumed t h a t the
one-dimensional ch an nel s to which motion of K+ ions i s confined are
di vi de d by im p e r f e c t i o n s i n t o segments of average le n g th L.
a
The h e i g h t s
of energy b a r r i e r s d i v i d i n g segments was taken to have a p r o b a b i l i t y
d e n s i t y given by.Woe xp (- a /k T m)
fo r
aq
s< a « a:
W(a ) =
(2 .7 )
0
oth e rw is e
where a^ -» <x>; t h i s c or re sp ond s to a p r o b a b i l i t y d e n s i t y of c l a s s ( i i )
[see E q . ( 2 . 5 b ) ] fo r the hopping r a t e between n e igh bor ing segments.
Using E q . ( 2 . 6 b ) , th e low-frequency c o n d u c t i v i t y a t te m pe rat u res T<Tm i s :
<*(w,T) = C(T) (jw )n(T)
(2 .8 a )
with a te m pe rat ure -d ep e nd e nt exponent given by:
n(T) =
1 - T/Tm
^
(2.8b)
1 + T/T
m
Experimental c o n d u c t i v i t y d a t a fo r K - h o l l a n d i t e , measured a t f r e q u e n c i e s
between 10 and 500 kHz and a t te m p e ra t u re s 190
the E q . ( 2 . 8 ) with Tm = 440 K.
to 253 K, was f i t t e d to
The model was l a t e r g e n e r a l i z e d to s i t u a ­
27
t i o n when th e chann el s were blocked by l e s s mobile ions and c a l l e d the
"moving-box" model [6 5] .
I t was s u c c e s s f u l l y used to f i t th e frequency-
dependent c o n d u c t i v i t y of K - p r i d e r i t e s [64].
Jonscher [78] compiled the a . c . c o n d u c t i v i t y d a t a for a range of
m a t e r i a l s , thought t o conduct by hopping charge movements, and observed
t h a t th e frequency dependence of c o n d u c t i v i t y followed a power-law
c(w) = o(o) + Awn
with n<l, t y p i c a l l y 0 . 8 .
( 2 .9 )
In a d d i t i o n a low frequency d i s p e r s i o n of the
d i e l e c t r i c p e r m i t t i v i t y , which m a n i f e s t i t s e l f as a weakly fr equencydependent c o n d u c t i v i t y (n as 0 . 1 ) , was noted fo r many hopping systems
[79].
Both kinds of be havior were i n t e r p r e t e d in terms of a u n i v e r s a l
model f o r d i e l e c t r i c response [80].
R ec e n tl y , Dissado and H i l l extended
t h e i r many-body th e o r y of d i e l e c t r i c r e l a x a t i o n [81] to a c l u s t e r of
hopping charge c a r r i e r s [82] and d e ri v e d a power-law frequency depen­
dence of c o n d u c t i v i t y .
In th e case of a hopping model with a l l s i t e s e q u i v a l e n t and no
i n t e r a c t i o n s between ions b e s id e s th e h a r d - c o r e r e p u l s i o n s , the conduc­
t i v i t y i s fr eq ue ncy -i nd epe nde nt [7 ] .
C on du ct iv it y becomes frequency-
dependent i f the s i t e s a r e not e q u i v a l e n t .
Richards [44,73] consi de red
hopping in a g e n e r a l t w o - s u b l a t t i c e s t r u c t u r e , with i n e q u i v a l e n t A and B
s i t e s , where only i n t e r s u b l a t t i c e ion hops were allowed.
The p o t e n t i a l
well of s i t e A i s assumed to be deeper than t h a t of s i t e B.
There a re
ZBA ^ype B_nei 9 hbors f ° r an A s i fce which a r e g e n e r a te d by v e c t o r s 8AB
and Z^0 type A-neighbors fo r s i t e B.
Using a decoupling approximation
for t h r e e - p a r t i c l e c o r r e l a t i o n s , Richards ob ta in e d an e xp re ss io n fo r the
28
c o n d u c t i v i t y in the form of E q .( 2 .3 ) with the c u r r e n t c o r r e l a t i o n f a c ­
t o r , f c , and th e e f f e c t i v e hopping time ,-r, given by:
f
o(0)
(n, - nB) X
= ------ = i ------------------- 2 ------ 2-----------------------------------------o(«)
Z - 1 - X[(nA - nB)(Z - 2) + (nA+nB- l ) z / 2 ]
T_1 = (WAB+WBA)[Z " 1 " x ( ( V nB)(Z ' 2) + ( V nB"1)z/23
(2.10 a)
(2.10b)
where nA and n0 are th e e q u i l i b r i u m occup ation numbers of s i t e s A and B,
r e s p e c t i v e l y , W^B and WBA a re A-»B and B-»A t r a n s i t i o n r a t e s , r e s p e c t i v e ­
l y , and
X = (W B A -W A B ) / ( W B A + W A B> -
Z = (ZA B + Z B A :)/2’
Z = ZA B - Z BA'
The above r e s u l t i s of i n t e r e s t h e r e , because the s i t e s fo r the Na+ ions
in the rhombohedral phase of NASICON c o n s t i t u t e a l a t t i c e of the type
consid ered by Richards [44].
The n e a r e s t - n e i g h b o r i n t e r a c t i o n s can be taken i n t o account by
in tr o d u c in g hopping r a t e s which a r e dependent on th e mode of occupation
of the n e a r e s t ne ighbors fo r the s i t e s between which given jumps can
occur.
There i s no unique way of doing t h i s and t h e e f f e c t of d i f f e r e n t
cho ice s fo r the hopping r a t e s on the c o n d u c t i v i t y was examined by Singer
and Peschel [83].
For c l a s s i c a l , a c t i v a t e d hopping, th e ch oice of r a t e s
depends on the i n t e r a c t i o n range.
Using only the n e a r e s t - n e i g h b o r
i n t e r a c t i o n s means t h a t the i n t e r a c t i o n s between s e c o n d - n e a r e s t neighbors w i l l be z e r o , i . e . , U(2a)=0 (a i s the n e a r e s t - n e i g h b o r d i s t a n ­
c e ) ; however, a p o t e n t i a l b a r r i e r between the f i r s t - and the second3
n e a r e s t - n e i g h b o r s can be e f f e c t e d , i . e . , U(^a) f 0. For very s h o r t - r a n g e
interactions,
U(^a) = 0, the c o n d u c t i v i t y i s freq u en cy -i nd ep en de nt .
When U(| a) f 0, the c o n d u c t i v i t y can be fr e q u e n c y - d e p e n d e n t.
29
Richards [73] has shown t h a t the dynamic c o n d u c t i v i t y fo r a l i n e a r
chain of e q u i v a l e n t s i t e s with n e a r e s t - n e i g h b o r i n t e r a c t i o n s between
hopping ions can be approximated by E q . ( 2 . 3 ) .
Singer and Peschel [83]
reached a s i m i l a r concl us ion using a continued f r a c t i o n expansion meth­
od.
The e f f e c t i v e hopping r a t e -t"* i s given by a combination of the
r a t e s fo r th e elementary hopping pro c e sse s and i n c r e a s e s s h a r p l y fo r
strong repulsions.
This means t h a t d i s p e r s i o n of th e c o n d u c t i v i t y
s h i f t s to hig he r f r e q u e n c i e s because of i n t e r a c t i o n s .
The the or y of l a t t i c e gas models has been extended by Beyeler e t a l
[72] beyond th e framework of the master e qua ti on approach to hopping.
Their tr e a t m e n t allow s one to includ e th e e f f e c t s of the i n t e r n a l dynam­
i c s of th e system on the t r a n s i t i o n r a t e s .
The e n t i r e frequency range
can be d e s c r i b e d with an effective-medium L i o u v i l l i a n used as a b a s i s
for th e memory f u n c t i o n approximation.
The c o n d u c t i v i t y i s ex pressed as
a sum of four te rms, th e f i r s t two r e p r e s e n t i n g th e s ta n d a r d master
eq ua ti on r e s u l t .
The t h i r d term a r i s e s from the coup lin g of t r a n s i t i o n
r a t e s to a l a t t i c e v i b r a t i o n a l mode which has r e l a x a t i o n time of the
o rd er of the mean r e s i d e n c e - t i m e .
the c o n d u c t i v i t y spectrum.
This adds an o s c i l l a t o r y s t r u c t u r e to
The fo u r th term d e s c r i b e s a s i t u a t i o n in
which a p a r t i c l e or a c o l l e c t i v e r e a c t i o n c o o r d i n a t e does not t h e r m a li z e
a f t e r a t r a n s i t i o n and m u l t i p l e hops may occur.
damping, t h i s may le ad to q u a s i - f r e e pr op a g a ti o n .
In th e case of low
The g e n e r a l formalism
of Beyeler e t a l [72] f i l l s , to some e x t e n t , the gap between the s t a n ­
dard m a s t e r - e q u a t i o n approach and q u a s i- c o n ti n u o u s d e s c r i p t i o n of p a r t i ­
c l e motion.
30
P i e t r o n e r o and S t r a s s l e r [84] c a l c u l a t e d the dynamic c o n d u c t i v i t y
fo r independent p a r t i c l e hopping on a l a t t i c e of e q u i v a l e n t s i t e s with
modulation of each hopping r a t e , r , by an independent harmonic o s c i l l a ­
t o r of frequency
and damping y.
They found, t h a t for flQ>r0 . fl0>Y.
the c o n d u c t i v i t y d e c r e a s e s with i n c r e a s i n g frequency a t f r e q u e n c i e s
below flQ.
This be havior i s not p o s s i b l e with the s ta n d a rd hopping
d e s c r i b e d by th e master e qu ati on
Continuous Motion Models
The aim of continuous models fo r f a s t i o n i c cond uctors i s to o b ta in
a u n i f i e d d e s c r i p t i o n of th e o s c i l l a t o r y and the d i f f u s i v e motions of
mobile io ns .
A phenomenological memory f u n c t i o n approach has been used by Bruesch
e t a l [85].
They p o s t u l a t e d a g e n e r a l i z e d Langevin eq ua ti on of the type
2 1
mx + mrQx + m<i>0 / M ( t - t ’ ) x ( t ' ) d t ' = f ( t )
(2.11)
where f ( t ) i s th e s t o c h a s t i c f o r c e ; th e memory f u n c ti o n M(t) should
pr o p e r ly reproduce the asy mp to tic be h a v io r , t h a t i s , E q .( 2 .1 1 ) should
reduce t o e q u a ti o n s f o r o s c i l l a t o r y motion with small
eq ua ti on
( t - t ’ ) and to an
for f r e e - d i f f u s i o n with l a r g e ( t - t ' ) . The s i m p l e s t
choice fo r
M ( t - t ' ) i s given by
M(t
- t ' ) = exp [ - ( t - t ' ) / T c ]
(2.12)
where t c de not es the t r a n s i t i o n time from o s c i l l a t o r y to d i f f u s i v e
be ha vi or .
(2 .1 2) i s :
The c o n d u c t i v i t y e x p re ss io n which foll ows from Eq. (2.11) -
31
ne 2
1
ne
2
(2 .13)
o(w)
id
o
ju + 1/ t
c
I t has been shown by Fulde e t a l [8 6 ] , t h a t th e memory f u n c ti o n
approach of Bruesch e t a l [85] i s e q u i v a l e n t to an approximate s o l u t i o n ,
ob ta in e d by the method of continued f r a c t i o n expansion of second order
fo r the c o n d u c t i v i t y of a Brownian p a r t i c l e moving in a p e r i o d i c poten­
t i a l , as d e s c r i b e d by th e follo win g Langevin eq uat io n:
(2 .1 4)
Thus, th e memory f u n c t i o n in E q . ( 2 . 1 2 ) , which has th e form of a r e t a r d e d
f r i c t i o n , p h y s i c a l l y s i m u l a t e s th e motion in a p e r i o d i c p o t e n t i a l .
C ond u ct iv it y of a p a r t i c l e undergoing Brownian motion in a p e r i o d i c
p o t e n t i a l , as d e s c r i b e d by Langevin eq u at i o n E q . ( 2 . 1 4 ) , can be for mally
c a l c u l a t e d from a v e l o c i t y - v e l o c i t y c o r r e l a t i o n f u n c t i o n expressed in
terms of th e t r a n s i t i o n p r o b a b i l i t y d e n s i t y between two p o i n t s in the
phase spac e. The t r a n s i t i o n p r o b a b i l i t y d e n s i t y obeys th e Fokker-Planck
e q u a ti o n .
D i e t e r i c h e t a l [87] c a l c u l a t e d t h e dynamic m o b i l i t y in the
s i n u s o i d a l p o t e n t i a l usi ng a s h o r t - t e r m expansion and a continued f r a c ­
t i o n expansion fo r t h e c o r r e l a t i o n f u n c t i o n .
For the case of small
f r i c t i o n a t an i n t e r m e d i a t e te m p e ra t u re , t h e m o b i l i t y has a pronounced
minimum a t a frequency around o n e - h a l f of th e frequency of small ampli­
tude o s c i l l a t i o n s in a s i n u s o i d a l p o t e n t i a l w e ll .
The occur ren ce of
t h i s minimum cannot be reproduced by the use of th e simple memoryf u n c t i o n approach due to Bruesch e t a l [85].
32
Some i n t e r e s t i n g t h e o r e t i c a l c a l c u l a t i o n s of io n i c c o n d u c t i v i t y have
been r e p o r t e d fo r m a n y - p a r t i c l e , continuous models. A one-dimensional
model, d e s c r i b e d by coupled Langevin e q u a ti o n , has baen s t u d i e d by
Ge isel [88] usi ng t h e fo ll ow in g Frenkel-Kontorova p o t e n t i a l :
V = V E c o s ( 2 n x . / a ) + a E ( x . +, - x. - b)^
i
i
(2 .15 )
Here th e p e r i o d , a , o f th e cage p o t e n t i a l may be d i f f e r e n t from the
n a t u r a l spacing between ions de fi n e d by the c o n c e n t r a t i o n n = l / b .
The
harmonic i n t e r a c t i o n s t r e n g t h i s measured by a c o r r e l a t i o n l e n g t h , k ~* =
2
2
ab /2fl kT.
The c o n d u c t i v i t y was c a l c u l a t e d a n a l y t i c a l l y usi ng a
c o n t i n u e d - f r a c t i o n expansion.
Strong dependence of th e c o n d u c t i v i t y on
c o n c e n t r a t i o n , ex pre sse d by the r a t i o a / b , was observed.
For a h a l f ­
f i l l e d l a t t i c e ( a / b = 0 . 5 ) , th e d . c . c o n d u c t i v i t y has a deep minimum
a r i s i n g from i n t e r a c t i o n s . The dynamic c o n d u c t i v i t y i s su ppressed for
fr e q u e n c i e s below t h e frequency, «o , of small o s c i l l a t i o n s in the w el ls
of a p e r i o d i c p o t e n t i a l .
For the c o n c e n t r a t i o n a/ b=0 .7 5, the d . c . and
low frequency c o n d u c t i v i t y i s enhanced by i n t e r a c t i o n s .
A s t o c h a s t i c Langevin dynamics s im u la ti o n of the c o r r e l a t e d io n ic
motion in a one-dimensional p e r i o d i c p o t e n t i a l has been performed by
Jacobson e t a l [ 8 9 , 9 0 ] ,
In t h e i r computer s i m u l a t i o n , random f o r c e s a re
ge ner at e d and th e coupled Langevin e q u a ti o n s a r e solved d i r e c t l y by
numerical methods.
The s t o c h a s t i c t r a j e c t o r i e s a r e analyzed and used to
c alc u la te appropriate c o rre la tio n functions.
Two types of i n t e r a c t i o n s
were include d: (1) a long- range Coulomb fo r ce and (2) a s h o r t range r
repulsive force.
~8
The dynamic c o n d u c t i v i t y has been c a l c u l a t e d fo r two
d e n s i t i e s of mobile io n s , p = 1/2 and p = 3/4, and the o th e r parameters
33
were chosen t o s i m u l a t e th e be havior of a-Agl and K - h o l l a n d i t e .
The
e f f e c t of Coulombic i n t e r a c t i o n s on the c o n d u c t i v i t y was found to be
much s t r o n g e r than t h e e f f e c t of th e s h o r t - r a n g e r e p u l s i o n s .
In the
case of p = 1/2, the d e n s i t y commensurate with th e pe rio d of the l a t ­
t i c e , th e low-frequency c o n d u c t i v i t y i s reduced more than an orde r of
magnitude due to th e Coulomb i n t e r a c t i o n s .
the f r e q u e n c i e s of t h e o rd e r 10
12
Hz.
The r e p o r t e d s p e c t r a cover
An i n v e s t i g a t i o n of t h i s kind for
the c o n d u c t i v i t y a t f r e q u e n c i e s below about 10*1 Hz would have re q u ir e d
p r o h i b i t i v e l y long s i m u l a t i o n tim es .
In s t o c h a s t i c , co nti nuo us models, th e v i b r a t i o n s of the l a t t i c e cage
a re t y p i c a l l y t r e a t e d as a he a t bath which g e n e r a t e s both vis co u s and
random f o r c e s .
Complete dynamics of an io n i c - c o n d u c t i n g s o l i d may be
sim ula ted by the method of molecular dynamics in which Newton's equa­
t i o n s of motion fo r s e v e r a l hundred i n t e r a c t i n g i o n i c p a r t i c l e s a re
solved nu m e ri c a l ly .
In p r a c t i c e , mole cular dynamics enables a stud y of
the phenomena o c c u r r i n g w i t h i n th e frequency range 10^* to 1 0 ^ Hz and
over d i s t a n c e s of t h e ord e r of 30&.
The r e s u l t s of s e v e r a l molecular
dynamics s t u d i e s of f a s t i o n i c cond ucto rs have now been r e p o rt e d
[9 1 , 9 2 , 9 3 ] .
A molecular dynamics c a l c u l a t i o n fo r s t r u c t u r e s as complex
as NASICON and for time s c a l e s co rre spo ndin g t o microwave f r e q u e n c i e s
have not been r e p o r t e d y e t .
I I I . MEASUREMENT AND ANALYSIS OF THE COMPLEX IMPEDANCE/ADMITTANCE
SPECTRA.
3.1. I n t r o d u c t i o n .
Measurements of the frequency-dependent complex impedance a r e
widely used fo r c h a r a c t e r i z a t i o n of e l e c t r o c h e m i c a l systems [9 4 ] , p a r ­
t i c u l a r l y s o l i d e l e c t r o l y t e s [ 9 5 ,9 6 ] . In t h e s i m p l e s t c a s e , when s u i t a ­
b l e i o n i c a l l y - r e v e r s i b l e e l e c t r o d e s a r e not a v a i l a b l e fo r a d . c . meas­
urement, th e bulk i o n i c c o n d u c t i v i t y of s o l i d e l e c t r o l y t e can be d e t e r ­
mined from measurement of i t s a . c . impedance, over an a p p r o p r i a t e f r e ­
quency range, using io n-b l o ck in g e l e c t r o d e s . In g e n e r a l , the complex
impedance spectrum of a s o l i d e l e c t r o l y t e / e l e c t r o d e s c e l l i s infl uen c ed
by: (1) p r o p e r t i e s of the i n t e r f a c e between e l e c t r o d e s and e l e c t r o l y t e ;
(2)
by ion t r a n s p o r t in th e e l e c t r o l y t e ; and ( 3 ) , in case of inhomoge-
neous m a t e r i a l s , by p o l a r i z a t i o n a t th e phase bo un da rie s. D i f f e r e n t
phenomena o f t e n dominate w it h in d i s t i n c t frequency ranges. When the
e xperim ental impedance spectrum has been measured over a wide frequency
range, i t may be p o s s i b l e to s e p a r a t e the i n f l u e n c e s of the v a r io u s
p ro ce sses in the system, fo r i n s t a n c e , th e ion t r a n s p o r t through the
g r a i n boundaries and w it h in the bulk in p o l y c r y s t a l l i n e s o l i d e l e c t r o ­
l y t e [97].
Measurement of the complex impedance s p e c t r a inv olv es many obser va ­
t i o n s in orde r to cover broad frequency ranges and automation of the
experiment i s very d e s i r a b l e . A c o m p u te r - c o n tr o ll e d impedance measuring
system was l a r g e l y re designed in course of t h i s work (see Se c tio n 3 .2 ) .
34
35
An improved design of the measuring system, t o g e t h e r with a p r e c i s e
c a l i b r a t i o n pro cedure, en sures r e l i a b l e impedance measurements in the
frequency range 1 to 5x10
when the sample impedance l i e s w it h in the
range 1 fl to 500 kft.
Analy sis of the impedance s p e c t r a i s g r e a t l y f a c i l i t a t e d by s u i t a ­
bl e g r a p h i c a l p r e s e n t a t i o n of d a t a . Several d i f f e r e n t kinds of graphs,
emphasizing c e r t a i n f e a t u r e s of th e s p e c t r a , have been employed. Most
commonly used a re th e complex-plane diagrams of the impedance, Z, and
the ad m itt a nce , Y=l/Z, with the measuring frequency as a v a r i a b l e p a r a ­
meter [9 7 ,9 8 ] . Sp e c tro s co p ic p l o t s of the complex d a t a in c e r t a i n r e p r e ­
s e n t a t i o n s ( e . g . the r e a l and the imaginary p a r t s of th e impedance, the
adm ittanc e or the complex modulus, M=iuCgZ), v e rs u s frequency on lo g a ­
r it h m ic s c a l e s a re p r e f e r e d by some a u th o rs [99,1 00] , Three-dimensional
p e r s p e c t i v e p l o t t i n g , introd uce d by Macdonald e t a l [101], combines the
two aforementioned kinds of graphs.
When a c e r t a i n model, t y p i c a l l y an e q u i v a l e n t c i r c u i t , i s p o s t u ­
l a t e d fo r the system under s tu dy , then some of i t s pa rameters can be
e s ti m ate d from th e c h a r a c t e r i s t i c f e a t u r e s of a p p r o p r i a t e graphs. A fter
s u b t r a c t i o n of the c o n t r i b u t i o n from the e s ti m a te d c i r c u i t elem ents , the
d a ta can be r e p l o t t e d in order to d i s p l a y p r o p e r t i e s of the remaining
p a r t of the system ( e . g . , s u b t r a c t i o n of the bulk r e s i s t a n c e may re ve al
e l e c t r o d e p o l a r i z a t i o n ) . A s e r i e s of g r a p h i c a l e x t r a p o l a t i o n s and sub­
t r a c t i o n s may lead to e st i m a t i o n of a l l r e l e v a n t pa rameters
[102,103,104]. Such a method of a n a l y s i s becomes d i f f i c u l t when the time
c o n s t a n t s a s s o c i a t e d with the i n d i v i d u a l p a r t s of the e l e c t r o c h e m i c a l
36
system a r e not s u f f i c i e n t l y d i f f e r e n t . A g r a p h i c a l method i s s u s c e p t i b l e
to a cumulation of e r r o r u n l e s s c e r t a i n i t e r a t i v e refinement of the
ge om et ri c al e s t i m a t e s i s employed [104,105]. Another l i m i t a t i o n comes
from th e f a c t t h a t only p a r t of the a v a i l a b l e in for ma ti on i s a c t u a l l y
used a t a time; fo r example the dependence on frequency i s not f u l l y
r e v e a le d when a complex plane p l o t i s examined, while the r e l a t i o n
between r e a l and imaginary p a r t s of a complex spectrum i s not a ppar ent
in s p e c t r a l p l o t s .
The method of n o n l i n e a r l e a s t sq ua re s can be a p p li e d to f i t a
complex f u n c t i o n of frequency, which r e p r e s e n t s t h e impedance of a
t h e o r e t i c a l model, to the experimental impedance spectrum. Such a n a l y s i s
p e rm its si mu ltaneous e s t i m a t i o n of a l l para mete rs and s u p p l i e s informa­
t i o n about th e u n c e r t a i n t y of the parameter e s t i m a t e s , as well as t h e i r
mutual c o r r e l a t i o n s . The goodness of f i t ob ta in e d with d i f f e r e n t models
can be compared, which provid es a q u a n t i t a t i v e c r i t e r i o n fo r judging the
a p p l i c a b i l i t y of each t h e o r e t i c a l model. Nonlinear l e a s t - s q u a r e s a n a l y ­
s i s of complex impedance/admittance s p e c t r a has been developed and used
by MacDonald and co-workers [10 6,107,108], and a l s o in our l a b o r a t o r y
[105].
As was po in te d out by Macdonald [106,107] , a p p l i c a t i o n of d i f f e r e n t
weighting schemes in th e sum of squared d e v i a t i o n s may lead to some
d i s p a r i t i e s in th e e s t i m a t e s of pa rameters. Here, we p o s t u l a t e use of
weights which a r e based on the r e s o l u t i o n of the measuring equipment.
Our weighting scheme i s d i r e c t l y r e l a t e d to the ex perimental procedure
used fo r impedance measurements and a s s u r e s t h a t th e r e s u l t s of a n a l y s i s
37
do not depend on how the da ta i s r e p r e s e n t e d in a complex spectrum,
i . e . , whether th e d a t a a r e expressed as th e complex impedance or as the
complex ad m itt a nce . The r e s u l t s of f i t t i n g usin g our we ightin g scheme
a r e the same (w ith in numerical e r r o r s ) i r r e s p e c t i v e l y of the manner of
r e p r e s e n t i n g the complex spectrum. We ta k e i n t o account c o r r e l a t i o n
between the e r r o r s of the r e a l and th e imaginary p a r t s of the complex
o b s er v ab le a t a given frequency, which may a r i s e in a t y p i c a l e x p e r i ­
ment. Because t h i s method a f f o r d s uniform t r e a t m e n t of a l l d a t a p o i n t s ,
s p e c t r a ex tending over a wide range of frequency and impedance can be
analyzed and r e s o l v e d .
A new FORTRAN program has been developed f o r a n a l y s i s of imped­
ance s p e c t r a in an i n t e r a c t i v e mode on a minicomputer. An improved
v e r s i o n of the Gauss-Newton-Marquardt a lg o r it h m a s s u r e s , in most c a s e s ,
convergence of th e n o n l i n e a r l e a s t - s q u a r e s r e g r e s s i o n , even when crude
i n i t i a l guesses of t h e pa rameters a r e made. This f e a t u r e of the modified
a l g o r i t h m r e l i e v e s t h e us er of u nd e rta kin g th e t e d i o u s ta s k of f i n d i n g
a c c u r a t e i n i t i a l e s t i m a t e s of the pa rame te rs . Several models of th e a . c .
response of e l e c t r o d e s / e l e c t r o l y t e c e l l s a r e i n c o r p o r a t e d in th e program
and many a d d i t i o n a l combinations can be ob taine d by f i x i n g a t c o n s t a n t
va lu e a s u i t a b l e s u b s e t of par am ete rs . Other models can be e a s i l y i n ­
cluded as new s u b r o u t i n e s . A c u r r e n t v e r s i o n of th e program i s imple­
mented on a PDP-11/34 minicomputer and i s equipped with c a p a b i l i t i e s for
g r a p h i c s scr ee n d i s p l a y and p l o t t i n g of d a t a , t o g e t h e r with f i t t e d
f u n c t i o n s , usi ng d i f f e r e n t forms of g r a p h i c a l p r e s e n t a t i o n .
38
3.2. Computer-Controlled Impedance Measuring System.
An impedance meter s u i t a b l e fo r c h a r a c t e r i z a t i o n of i o n i c conduc­
t o r s should cover a broad range of frequency and impedance and o p e r a te
a t low and c o n t r o l l e d l e v e l s of the a . c . v o l t a g e a p p l i e d to th e sample.
At hi g h e r f r e q u e n c i e s (*>100 kHz) the r e s u l t s of measurement can e a s i l y
be d i s t o r t e d by s p u r io u s c a p a c i t a n c e s or in du ctan ces of l e a d s , r e f e r e n c e
r e s i s t o r s , c onnec to rs e t c . and car e should be taken to reduce t h i s
effect.
A c o m p u t e r - c o n t r o l l e d impedance measuring system has been p r e v i o u s ­
ly developed in our l a b o r a t o r y [109]. The a v a i l a b l e frequency range was
5
from 1 to 7x10 Hz. The o p e r a t i o n was based on a d i r e c t measurement of
th e a . c . v o l t a g e a c r o s s the unknown impedance and an independent meas­
urement of c u r r e n t pa ss in g through a r e f e r e n c e r e s i s t o r connected in
s e r i e s with th e unknown impedance. The a . c . s i g n a l was d e t e c t e d by a
d i f f e r e n t i a l a m p l i f i e r connected to th e two ends of th e unknown impedan­
ce ( t h e c e l l of i n t e r e s t ) through a p a i r of video p r e a m p l i f i e r s . In
p r i n c i p l e , t h i s scheme could o f f e r th e b e n e f i t s of a fo ur -p ro b e measure­
ment, but i t s p r a c t i c a l r e a l i z a t i o n s u f f e r e d from high input c a p a c i t a n ­
ces and poor s t a b i l i t y of th e p r e a m p l i f i e r s . In e f f e c t , impedances
l a r g e r than *>10 kfl could not be measured a t fr e q u e n c i e s in th e upper
p a r t of the a v a i l a b l e range. In order to extend the u s e f u l impedance
range a t high f r e q u e n c i e s , a simpler p r i n c i p l e of measurement was em­
ployed and the c r i t i c a l p a r t s of th e system were red esi gned .
The p r i n c i p l e of o p e r a ti o n i s as fo ll ow s . The a . c . v o l t a g e from a
programmable frequency s y n t h e s i z e r (Rockland 5100) i s a p p l i e d to a
39
r e f e r e n c e r e s i s t o r which i s connected in s e r i e s with the unknown imped­
ance (se e F i g . 3 . 1 ) . The o th e r end of th e unknown impedance i s connected
to the system ground. Two a . c . v o l t a g e s , measured r e l a t i v e to the
ground, a r e compared: ( 1 ) the s y n t h e s i z e r s i g n a l a p p li e d to th e r e f e r e n ­
ce r e s i s t o r , V
i s fed through a p a s s i v e probe to the r e f e r e n c e input
of a g a in -p h a se meter (Dranetz 305-PA-3009A); (2) the v o l t a g e a t a
connector between th e r e f e r e n c e r e s i s t o r and the unknown impedance, V5
i s probed by a p a s s i v e v o l t a g e probe (T ekt ro nix P6008, 10X, lOMft),
a m p l i f i e d by t h e p r e a m p l i f i e r of an o s c i l l o s c o p e (modified Tekronix
2213) and a p p l i e d to th e s i g n a l inp ut of th e ga in -p ha se meter. The
s i g n a l ( 2 ) i s d i s p l a y e d by th e o s c i l l o s c o p e , which permi ts d i r e c t moni­
t o r i n g of th e a . c . v o l t a g e a p p l i e d to th e sample dur ing measurement. The
g a in -p h a s e meter pro v id e s d i g i t a l ou tp u t of the phase d i f f e r e n c e between
V and V , e=arg(V )- ar g( V ) , and two analo g o u t p u t s which d e l i v e r d . c .
L
5
w
l
v o l t a g e s p r o p o r t i o n a l to the am pli tud es of th e a . c . s i g n a l s :
|Vf | and
IV | . The d . c . v o l t a g e s a r e d i g i t i z e d by a m ul ti c h a n n el a n a l o g - t o d i g i t a l c o n v e r t e r (Northern Tracor NS-626). A minicomputer ( D i g i t a l
PDP8 / e ) r e c e i v e s t h r e e numbers |VL | , |V | and 0. The unknown impedance,
Zu , i s r e l a t e d to th e measured v o l t a g e s by a p r o p o r t i o n a l i t y r e l a t i o n
V
|V I
Z
_s =
exp(j0 ) = —
Vr
|Vr l
Zu+R
( 3. 1)
The r e f e r e n c e r e s i s t a n c e , R, i s s e l e c t e d by th e computer program
from 12 r e s i s t o r s (300kft to 1 ft in a 3 0 0 , 1 0 , 3 , . . . sequence) a v a i l a b l e in
a s w it ch- box , in such way t h a t th e v o l t a g e r a t i o |V_|/ |V -V | i s on the
O
r
S
order of u n i t y . The d e t a i l s of the impedance matching a lg o r it h m are
40
operators
console
co n tro ller
in terfa ce
magnetic disk
data storage
printer
computer
(PDP B/e)
frequency
synth esizer
(Rockland 5100)
d ig itiz e d
phase e
ivr l
IMS v o lta g e
probe
gain-phase
reference meter
(Dranetz
305-PA-3Q09A)
reference
r e s isto r s
(switchbox)
analog
amplitudes
IV.
ADC converter
(Northern NS636)
10MS voltage
probe
(Tektronix
P6008)
unknownw
impedance
(sample)
Zu
sig n a l
pream plifier
(Tektronix 2213)
F ig .3 .1 . Schematic diagram of the com puter-controlled impedance
measuring system.
41
given in Appendix A. Only one r e s i s t o r a t a time i s connected in t o the
s i g n a l path by an a r r a y of low c a p a c i t a n c e r e l a y s ope ra te d by the com­
p ut e r v i a a c o n t r o l l e r - i n t e r f a c e u n i t . The computer s e t s both the f r e ­
quency and the amplitude of the s i n u s o i d a l s i g n a l ge ne ra te d by the
s y t h e s i z e r . At each frequency t h e ou tp u t a t t e n u a t i o n of the s y n t h e s i z e r
i s a d j u s t e d in such way t h a t the amplitude of th e v o l t a g e over the
sample remains approximately c o n s t a n t (w it h i n +1 0 $ of one of t h r e e
p r e s e t l e v e l s : 25, 100, 450 mV).
The simple r e l a t i o n between th e measured s i g n a l s and the unknown
impedance given by E q . ( 3 . 1 ) does not ta k e i n t o account e i t h e r s pu ri ou s
c a p a c i t a n c e s and in duct an ces a s s o c i a t e d with the r e f e r e n c e r e s i s t o r s ,
le a d s and co n n ec to rs , or o th e r s o u rc es of r . f . le a k s to the ground,
which cannot be n e g le c te d a t high f r e q u e n c i e s . The path of the a . c .
s i g n a l in the system i s more r e a l i s t i c a l l y modelled by an e q u iv a l e n t
c i r c u i t pre se nte d in F i g . 3.2 and th e a c t u a l c a l c u l a t i o n s of Zy a re made
a c c o rd in g ly . For measurements on samples of m a t e r i a l s , the e q u i v a l e n t
impedances of the sample ho ld e r a r e inc luded in the v a lu es of parameters
of the c i r c u i t in F i g . 3.2 and the c a l c u l a t e d impedance r e p r e s e n t s only
th e sample with the d e p o s it e d e l e c t r o d e s .
Values of the parameters of the e q u i v a l e n t c i r c u i t r e p r e s e n t i n g the
s i g n a l path in the measuring equipment a r e e s t a b l i s h e d by a c a l i b r a t i o n
procedure which invol ve s measurements of impedances of w e l l - c h a r a c t e r ­
ized r e s i s t o r s and c a p a c i t o r s followed by a n o n l i n e a r l e a s t - s q u a r e s
f i t t i n g . As an i n i t i a l s t e p of the c a l i b r a t i o n , the gain-p ha se c ha ract e r i c t i c of the s i g n a l channel ( i n c l u d i n g th e v o l t a g e probe and pre-
42
generator
Rf - reference r e s ito r
Lf - inductance in s e r ie s with the reference r e s isto r
- capacitance across the reference r e s is to r
Rp - resista n ce o f the v o ltage probe
Cg - spurious capacitance of le a d s, connectors, voltage probe, e tc .
I*s - inductance o f leads
Rs - resista n ce o f leads
- unknown impedance (sample)
F ig .3 .2 . Equivalent c ir c u it representing the a .c . sig n a l path in the
impedance measuring systqm.
43
a m p l i f i e r ) r e l a t i v e to the r e f e r e n c e channel i s recorded over the e n t i r e
frequency range using a
unknown and th e
1 0 0
1 0 0
ft c a l i b r a t i o n r e s i s t o r in p la ce of the
fl r e f e r e n c e r e s i s t o r (t h e e f f e c t s of s p u r io u s imped­
ances a re n e g l i g i b l e in t h i s c a s e ) . A n a l y t i c a l e x p r e s s i o n s (sums of
s e v e r a l power terms and a loga rit hm of frequency) a r e then f i t t e d to the
gain and to the phase c h a r a c t e r i s t i c s of the equipment. During a l l
subsequent measurements, c o r r e c t i o n s of the r a t i o of am pl itu d e s,
IV | / 1 V | , and of the phase d i f f e r e n c e ,
S
L
8
, a r e c a l c u l a t e d acc ording to
the f i t t e d e x p r e s s i o n s . The c a l i b r a t i o n parameters ( c o e f f i c i e n t s of the
e x p r e s s i o n s which i n t e r p o l a t e the gain and th e phase c h a r a c t e r i s t i c and
pa rameters of the e q u i v a l e n t c i r c u i t ) a r e s t o r e d on a d i s k e t t e ( f i l e
CALIBR.DA) and a r e r e c a l l e d by the program c o n t r o l l i n g t h e system be fo re
measurements a r e made. The d e t a i l s of the c a l i b r a t i o n procedure are
given in Appendix A.
The measurements a re made under c o n t r o l of a program AIMESR. During
normal o p e r a t i o n , th e program r e q u i r e s from the user only an i n i t i a l
s e t t i n g of th e foll owing pa rameters: s t a r t i n g and ending frequency,
l o g a r i t h m i c or l i n e a r frequency scan, number of measuring f r e q u e n c i e s ,
l e v e l of th e a . c . s i g n a l a p l l i e d to th e sample and i n i t i a l s e t t i n g of
the r e f e r e n c e r e s i s t o r . A d d iti on a l c o n t r o l of th e system during oper a­
t i o n i s provided through the f r o n t panel sw it ch e s of the PDP8 / e mi ni ­
computer. A f te r each change of the frequency, the s i g n a l a t t e n u a t i o n or
the r e f e r e n c e r e s i s t o r , th e program g e n e r a t e s time d e la y s in order to
o b t a i n s t a b l e r e a d i n g s . At each measuring frequency, the computer reads
the v a lu e s of |V I, |V | and 9 a t o t a l of f i v e times and ta k e s the
I
S
44
a v e r a g e s . When th e measured v a lu e s of any of the input parameters are
s c a t t e r e d around th e average more than a p r e s e t l i m i t , the measurements
a r e r e p e a t e d u n t i l t h e r e a di ng s s t a b i l i z e (maximum 5 t i m e s ) . The meas­
ured impedances, c o r r e c t e d acc ord ing to th e c a l i b r a t i o n scheme, a re
p r o g r e s s i v e l y p r i n t e d out duri ng measurements and a r e s t o r e d on a
d i s k e t t e a f t e r completion of a frequency s c a n . The s t o r e d impedance
s p e c t r a can be t r a n s f e r e d to a l a r g e r minicomputer (PDP11/34) where a
n o n l i n e a r l e a s t - s q u a r e s a n a l y s i s , g r a p h i c s s cr ee n d i s p l a y and p l o t t i n g
can be accomplished usin g th e program FIRDAC d e s c r i b e d in the following
sections.
3 .3 . Nonlinear Le a st - S a u a r e s A nal ys is of Complex Sp e c tra using Weights
Based on I n s t r u m e n t a l R e s o l u t i o n .
Let us c o n s i d e r a complex q u a n t i t y , y, which depends on th e an gul ar
fr eq ue nc y , w, and i s measured d i r e c t l y or i n d i r e c t l y . The ex perimental
spectrum c o n s i s t s of m p a i r s y . ,
mental e r r o r s , e ^
Measured v a lu e s y. c o n ta i n e x p e r i ­
thus,
yi = y i + e i ’
i = 1 »2 >•••»">
( 3 .2 )
where y? d e note s t h e t r u e v a lu e of y^.
In t h e f o ll o w i n g a n a l y s i s , i t i s assumed t h a t th e f r e q u e n c i e s (u>.)
a r e known e x a c t l y and, t h e r e f o r e can be t r e a t e d as independent parame­
t e r s . In a t y p i c a l experiment, frequency i s measured with much high er
accura cy than y ( e . g . the complex impedance) and n e g l e c t i n g e r r o r s of w
is ju s tifie d .
I f i t i s d e s i r e d to t r e a t frequency as an oth er dependent
v a r i a b l e , measured with e r r o r s , then one has to deal with parameter
45
}
e s t i m a t i o n in a n o n l i n e a r i m p l i c i t model [110]. I t i s , however, d i f f i ­
c u l t to o b t a i n convergence of the l e a s t - s q u a r e s a lg o ri th m for such model
u n l e s s a c c u r a t e i n i t i a l v a lu e s of th e pa rameters a r e known; a c c o r d i n g l y ,
i t would be s t i l l r e q u i r e d t h a t an e s t i m a t i o n of the parameters be made
f i r s t by a n o n l i n e a r l e a s t - s q u a r e s f i t t i n g usi ng an e x p l i c i t model, as
d e s c r i b e d he re .
We assume t h a t t h e measured q u a n t i t y y. i s a complex f u n c t i o n of
frequency w. and r pa ram ete rs X={x^,X2
y i " e i = y i = f ( V x) ’
i = 1 ’2
xr ):
m
( 3 - 3)
Our goal i s to e s t i m a t e th e para mete rs X and to f in d out how well the
model f u n c t i o n f( u ,X) f i t s th e exper imen ta l d a t a .
As has been d e s c r i b e d by Sheppard e t a l [111,112] and by Macdonald
and Garber [106], complex d a t a can be f i t t e d us in g a s t a n d a r d , r e a l
a r i t h m e t i c , n o n l i n e a r l e a s t - s q u a r e s a l g o r i t h m s . The l e a s t - s q u a r e s o b j e c ­
t i v e f u n c t i o n , Q, which i s minimized in course of f i t t i n g , must include
s q u ar es of both r e a l and imaginary p a r t s of th e d e v i a t i o n s of the mea­
sured v a lu e s from t h e v a lu es of th e model f u n c t i o n ,
m
Q = Y . [wi {Re[yr f ( V X)] )2 + Mi t l « n [ y i - f ( ® i . X ) ] } 2 ]
(3.4)
i= l
where w^, w"^ a r e weights a s s o c i a t e d with r e a l and imaginary p a r t s of
i - t h d a t a . The we ights should be chosen to be i n v e r s e l y p r o p o r t i o n a l to
the v a r i a n c e s of e r r o r s [113].
In E q . ( 3 . 4 ) , t h e r e a l and imaginary p a r t s of th e complex measured
q u a n t i t y , y ^ a r e t r e a t e d as u n c o r r e l a t e d o b s e r v a t i o n s . Absence of
c o r r e l a t i o n ca nnot, however, be assumed when the r e a l and imaginary
46
p a r t s of y a r e not measured d i r e c t l y . In ord er t o p a r t i c u l a r i z e t h i s
d i s c u s i o n , c o n si d e r the measurements of complex impedances made with a
v e c t o r impedance meter ( e . g . Hewlett Packard 4800A). The ins trument
g iv e s s e p a r a t e read in gs of th e magnitude of impedance, | Z | , and phase
a n g l e , d>. Deter minat ions of |Z| and 0 a re performed by d i f f e r e n t p a r t s
of ins tru me nt c i r c u i t r y . We can t r e a t |Z| and <& as d i r e c t l y measured
q u a n t i t i e s and assume t h a t t h e i r e r r o r s a r e not c o r r e l a t e d , t h a t i s ,
<e I Z | , e <t>>=0 . We cannot a s s e r t t h a t the e r r o r s
6
| Z| .
a re normally
d i s t r i b u t e d and t h a t t h e i r v a r i a n c e s a r e known e x a c t l y .
I t does not seem p l a u s i b l e t o unde rta ke s t a t i s t i c a l e s t i m a t i o n of
t h e s e v a r i a n c e s by r e p e a t i n g measurements on a c e r t a i n RC c i r c u i t , a t a
given frequency, because t y p i c a l r e s o l u t i o n of th e measuring instrument
does not permit one to observe random changes in th e re ading and un cer ­
t a i n t i e s of the ins tru me nt c a l i b r a t i o n p r e v a i l over random e r r o r s .
In
ord er to o b ta in a s t a t i s t i c a l sample fo r e s t i m a t i o n of v a r i a n c e of
d e v i a t i o n s from " r e a l " v a l u e , one would have to r e p e a t measurements
usi ng la r g e number of in s tr u m e n ts of the same kind. This i s not f e a s i b l e
so we have to r e s o r t to making c e r t a i n approxima tions for the v a r i a n c e s .
From the s p e c i f i c a t i o n s and performance of th e measuring equipment, one
can e s t i m a t e the accuracy of th e d i r e c t l y measured q u a n t i t i e s , a |Z| and
a<
j>.
For s i m p l i c i t y , we w i l l assume t h a t a | Z | / | Z | and ao ar e c o n s t a n t ,
but t h e i r v a r i a t i o n s with frequency and range of | Z | , or 4>, can a l s o be
t r e a t e d . The square of the accuracy l i m i t i s used to approximate the
v a r i a n c e s of e r r o r s ; t h u s ,
47
<e| 2 t ' e IS|> * ( 4 U I ) 2
<e» ' e» > * ( M ) 2
(3 5)
The r e a l and imaginary p a r t s of the impedance a r e c a l c u l a t e d from
Re(Z) = IZIcoso
Im(Z) = |Z|sin<>
( 3 .6 )
Applying a l i n e a r approximation for p ro pa ga tio n of e r r o r s [114], we
o b ta in e r r o r s of the r e a l and imaginary p a r t s of Z from e r r o r s in |Z|
and <t> to be:
eReZ ■ T T z f ^ m
+
= e U | cos* - ‘ *1 *!•»»«
e ImZ * e t 2 | s i n ® + e« l z l cos®
<37)
A c ovar ian ce ma tri x of e r r o r s of Re(Z) and Im(Z) can be expressed by the
v a r i a n c e s of e r r o r s of |Z| and <|>,
C11 = <eReZi , e ReZi7 = <e| Z | ’e | z | >cos2V % ’V
| z i |2sin2*i =
* |z. | 2 [(aZ/Z)2c os 20 . + (A0 ) 2s i n 2<t>. ]
(38)
°22 * <eImZi, e ImZi3 ’ l z i l2 [ U V Z j W c j + (4* ) 2cos2. , ]
c 12 = <eReZi, e ImZi> = U t
= c*j
The sum of the squared d e v i a t i o n s [s e e E q . ( 3 . 4 ) ] can now be r e ­
placed by the e x p re ss io n
m
Q = ^ ] [ g } 1 {Re[Zr f z (o)i ,X)]}2 + g}2 {lm[Zi - f z (« i ,X)]}2 +
i= l
+ 2gJ2Re[Zi - f z (wi ,X)] Im[Z. - f z ( w. , X)]]
(3 .9 )
where g ^ a r e elements of the in v e rs e G*=(C*)- '*' of the c ov arian ce ma tri x
given for each d a ta p o in t by E q . ( 3 . 8 ) , the s u b s c r i p t Z i s used to s t r e s s
t h a t the model f u n c t i o n i s c a l c u l a t e d in an impedance r e p r e s e n t a t i o n . In
43
v e c t o r n o t a t i o n , the o b j e c t i v e fu n c t i o n can be w r i t t e n as
Q(X) = [ Y-F(X)]T
( 3 .1 0)
[Y-F(X)]
where Y = {Re(21),Im(Z1),Re(Z2 ) f Im(Z2 ) , . . . ,Re(Zm) , Im(Zm) J ,
r t. a ^
^
#
1 z
in *
* **||ii 2 ' n\ * *J J
and the only nonzero elements of the 2m x 2m weight m a tr ix Gy a re
The co va ri an c e s <eDo7, e T _> a r e , in g e n e r a l , nonzero. As can be
ReZ’ ImZ
seen from E g . ( 3 . 8 ) , when aZ/Z=ao=D they a r e equal zer o. Consequently,
terms with g j 2 d is a p p e a r from E q . ( 3 . 9 ) and
9n
= 4
= (D' Z I > ' 2 •
«12 = 0
(3.12)
This i s the s i m p l e s t weighting scheme fo r complex impedance d a t a which
s t i l l provid es re a so na bl e s c a l i n g of the d e v i a t i o n s when the d a t a
extend over a wide range of magnitudes of | Z | .
When fo rm ul at io n of th e problem in a r e p r e s e n t a t i o n o th e r than
impedance i s d e s i r e d , then a p p r o p r i a t e weights can be c o n s t r u c t e d by a
procedure s i m i l a r to t h a t given in E q s . ( 3 . 5 ) —( 3 . 9 ) . In p a r t i c u l a r , i f
e r r o r s of |Z| and <j> a r e not c o r r e l a t e d , th e weight ma tri x for complex
ad mittance r e p r e s e n t a t i o n i s simply r e l a t e d to t h a t fo r complex imped­
ance as foll ows:
ry
Y 2i-1,2i-l
Gy
Y2i , 2 i -1
(3 .13)
I t can be shown t h a t , when the weight m a t r i c e s fo r l e a s t - s q u a r e s
49
f i t t i n g of d a t a in d i f f e r e n t numerical r e p r e s e n t a t i o n s a re c o n s t r u c t e d
based on th e same assumption about v a r i a n c e s of d i r e c t l y measured quan­
t i t i e s , th e r e s u l t i n g o b j e c t i v e f u n c t i o n s Q(X) a re approximately equal
( a t l e a s t up to the second order terms in d e v i a t i o n s of measured v a lu e s
from the v a l u e s of th e model f u n c t i o n ) . This implie s t h a t , i f the model
fu n c t i o n re a son ab ly well r e p r e s e n t s th e d a t a and the d e v i a t i o n s ar e
s m a l l, r e s u l t s of l e a s t - s q u a r e s a n a l y s i s do not depend on r e p r e s e n t a ­
t i o n to which the d a t a and the model f u n c ti o n a re transformed. On the
c o n t r a r y , when a r b i t r a r y weights a p p li e d fo r f i t t i n g in d i f f e r e n t
r e p r e s e n t a t i o n s a re not r e l a t e d to each o t h e r , the n, in g e n e r a l , the
o b j e c t i v e f u n c t i o n s Q(X) a r e v a s t l y d i f f e r e n t and r e s u l t s of the a n a l y ­
s i s may d i s a g r e e s u b s t a n t i a l l y . I t should be noted t h a t the complex
spectrum can be analyzed in the same r e p r e s e n t a t i o n as a re th e measured
d a t a ; in our case | Z | , 4> [with weights accord ing to the v a r i a n c e s of
E q . ( 3 . 5 ) ] . This approach has not been p u r s u i t here because computation
of complex model f u n c t i o n s and t h e i r d e r i v a t i v e s in p o la r c o o r d i n a t e s
i s more complicated than when r e a l and imaginary p a r t s a re used.
Engstrom and Wang [115] f i t t e d t h e i r da ta re p r e s e n te d as s i g n a l ampli­
tude and phase d i r e c t l y measured by a network a n a l y z e r , but used u n i t y
weights.
Having decided t h a t th e form of the o b j e c t i v e f u n c ti o n Q(X) i s
t h a t given by E q . ( 3 . 1 0 ) , we must f in d v a lu e s of parameters X for which
Q(X) i s a minimum. Since the model f u n c t i o n f(w,X) i s not l i n e a r in the
parameters X, min imization has to be achieved by an i t e r a t i v e a l g o ­
rithm. The Gauss-Newton method i s based on l i n e a r i z a t i o n of the func-
50
t i o n s f ( u . , X ) around c u r r e n t v a lu e s of th e parameters X, followed by
s o l v i n g such c r e a t e d l i n e a r l e a s t s q u ar es problem. For th e kt h s t e p ,
the i t e r a t i o n c o r r e c t i o n s to the para mete rs a re ob ta ine d in the form
[113]
Xk+1 ' Xk = *Xk =
[Y- F( xk )]
(3.14)
where A i s the Jac ob ia n m a tr ix of p a r t i a l d e r i v a t i v e s
3 R e [ f ( u ., X ) ]
a
2 i-l,j
3Xj
3 I m [ f( u ., X )]
X=Xk
a,
21 • 3
(3 .1 5)
31!j
X=X,
'k
Depending on t h e degree of n o n l i n e a r i t y of f(w,X) and the d i s ­
placement of X^ from t h e minimum of Q(X), c o r r e c t i o n s given by
E q. ( 3. 1 4) may r e s u l t in a d e c r e a s e or an i n c r e a s e of the o b j e c t i v e
f u n c t i o n Q(X). M o d i f i c a t i o n s of th e a l g o r i t h m a re ne ce s sa ry to a s s u r e
convergence. The a l g o r i t h m implemented in our program combines the
Gauss-Newton method with an a d j u s t a b l e s t e p and the Marquardt method
[116] (see Appendix B f o r d e t a i l s ) . Whereas our a lg o ri th m has been de ­
veloped fo r e f f i c i e n t f i t t i n g of complex s p e c t r a on a minicomputer, i t
does have f e a t u r e s found in g e n e r a l purpose n o n l i n e a r , l e a s t - s q u a r e s
r o u t i n e s a v a i l a b l e on mainframe computers [117,11 8,119]. P r o v is io n s for
f i x i n g v a lu es of th e para mete rs and s e t t i n g lower and upper bounds on
the parameters a re a l s o in cl uded .
A fte r a minimum, Q, of th e o b j e c t i v e f u n c t i o n i s found, the f o l ­
lowing r e s u l t s a r e provided:
( i ) e st i m a te d v a l u e s of p a ra m et e rs , x^, j = l , 2 , . . . r
( i i ) u n c e r t a i n t y of the parameter e s t i m a t e s
51
AXj = / b j j Q / ( 2 m - r )
where
(3.16)
B = {bj .} = (ATGyA)
In the l i n e a r c a s e , with a normal d i s t r i b u t i o n of th e e r r o r s in the
measurements, x ^+2ax^ a r e approximate
the parameter j when o th e r
95%confidence l i m i t s for
para mete rs a r e held c o n s t a n t .
( i i i ) c o r r e l a t i o n m a tr ix of parameters
cn
=
^
j—
(3.17)
/£jj
which has s t r i c t s t a t i s t i c a l meaning in th e l i n e a r case with a
normal d i s t r i b u t i o n of e r r o r s .
( i i i i ) weighted root-m ean -sq ua re r e s i d u a l
Rmsr = v/§/(2m-r)
(3.18)
whose expected val ue i s u n i t y when th e model f i t s t h e d a ta and
the m a tri x of th e weighting f a c t o r s i s th e in v e r s e of the t r u e
c o r r e l a t i o n m a tr ix of random e r r o r s [113].
As can be seen from E q .( 3. 10 ) and E q s . ( 3 . 1 6 - 3 . 1 7 ) , e st i m a te d
u n c e r t a i n t i e s of th e para mete rs and t h e i r c o r r e l a t i o n m a tri x a re not
changed when a l l elements of the m a tr ix of th e weighting f a c t o r s a re
m u l t i p l i e d by the same c o n s t a n t f a c t o r . They depend only on the r e l a ­
t i v e magnitudes of th e weights a s s ig n e d to the i n d i v i d u a l d a ta p o i n t s .
On th e o th e r hand, t h e weighted root- me an -squar e r e s i d u a l Rmsr of
E q . ( 3 .1 8 ) i s p r o p o r t i o n a l to th e f a c t o r m u l t i p l y i n g the m a tri x of
weig hting f a c t o r s . T h e re fo re , i t does not make sense to compare v a lu es
of Rmsr r e s u l t i n g from the use of d i f f e r e n t weighting schemes for
purposes o th e r than e s t a b l i s h i n g con ven ient magnitudes for the
52
weighting f a c t o r s . Comparison of u n c e r t a i n t i e s of the parameter e s t i ­
mates and t h e i r c o r r e l a t i o n s i s , however, u s e f u l as a c r i t e r i o n for
s e l e c t i n g an a p p r o p r i a t e weighting scheme.
Our approximation of the v a r i a n c e s of the d i r e c t l y observed ex per­
imental q u a n t i t i e s based on th e a c c u r a c i e s of the v a r i o u s in s tru m e nts
cannot be s t r i c t l y j u s t i f i e d on the b a s i s of a s t a t i s t i c a l reasoning
and s t a t i s t i c a l meaning of the r e s u l t s ( i i ) - ( i i i i ) , which are based
on maximum l i k e l i h o o d argument or on t h e Gauss-Markov theorem [113],
must be t r e a t e d with a dose of s k e p t i c i s m . S t i l l th e r e s u l t s ( i i ) to
( i i i i ) provide very u s e f u l in fo rm at io n about v a l i d i t y of th e model
f u n c t i o n and the meaning of pa ra m et e rs . Large u n c e r t a i n t y ( i i ) of a
parameter (say AXj/Xj>0.2) i s a warning t h a t parameter x^ i s not p r e ­
c i s e l y e s ti m at e d because e i t h e r th e d a t a do not c o n ta i n enough informa­
t i o n ( e . g . , the frequency range i s not l a r g e enough) or the model which
in c l u d e s t h i s parameter i s not adeq ua te. The c o r r e l a t i o n m a tri x ( i i i )
allo ws us to s i n g l e out p a i r s of pa ram ete rs which a re not well define d
i n d i v i d u a l l y . I f c ^ j - l , the o b j e c t i v e f u n c t i o n Q(X) does not change
when both x^ and x^ a r e i n c re a s e d s im u lt a n e o u s ly by small amounts
p r o p o r t i o n a l to t h e i r r e s p e c t i v e co nfid e nce i n t e r v a l s . Values of the
root-mean-square r e s i d u a l Rmsr can be used to a s s e s s the v a l i d i t y of
d i f f e r e n t models. An F - t e s t fo r goodness of f i t [120] i s s t r i c t l y v a l i d
only in the l i n e a r case with a normal d i s t r i b u t i o n of e r r o r s . Neverthe­
l e s s , l a r g e d i f f e r e n c e s in the Rmsr v a l u e s ob ta in e d with the i n d i v i d u a l
models c l e a r l y i n d i c a t e t h a t the model y i e l d i n g the s m a l l e s t Rmsr
a g re e s b e t t e r with the experiment. G e n e r a l l y , th e b e s t model from the
53
s t a n d p o i n t of l e a s t sq ua re s a n a l y s i s , i s one which g i v e s lowest Rmsr
with a l l parameters having small u n c e r t a i n t i e s and a minimum number of
high c o r r e l a t i o n s between pa rameters.
3.4. Comparison of D i f f e r e n t Weighting Schemes.
From the p o in t of view of th e numerical procedure of n o n li n e a r
l e a s t s q u a r e s , use of d i f f e r e n t weighting f a c t o r s in th e sum of the
squared d e v i a t i o n s i s e q u i v a l e n t to a s c a l i n g of the d a ta and th e model
f u n c t i o n in such a way t h a t the expected e r r o r s of d i f f e r e n t d a ta
become of th e same order of magnitude. When terms due to c o r r e l a t i o n s ,
e . g . , g j 2 in E q . ( 3 . 9 ) a r e n e g l e c t e d , use o f we ightin g f a c t o r s c a n be
re p la c e d by m u l t i p l i c a t i o n of a l l d a t a and the model f u n c t i o n s by the
s q u a r e - r o o t of the co rresponding s t a t i s t i c a l weight. As long as such
s c a l i n g of d a t a does not s i g n i f i c a n t l y a l t e r t h e i r r e l a t i v e magnitudes,
no dra matic change of e st i m a te d parameters i s expected. With u n i t y
weights a l l r e s i d u a l e r r o r s a r e expected to be of th e same or der of
magnitude. When numerical v a lu e s of th e d a t a extend over s e v e r a l o rd e rs
of magnitude, then th e p a r t of th e d a t a having l a r g e a b s o l u t e v a lu e s i s
f i t t e d with much s m a l le r r e l a t i v e e r r o r than th o s e d a t a which a r e
ord e rs of magnitude s m a l l e r . Such a s i t u a t i o n i s t y p i c a l for complex
admittance and impedance s p e c t r a of s o l i d e l e c t r o l y t e s with i o n i c a l l y blocking (or p a r t i a l l y - b l o c k i n g ) e l e c t r o d e s . For a frequency range of a
few decades, the a b s o l u t e value of impedance i s o f t e n s e v e r a l o r d e r s of
magnitude higher a t th e low-frequency end of the spectrum than a t the
h i g h e s t f r e q u e n c i e s . Obviously, the r e v e r s e i s t r u e for admittance
54
d a t a . Least sq uar es f i t t i n g of impedance d a ta with u n i t y weights empha­
s i z e s s t r o n g l y the low frequency p a r t of the spectrum and gi ve s good
e s t i m a t e s of the parameters dominating a t low f r e q u e n c i e s , while p a r a ­
meters governing in th e hi gh-f re que nc y p a r t of the spectrum a r e e s t i ­
mated with g r e a t u n c e r t a i n t y or e f f e c t i v e l y not e st i m a te d a t a l l . When
da ta a re transformed to the adm ittanc e r e p r e s e n t a t i o n , r e l a t i v e a c c u r a ­
c i e s of f i t a r e the r e v e r s e of t h e impedance s i t u a t i o n .
F i t t i n g of the
whole spectrum i s g e n e r a l l y not p o s s i b l e with u n i t y weights.
Several weighting-schemes fo r complex impedance/admittance d a t a
proposed in th e l i t e r a t u r e a r e aimed a t s c a l i n g th e d a t a . Macdonald and
coworkers [106,107] compared the case of u n i t y weights with a weight
(
p r o - p o r t i o n a l to d i f f e r e n t powers of th e numerical value of the i n d i ­
v id ua l d a ta p o i n t , ta k i n g indep enden tly r e a l and imaginary p a r t s .
Weights which a re based on the assumption t h a t th e random e r r o r s of the
r e a l and imaginary p a r t s of a d a t a p o in t a r e not c o r r e l a t e d and a re
p r o p o r t i o n a l to t h e i r i n d i v i d u a l magnitudes were found to give good
e s t i m a t e s of the pa rameters [107], In the n o t a t i o n of E q . ( 3 . 9 ) such
weights a re
9^ = P[Re(yi ) ] - 2 ,
g \2= p[ I m ( y . ) r 2,
gj 2= 0
(3.19)
with a p r o p o r t i o n a l i t y c o n s t a n t p.
Recently Boukamp [121] r e p o r t e d n o n l i n e a r l e a s t sq ua re s f i t t i n g of
impedance/admittance d a t a which were a p p a r e n t l y weighted by th e in v e r s e
of t h e i r a b s o l u t e v a l u e s . Thus,
(3.20)
D if f e r e n c e s between the r e s u l t s of f i t t i n g d a t a r e p r e s e n t e d as
impedance and adm ittanc e were r e p o rt e d in both c as e s. Disagreement i s
i n e v i t a b l e between e s t i m a t e s of th e parameters ob tained by f i t t i n g in
d i f f e r e n t r e p r e s e n t a t i o n s when the a p p l i e d weights a re not r e l a t e d in a
way a s s u r i n g approximate eq uiv alenc e of the o b j e c t i v e f u n c t i o n s . In
order to demonstrate the e f f e c t s of weighting on a n o n li n e a r l e a s t sq ua re s e s t i m a t i o n of pa ram ete rs , we p r e s e n t two examples: (1) the
f i r s t in vol ve s s y n t h e t i c d a ta g e n er at e d by computer,- and (2) the second
inv ol ve s ex perimental impedance spectrum of an RC c i r c u i t .
S y n th e ti c d a ta were ge ne r at e d by a computer for the c i r c u i t given
in F i g . 3 .3 , which had been used by Boukamp [121] to f i t the impedance
spectrum of a p o l y c r y s t a l l i n e Na2 SnS3 sample with evaporated gold e l e c ­
t r o d e s . Values of th e 11 parameters shown in F i g . 3.3 a r e equal to those
e s ti m a te d by Boukamp in the course of f i t t i n g d a ta in an impedance
r e p r e s e n t a t i o n . P a r a m e t r i z a t i o n of c o n s t a n t phase elements i s accom­
p l i s h e d here by means of Eq. ( 3 . 2 2 ) , which d i f f e r s from t h a t used by
Boukamp in the d e f i n i t i o n of the exponent. Data ge ner ate d fo r 45 f r e ­
quencies e q u a ll y -s p a c e d on a lo g a r it h m i c s c a l e between 1.3 mHz and 100
kHz were re p r e s e n te d as frequency, th e a b s o l u t e value and the phase of
th e impedance, and were rounded to 4 decimal d i g i t s .
In order to i n t r o ­
duce e r r o r s of magnitude s i m i l a r to th e r e s i d u a l e r r o r s r e p o rt e d by
Boukamp, va lu es of |Z| and ao were f u r t h e r rounded o f f to the n e a r e s t
p o in t on a d i s c r e t e g r i d c o n s t r u c t e d in such a way, t h a t the s m a l l e s t
r e s olv ed increments were 1% of th e magnitude for the a b s o l u t e valu e of
impedance, | Z | , and 1.5 degrees for th e phase an gle. Roundoff e r r o r s ,
r a t h e r then pseudo-random e r r o r s , were chosen to s im ul ate in s t r u m e n t a l
56
CPgb**3.05x10
ANgb«0.274
Rbu«823..kft
Cd1 -2 0 0 . nF
rA A A r
VW-
Rgb-14.MJ1
CPto“ l .9x10
A N to-0.47
-A A /V
=9—
C in s-S . 8pF
CPad»2.13x10
ANad-0.31
-7
P ig .3 .3 . Equivalent c ir c u it used for computer generation of sy n th etic
impedance data. The valu es of parameters are as estim ated by
Boukamp for a p o ly c r y sta llin e Na^SnS^ sample [2 8 ]. Admittance
o f a constant phase elem ent, represented by crosshatched capa­
c it o r , i s Y = CP(j»)
1-AN
Cgb»9.93nF
Cdl«103.5nF
Rbu»10.02A
M A /V
~AAAr lJW\r
R gb-947.il
R ct»301.kJl
F ig .3 .4 . Schematic arrangement o f the t e s t RC c ir c u it . Displayed values
of r e sista n c e s and capacitances were measured using a d ig it a l
multimeter and a capacitance bridge, r e s p e c tiv e ly .
57
r e s o l u t i o n s in order to show t h a t a l e a s t - s q u a r e s a n a l y s i s with re a so n­
a b le s t a t i s t i c a l weights can be e f f e c t i v e when th e e r r o r s a r e not
randomly d i s t r i b u t e d . Table 3.1 l i s t s th e r e s u l t s of our f i t t i n g , the
row Rep g iv e s the r e p r e s e n t a t i o n used (A-admittance, I- im p e d a n c e ), the
row Wts g iv e s the weighting schemes d e s c r i b e d below. The r e l a t i v e
u n c e r t a i n t i e s (e xpr ess ed in p e r c e n t of the magnitude) a re l i s t e d below
th e e s ti m a te d v a lu e s of the pa rame te rs .
Estima tes of t h e pa rameters and t h e i r u n c e r t a i n t i e s ob ta in e d in
the admittanc e and impedance r e p r e s e n t i o n s were e q ua l, a t l e a s t up to 4
s i g n i f i c a n t f i g u r e s , when th e weights were c o n s t r u c t e d acc ording to
E q s . ( 3 . 5 ) —(3 .9 ) us in g aZ/Z=0.005, A(D=0.75deg which r e f l e c t the r e s o l u ­
t i o n of s y n t h e t i c d a t a (Wts=R), and when s i m p l i f i e d weight f a c t o r s of
E q . ( 3 . 1 2 ) , (Wts=S) were used. With th e weights i n d i c a t e d in E q . ( 3 . 2 0 ) ,
(Wts=B), behavior of th e e s t i m a t e s r e p o r t e d by Boukamp was q u a l i t a t i v e ­
ly reproduced. When e s t i m a t e d in ad mi tta nce r e p r e s e n t a t i o n , the parame­
t e r s which were meant to d e s c r i b e the bulk of e l e c t r o l y t e (Rbu, CPto,
ANto, CAto) and the g r a i n boundary c a p a c i t a n c e (Cpgb, Angb) have small
u n c e r t a i n t i e s , while the g r a i n boundary r e s i s t a n c e (Rgb) and the param­
e t e r s of the e l e c t r o d e s (CAdl, CPad, ANad, Ret) have l a r g e r u n c e r t a i n ­
t i e s . The r e v e r s e i s t r u e of f i t s in th e impedance r e p r e s e n t a t i o n .
Si m il ar behavior i s observed to a much hig he r degree fo r r e s u l t s ob­
t a i n e d using u n i t y weights (Wts=E), where not only the e st i m a te d u n c er ­
t a i n t i e s a r e very high but a l s o e s t i m a t e s d i f f e r from th e parameter
v a lu e s used for g e n e r a t i o n of d a t a . Such a tr e nd can be e xpla in ed by
n o ti n g t h a t weights ^ | Z | _1 [ Eq. ( 3 . 2 0 ) ] a re "halfway" between u n i t y
58
Table 3.1
R e s u l t s of n o n l i n e a r l e a s t - s q u a r e s f i t t i n g with d i f f e r e n t weights for
the s y n t h e t i c impedance spectrum ge n er at e d fo r th e c i r c u i t of F i g . 3.3.
The e st i m a te d u n c e r t a i n t y (e xpre sse d in p e rc e n t of the val ue of a p a r a ­
meter) i s given below each e s t i m a t e of a parameter.
Wts
R
S
B
B
E
E
P
P
Synth.
R ep r.
A ,1
A ,1
I
A
I
A
I
A
8 .23
xlO
Rbu
8 .18
0 .5%
8..19
0 .7%
8.,19
3. 1%
8. 18
0. 5%
8.20
21.%
8. 16
0. 6%
8,.27
0,.7%
8.,23
0..6%
1..69
9..5%
1..78
1 3 .%
1. 75
69.%
1. 74
8. 2%
0.94
625.%
1. 70
7. 6%
2..29
14.%
1. 77
12.%
0..458
1,.9%
0.,462
2..4%
0. 461
14.%
0. 461
1. 5%
0.395
165.%
0. 459
1. 4%
0.,485
2..5%
0. 463
2. 4%
3.,60
3. 2%
3. 60
3.,4%
3. 56
23.%
3. 59
1. 5%
2.12
670.%
3. 58
1. 5%
3..83
3. 5%
3. 62
3. 8%
1 90 Q CPto
xlO
ANto
0..470
3. 80 l o Cins
xlO
1..4° ?
xlO
Rgb
1. 40
1., n
1.,41
1.,6%
1. 40
2. 2%
1. 41
4. 1%
1.34
6.2%
1. 47
14.%
1. 43
2. .1%
1. 45
2. U
3,.05
XlO-8
CPgb
3.,06
0..81
3.,05
1. 1%
3. 04
3. 6%
3. 06
1. 5%
3.01
18.%
3. 09
2. 9%
3..03
1. 5%
3. 09
1. 3%
0..274
ANgb
0. 275
0. 6$
0. 275
1. 0%
0. 274
2. 8%
0. 275
1. 2%
0.267
14.%
0. 278
1. 9%
0. 275
1. 2%
0. 279
1. 1%
Cdl
2. 00
2. 0%
2. 05
3. 0%
2. 01
3. 4%
2. 06
9. 6%
1.87
7.6%
2. 23
42!.%
2. 10
4. 1%
2. 11
4. 2%
CPad
2. 20
1. 5$
2. 19
1. 8%
2. 17
1. 0%
2. 68
12.%
2.17
1.8%
2. 30
120.%
2. 19
2. 5%
2. 17
2. 7%
0..310
ANad
0. 303
1. 2%
0. 304
1. 3%
0. 306
0. 6%
0. 304
9. 5%
0.308
0.7%
0. 292
100.%
0. 304
1. 9%
0. 305
1. 9%
1,.80
xlO
Ret
1.,85
2. 3%
1. 87
3. 3%
1. 83
2. 2%
1. 88
15i . 6 %
1.76
3.1%
2. 06
105.%
1. 90
4. 7%
1. 88
4. 9%
2.
00
7
13
7
xlO
2.
xlO
weights and weights «»|Z|
_2
[ Eq. ( 3 . 1 2 ) ] , th e l a t t e r a s s u r i n g proper
s c a l i n g of d a ta . R e s u l t s of an e s t i m a t i o n with the weights given in
E q . ( 3 . 1 9 ) . (Wts=P) were good and e x h i b i t e d only small disagreement
between the adm ittanc e and impedance f i t s .
I t i s a l s o worth noting t h a t
r e s u l t s obtained using weights which reproduced th e e r r o r s of the syn­
t h e t i c d a ta ( aZ/Z=0.005, a<d=0.75 deg, Wts=R), were only ma rg in ally
d i f f e r e n t from th os e obta in e d with si m ple r weights ->|Z|
_2
[Wts=S,
E q . ( 3 . 1 2 ) ] . The l a t t e r weight f a c t o r s were i d e n t i c a l to Wts=R weights
with aZ/Z=0. 0 1 , Ad>=0.573 deg = 0.01 rad.
Impedance of the c i r c u i t composed of r e s i s t o r s and c a p a c i t o r s ,
shown in F i g . 3.4 was measured with Hewlett Packard 4800A ve ct o r impe­
dance meter. T h i r t y f i v e measurements were taken a t 30 fr e q u e n c i e s
approximately e q u a ll y -s p a c e d on l o g a r i t h m i c s c a l e between 5H2 and
500kHz (two r e a d in g s were taken a t the same frequency every time the
frequency or the impedance range was changed). The r e s i s t o r s and capac­
i t o r s were f i r s t measured s e p a r a t e l y usi ng a d i g i t a l multim eter and
General Radio Capacitance Bridge 1615A, r e s p e c t i v e l y . The impedance of
each element alo ne was a l s o measured over the whole frequency range
with a v e c t o r impedance meter in or der to check t h a t r e a c t i v e compo­
n en ts of impedance of r e s i s t o r s and l o s s e s in c a p a c i t o r s were n e g l i g i ­
b l e . The c i r c u i t given in F i g . 3.4 has been used by Bauerle [97] as a
simple model of a p o l y c r y s t a l l i n e oxygen ion conductor with p a r t i a l l y
blocking e l e c t r o d e s . Here, v a lu es of the r e s i s t o r s and c a p a c i t o r s were
chosen to o b t a i n impedance spectrum ext end in g over four o rd e rs of
magnitude in | Z | .
60
Table 3.2
R e s u l t s of n o n l i n e a r l e a s t - s q u a r e s f i t t i n g with d i f f e r e n t weights for
th e exper iment al impedance spectrum of t h e RC c i r c u i t of P i g . 3 .4 . The
e st i m a te d u n c e r t a i n t y i s given below each e s t i m a t e of a parameter ( in
p e rc e n t of th e value of a parameter)
Wts
S
R1
R2
E
E
P
P
Rep
I, A
I, A
I, A
I
A
I
A
Nominal
valu e
AZ/Z
4>
0.01
0.01
0.005
0.02
0.02
0.005
10.02
(A)
Rbu
10.46
2.6*
11.02
4.3*
10.41
2.2*
33.2
670.*
10.12
0.5*
10.63
2.1*
10.45
0.3*
997.
(A)
Rgb
995.
0.4%
989.
0.3%
1002.
0.8*
1111.
23.%
956.
1.4%
1004.
0.8*
995.
0.8*
9.93
(nF)
Cgb
9.68
0 .4*
9.68
0.3*
9.72
0.8*
10.5
66.*
9.59
0.4*
9.64
0.7*
9.69
0.9*
103.5
(nF)
Cdl
104.7
0.2*
104.4
0.2*
105.1
0.6%
106.1
0.3*
103.4
3.5*
104.6
0.6%
105.5
0.5%
301.
(kfl)
Ret
295.
0.9*
296.
0.8*
301.
1.0*
303.4
0.4*
151.
145.*
293.
1.5*
297.
1.1*
Rmsr
0.984
1.054
1.610
The r e s u l t s of the f i t t i n g s with d i f f e r e n t weights a r e l i s t e d in
Table 3.2. Estimates of the parameters a re given along with t h e i r r e l a ­
t i v e u n c e r t a i n t i e s . Values of the roo t-m ean-square r e s i d u a l Rmsr a re
included for f i t s employing r e s o l u t i o n weights of E q s . ( 3 . 4 ) - ( 3 . 8 ) with
d i f f e r e n t parameters aZ/Z,
a <p
.
The Rmsr i s expected to be equal 1 when
weights a re based on the v a r i a n c e s of e r r o r s . Values of th e Rmsr ob­
t a i n e d from f i t t i n g th e measured spectrum fo r a known c i r c u i t can be
used to e st i m a t e the magnitude of aZ/Z and
Ad>.
The r a t i o (aZ/Z)/ao>
which giv es the lowest u n c e r t a i n t i e s of a l l parameters i s expected to
b e s t r e f l e c t the t r u e d i s t r i b u t i o n of measurement e r r o r s . From Table
3 .2 , i t appears t h a t the f i t with aZ/Z=0.01, a<>=0.573deg=0.01rad
(Wts=S) provides the most c o n s i s t e n t weighting f a c t o r s . I n c i d e n t a l l y ,
th e se weighting f a c t o r s a re i d e n t i c a l with the s im ple r weights HZ!
-2
given in Eq. ( 3 . 1 2 ) with D=0.01.
Consistency of any n o n li n e a r l e a s t - s q u a r e s f i t t i n g may be examined
by a n a l y s i s of i t s r e s i d u a l e r r o r s , t h a t i s , th e
d i f f e r e n c e s between
th e measured and the f i t t e d v a lu e s . Residual e r r o r s , r e p r e s e n t e d as
r e l a t i v e e r r o r s of |Z| expressed in pe rc e n ta g e s and the e r r o r s of phase
expressed in de gre es , a re p l o t t e d in F i g . 3.5 fo r the f i t usin g weights
HZ I
_2
(Wts=S). The r e s i d u a l e r r o r s of
|Z| and
seem to have r o o t -
mean-square va lu es approximately given by th e assumed
r e s o l u t i o n of
measurements. C o r r e l a t i o n s between e r r o r s of th e measurements a t ne igh ­
boring f r e q u e n c i e s a r e a ppar ent : r e s i d u a l e r r o r s have the same sign for
s e v e r a l co nse c utiv e f r e q u e n c i e s . Such c o r r e l a t i o n s a r i s e from the f a c t
t h a t s y s te m a ti c e r r o r s due to im perfect c a l i b r a t i o n a re s i m i l a r a t
100*err(Z)/Z
err (Phase)
/deg
62
o. a
0.0
0.0
0.6
1.2
1.8
2.4
3.0
3.6
4.2
4.8
S. 4
6.0
lo g ( f )
F ig .3 .5 . Residual errors of the absolute value and the phase of impedan­
ce obtained from nonlinear lea st-sq u a res f it t i n g of the theore­
t i c a l impedance of an RC c ir c u it arranged as in F ig .3 .4 . to the
measured spectrum. Heights given by E q .(3.12) were used (Wts=S
in Table 3 .2 ).
63
ne igh borin g f r e q u e n c i e s , an e f f e c t which has not been taken i n t o a c ­
count in our a n a l y s i s . R e s u l t s of the f i t a r e a l s o shown in F i g . 3 .6 ,
where d a t a a re r e p r e s e n t e d by p o i n t s and the f i t t e d response by c o n t i ­
nuous l i n e s . The r e s u l t s of th e f i t in adm ittance and impedance r e p r e ­
s e n t a t i o n were i d e n t i c a l , to a t l e a s t 3 s i g n i f i c a n t f i g u r e s , when
weights were p r o p e r l y trans for med between r e p r e s e n t a t i o n (Wts=Rl, R2,
S). F a i l u r e of th e f i t s with u n i t y weights
(Wts=E) u nd ersco res the need
fo r proper s c a l i n g of d a t a .
The p r o p o r t i o n a l weights
r e s u l t s . As long as th e phase
of Eq .( 3 .1 9 ) (Wts=P) ag ain y ie ld e d good
a ngle of th e complex d a t a does not l i e in
the v i c i n i t y of 0° and +90°, p r o p o r t i o n a l weights seem t o perform
s a t i s f a c t o r i l y . When th e phase ang le approaches 0 or +90 d e gre es , the
imaginary or r e a l component goes to z e r o , and acc ord ing t o E q . ( 3 . 1 9 ) ,
th e corr esp ond in g weight d i v e r g e s . Even i f a zero value i s not reached,
th e r e s u l t i n g heavy w ei ght ing of th e v a n is h i n g components of the com­
plex datum i s not j u s t i f i e d , u n l e s s p r e c i s e and d i r e c t measurement of
t h i s component i s made (as in t h e case of l o s s ta n g e n t measurement with
a capacitance bridge).
3 .5 . Models of the A.C. Response of S o li d E l e c t r o l y t e / E l e c t r o d e s C e l l s .
Seven model f u n c t i o n s a r e i n c o r p o r a t e d in th e computer program
FIRDAC.
The common f e a t u r e of th os e f u n c t i o n s i s the a b i l i t y to model
the de pre sse d s e m i c i r c l e s and i n c l i n e d sp u rs o ft e n observed in complex
plane p l o t s of impedance or a d m itt a nce . F r a c t i o n a l powers of the f r e ­
quency a re used for t h i s purpose e i t h e r in Cole-Cole l i k e e x p re s s io n s
64
3.2
8.0
2.8
7.0
6.0
2.0
S.0
SuscQptance
/10*z fl'1
2. 4
4.0
3.0
2.0
0. 4
—1 0 . 0 * -
8.0
0.0
1.0
2.0
3.0
4.0
S. 0
6.0
7.0
8.0
S. 0
Conductance / lO '4 fl"1
F ig .3 .6 . Complex plane p lo ts of admittance for the c ir c u it of F ig .3 .4 .
The measured impedances are represented by p o in ts, the f it t e d
response (Wts=S f i t of Table 3 .2 ) by continuous lin e s with tic k
marks at measuring freq uen cies.
(a) the whole measured spectrum 5 Hz to 500 kHz,
(b) enlarged low frequency part 5 Hz to 12 kHz.
65
[122].
R
Z = -------------_ _
l + ( j n 0)
(3 .21 )
or in the admittance of c o n s t a n t phase elements - CPE [123,124]
Y = C P( ju )1_AN = CP w1_AN [cosj(l-AN) + jsinjCl-AN)]
where R, a , tq and CP, AN a r e r e a l c o n s t a n t s .
(3.22)
With AN=0, Eq. ( 3 . 2 2 )
y i e l d s th e ad mi tta nce of a p e r f e c t c a p a c i t o r . The ad mittan ce given by
E q .( 3 .2 2 ) obeys the Kramers-Kronig r e l a t i o n between the r e a l and the
imaginary p a r t s [125], The un frequency dependence of ad mittance c o r r e ­
sponds to the time domain c u r r e n t t r a n s i e n t response on a p l i c a t i o n of a
c o n s t a n t v o l t a g e of the form t “n . The use of a c o n s t a n t phase angle
i m plie s t h a t the r a t i o of the average energy s t o r e d to th e energy
d i s s i p a t e d per c yc le i s freq ue ncy -i nde pe nde nt.
Frequency-dependence of d i e l e c t r i c p e r m i t t i v i t y , le a d i n g to
E q . ( 3 . 2 2 ) , has been observed in a v a r i e t y of m a t e r i a l s and proposed to
be "The Univer sal D i e l e c t r i c Response" a r i s i n g from c o o p e r a t i v e manybody i n t e r a c t i o n s in the s o l i d s [7 8 ,8 0 ,1 2 6 ] . In the re c e n t th e o ry
of
d i e l e c t r i c r e l a x a t i o n developed by Dissado and H i l l [ 81 ,1 27 ] , f r a c t i o n ­
a l power-dependence of the p e r m i t t i v i t y on frequency i s ob ta in e d asymp­
t o t i c a l l y , both for f r e q u e n c i e s above and below the l o s s peak. Values
of the exponents a r e r e l a t e d to the energy spectrum of weak, coopera­
t i v e i n t e r a c t i o n s in the m a t e r i a l [128], There e x i s t s exper imen tal
evidence t h a t s t r o n g low-frequency d i s p e r s i o n of s u s c e p t i b i l i t y of the
form of E q . ( 3 . 2 2 ) , with AN approaching 1, i s t y p i c a l for m a t e r i a l s in
which hopping of the charge c a r r i e r s c o n t r i b u t e s to the d i e l e c t r i c
66
r e l a x a t i o n [ 7 9 ,8 2 ] . Frequency-dependent c o n d u c t i v i t i t y , le ad in g to
admittance given by E q . ( 3 . 2 2 ) , has been ob ta in e d by Bernasconi e t a l .
[76,77] in the th e o ry of hopping in a d i s o r d e r e d , one-dimensional
system with random d i s t r i b u t i o n of th e h e i g h t s of th e energy b a r r i e r s .
Constant ph ase -a ngl e impedance has been a l s o a s s o c i a t e d with a " d i s ­
persed" i n t e r f a c e and has been observed fo r porous and rough e l e c t r o d e s
[123]. Recently the adm ittanc e given by Eq .( 3 . 2 2 ) has been de ri v e d in
th e th e o ry of t r a n s p o r t through a " s e l f s i m i l a r " i n t e r f a c e [129], which
i s based on the concept of f r a c t a l geometry [130], D if f us io n c o n t r o l of
the t r a n s p o r t le a d s to an impedance given by Eq. ( 3 . 2 2 ) , with AN=0.5
corre spo ndi ng to an i n f i n i t e Warburg impedance [131].
The foreg oin g l i s t of proposed j u s t i f i c a t i o n s for c o n s ta n t- p h a s e
ad mittan ce with a f r a c t i o n a l power dependence on frequency s u gg es ts
t h a t no unique i n t e r p r e t a t i o n can be given for CPE un le s s some a d d i ­
t i o n a l in fo rm at io n about the system under study i s a v a i l a b l e . In the
followi ng models of s o l i d e l e c t r o l y t e / e l e c t r o d e s c e l l s , the CPE [and
Cole-Cole l i k e e x p r e s s i o n s , e . g . , Eq. ( 3 . 2 1 ) ] a re used as convenient
phenomenological means fo r p a r a m e t r i z a t i o n of th e ex perimental imped­
ance s p e c t r a . A n a ly si s of the v a r i a t i o n s with ex perimental c o n d i t i o n s
(t e m p e r a tu r e , atmosphere, dimensions) of the CPE parameters (CP,AN)
e st i m a te d fo r a s e t of s p e c t r a measured with c e r t a i n m a t e r i a l may,
however, lead to a co nc lu si on about the ph y s ic a l meaning of given
element in th e model.
Constant phase elements a r e r e p r e s e n t e d in e q u i v a l e n t c i r c u i t d i a ­
grams as c ro s s h a tc h e d c a p a c i t o r s . An acronym of a model f u n c ti o n used
67
in the program gi v e s th e maximum number of parameters and a unique
t h r e e - c h a r a c t e r name. Acronyms of th e parameters a r e t y p i c a l l y given as
a combination of one or two c a p i t a l l e t t e r s d e f i n i n g th e type of param­
e t e r [R r e s i s t a n c e , C or CA i d e a l c a p a c i t a n c e , CP p r e e x p o n e n ti a l f a c t o r
of CPE, AN exponent of CPE acc ord ing to E q .( 3. 22) and two small l e t t e r s
marking the t y p i c a l p h y s ic a l o b j e c t or phenomenon modelled by given
parameter (bu - bulk of e l e c t r o l y t e , gb - g r a i n b o un da ri e s, e l - e l e c ­
t r o d e s , ad - a d s o r p t i o n , e t c . ) . When a model i s used to f i t experimen­
t a l spectrum, the a c t u a l p h y s i c a l i n t e r p r e t a t i o n of a parameter may not
be r e l a t e d to i t s acronym.
Lengthy formulas for the impedance as a fu n c t i o n of frequency a re
omitted for more complex c i r c u i t s . They can be simply de ri v e d by adding
impedances for p o r t i o n s of th e c i r c u i t connected in s e r i e s and adding
adm ittanc es fo r the p a r a l l e l bran ch es. M ulti-ele men t networks of models
13C0M, 17T0T, 12C0R and 18MAC a r e not meant to be used in t h e i r whole
complexity. They a re provided as f l e x i b l e frameworks fo r the c r e a t i o n
of a v a r i e t y of subnetworks by e l i m i n a t i o n of c e r t a i n elements or
branches,- t h i s i s accomplished by f i x i n g a p p r o p r i a t e para mete rs. A
s h o r t - c i r c u i t may e a s i l y be in tr od uc e d in p la ce of element of any kind
by f i x i n g some pa ra m ete rs , fo r example, f i x i n g the r e s i s t a n c e a t zero
(or very small v a l u e , e . g . 10
-6
g ) , the c a p a c it a n c e a t a very high value
( e . g . 1000F when lowest frequency i s 1Hz). The CPE - preexponent a t a
5
high val ue (CP=10 ) , the exponent AN=1. S i m i l a r l y , each element may be
replaced by an o p e n - c i r c u i t . The number of s im ul ta neo us ly est im at e d
parameters i s l i m i t e d to f i f t e e n by the program.
68
3 . 5 . 1 . Sum of Two Cole-Cole l i k e Formulas fo r Impedance - 7INC.
The complex impedance i s given by
R1
Z(w) = Rif + --------------------- +
fc)^
1+
where R i f ,
R2
(3.23)
1 +(J*s)1_“2
Rl, R2, u^=OMl, v>2=OM2, a^=ALl, «2=AL2 a r e th e seven r e a l
parameters.
When the r e l a x a t i o n f r e q u e n c i e s w^,
differ significantly
[u^/«2>100] the impedance given by Eq .( 3 .2 3 ) i s re p r e s e n te d by two
de pressed s e m i c i r c l e s in a complex plane p l o t , each s e m i c i r c l e c o r r e s ­
ponding to one Cole-Cole term. E x t r a p o l a te d i n t e r c e p t s with th e r e a l
a x i s a r e R i f , Rif+Rl and Rif+Rl+R2. The d e p re s s io n a n g le s of the semi­
c i r c l e s a re ot^n/2 and 0 ^ / 2 . The impedance of th e
F i g . 3 .7 i s
c i r c u i t shown in
given by E q . ( 3 . 2 3 ) . The r e l a x a t i o n f r e q u e n c i e s Uj, <i> 2 a r e
r e l a t e d to parameters of c o n s t a n t phase elements by
(Vl)
= CPlxRl
(ot7_1)
«2
= CP2xR2
(3. 24)
K l e i t z and Kennedy [104] pr e s e n te d a g r a p h i c a l method fo r a n a l y s i s
of impedance diagrams correspo nding to Eq .( 3 .2 3 ) based on i t e r a t i v e
g e o m e t r i c a l - f i t t i n g of each of the two s e m i c i r c l e s , when the o th e r
Cole-Cole term i s s u s t r a c t e d from the t o t a l impedance. This method was
improved by Tsai and Whitmore [105] by means of n o n li n e a r l e a s t - s q u a r e s
f i t t i n g of s e m i c i r c l e s . Nonlinear l e a s t - s q u a r e s f i t t i n g of th e e n t i r e
spectrum gener ate d acc ording to Eq .( 3 .2 3 ) can, however, be accomplished
s t a r t i n g from crude i n i t i a l gue sses for pa rameters which can be made
°—vw—
69
CP2
|AL2
CPl
ALl
R if
R2
Rl
u \M r J
F ig .3. 7. C ircuit including constant phase elements which i s equivalent
to the sum of two Cole-Cole lik e expressions for the impedance
given by Eq.( 3 .2 3 ).
<4)CPgb
(6)ANgb
Model 6CFC
•—| —VW
(3)CPel
(5)ANel
(l)Rbu
4A /W
(2)Rgb
F ig .3. B. Generalized four-element equivalent c i r c u i t 6 CFC.
Model 13COM
(6)CPgb
(7)ANgb
(4)CPbu
(5)AMbu
(13)CAgb
(12)CAbu
(8)CPel
(9)ANel
(3)Rad
(10)CPad
(ll)ANad
t
(2)Rgb
“ VVV"
(l)Rbu
Fig.3. 9. Generalized equivalent c i r c u i t 13C0M; nine elements, three sec­
tio n s connected in s e r i e s .
70
upon visual inspection of the complex plane plot of impedance, without
invoking the above mentioned ite r a t iv e geometrical schemes.
3 .5 .2 . Two Depressed Arcs in Complex Plane Admittance Plot - Empirical
Formula 6 TSA.
The expression for complex admittance i s given by [12]
1
1
Y(«) =
<^-1
Rbu+Rgb
l+[jwCdl(Rbu+Rgb)]
^ 25)
Rgb
1
Rbu(Rgb+Rbu)
l + [ jwCgb Rbu Rgb/(Rbu+Rgb)]
«9- l
where Rb, Rgb, Cdl, Cgb, 0$ctj=AL2$l, 0 $<*2 =AL2^ are s ix real parameters.
Equation (3.25) i s the sum of two Cole-Cole lik e expressions
[Eq.(3.21)] for admittance. With s u f f ic ie n t separation of the time con­
s ta n ts, Cdl(Rb+Rgb) >> Cgb Rb Rgb/(Rb+Rgb), i t i s represented by two
depressed arcs in a complex plane p lot of admittance. The extrapolated
in tercepts with real axis are 0, 1/Rbu, and approximately l/(Rbu+Rgb).
Equation (3.25) has been used as an empirical expression for the admit­
tance of p o ly cry stallin e NASICON p e l le t s with ion-blocking platinum
electrodes [105], where i t has been proposed that Eq.(3.25) i s a
gene­
ra liza tio n of the RC c ir c u it arranged as in F ig .3.4 with Ret replaced
by an open c ir c u it . Such a c ir c u it has been used as a model of poly­
c r y s ta llin e s o lid e le c t r o ly te with blocking electrodes [39,41,132]. It
has to be pointed out, however, that in the lim it 0^= 0 , ot2 =0 admittance
given by Eq.(3.25) i s not exactly equal to the admittance of th is RC
c ir c u it . The relation between parameters used in Eq.(3.25) and the
71
elements of the e q u i v a l e n t c i r c u i t of F i g . 3.4 i s not w e l l - e s t a b l i s h e d .
N e v e r t h e le s s , E q .( 3 .2 5 ) proved to be q u i t e u s e f u l f o r e x t r a c t i o n of the
r e s i s t a n c e s of bu lk, Rbu, and g r a i n bou n d a ri e s, Rgb, from ex perimental
da ta e x h i b i t i n g s i g n i f i c a n t o v e rl a p between two de pre sse d a r c s in the
complex plane p l o t of adm ittanc e [105].
3 . 5 . 3 . Gen eralized E qui val en t C i r c u i t . Four Elements - 6CFC.
The c i r c u i t of F i g . 3.8 i s a g e n e r a l i z a t i o n of the RC c i r c u i t o ft e n
used as simple model of a p o l y c r y s t a l l i n e e l e c t r o l y t e with blocking
e l e c t r o d e s [39,41,132] or s i n g l e c r y s t a l s o l i d e l e c t r o l y t e with an ad­
sorbed i n t e r f a c e l a y e r a t the e l e c t r o d e [133]. C ap a c it o rs a re rep laced
by c o n s t a n t phase elements in order to o b t a i n b e t t e r agreement with
experimental d a t a . Impedance of t h i s c i r c u i t i s given by
1
Rgb
(3.26)
Z(w) = Rbu +
CPel(jw)
The s i x parameters a r e Rbu, Rgb, CPel, CPgb, ANel, ANgb (O^ANeKl,
O^ANgb^l). Equation (3 .2 6) giv e s th e impedance of an i d e a l RC c i r c u i t
when ANel=0 and ANgb=0.
3 . 5 . 4 . Generalized Eq uiv ale nt C i r c u i t . Nine Eleme nts . Three S e c ti o n s in
S e r i e s - 13C0M.
The c i r c u i t of F i g . 3.9 c o n s i s t s o f t h r e e s e c t i o n s , connected in
s e r i e s . The l e f t m o s t s e c t i o n in F i g . 3 .9 , made of r e s i s t o r Rad and two
c o n s t a n t phase elements (CPel,ANel) and (CPad.ANad), i s a g e n e r a l i z a ­
t i o n of the Randles [94,134] e q u i v a l e n t c i r c u i t for an e l e c t r o d e with
a d s o r p t i o n pro c e ss . The element (CPel,ANel) r e p l a c e s the i d e a l do ub le ­
l a y e r c a p a c i t o r and the element (CPad.ANad) s u b s t i t u t e s for an i n f i n i t e
Warburg impedance. The two remaining s e c t i o n s , each c o n s i s t i n g of a
r e s i s t o r p a r a l l e l to a CPE and an i d e a l c a p a c i t o r , may r e p r e s e n t the
r e s i s t i v e and p o l a r i z a t i o n e f f e c t s of the g r a i n bo und ar ies and the bulk
of th e p o l y c r y s t a l l i n e m a t e r i a l . I t has been observed in s e v e r a l c ase s
t h a t a p a r a l l e l combination of an i d e a l c a p a c i t o r and a CPE gi ve s
b e t t e r agreement with th e ex perimental d a t a than e i t h e r of the two
alo ne.
The fr equency-dependent impedance of the c i r c u i t of F i g . 3.9
is
given by
CPad (jw) 1-ANad
Z(«) =
C p e l ( j u ) 1_ANel +
-1
1 + Rad CPad (jto) 1-ANad
+
Rgb
+ --------------------------------------------------------------- +
1 + Rgb CAgb j» + Rgb CPgb ( j w ) i-AW9D
(3 .27)
Rbu
1 + Rbu CAbu ju
+ Rbu CPbu ( j w ) 1~ANbU
where th e exponents ANel, ANad, ANgb, ANbu l i e between 0 and 1.
3 . 5 . 5 . Generalized Equi val en t C i r c u i t . Seventeen Elements. Four Sec­
t i o n s in S e r i e s - 17T0T.
The c i r c u i t in F i g . 3.10 i s an extended v e r s i o n of th e c i r c u i t
13C0M of F i g . 3.9. In the s e c t i o n r e p r e s e n t i n g e l e c t r o d e s , i t c o n t a i n s
an a d d i t i o n a l p a r a l l e l c a p a c i t o r CAdl which, t o g e t h e r with CPE (CPel,
ANel), may be used to model a combination of an i d e a l and a d i s p e r s i v e
d o u b l e - l a y e r . An a d d i t i o n a l s e c t i o n i s connected in s e r i e s , which
73
(15)CAdl
(9)CPgb
(10)ANgb
(5)CPbu
(6)ANbu
|8 |___
(2)CfbO
(3)ANbO
(13)CPad
(14)ANad
(12)Rad
(16)CPel
(17)ANel
(11)CAgb
(8)Rgb
(7)CAbu
*J\AAr*
(l)RbuO
(4)Rbu
Model 17TOT
F ig .3.10. Generalized equivalent c i r c u i t 17T0T; twelve elements, four
se c tio n s connected in s e r ie s .
Model 12C0R
(3)CPgb
(4) ANgb
(9)CAdl
(7)CPad
(8)ANad
(l)Rbu
(5)CAgb
(2)Rgb
(6)Rad
(10)CPel
(11)ANel
m
(1 2)G le-l/R le
F ig .3.11. Generalized equivalent c i r c u i t 12C0R; nine elements.
t
74
c o n s i s t s of a r e s i s t o r , RbuO, in p a r a l l e l with a CPE (CPbO.ANbO). This
s e c t i o n may be u s e f u l in s i t u a t i o n s when t h r e e s e p a r a b l e , d i s p e r s i v e
phenomena m a n if e s t i t s e l f in the system, such as the bulk e f f e c t s ,
p o l a r i z a t i o n of the g r a i n bounda ries and an adsorbed i n t e r f a c e l a y e r on
the s u r f a c e of e l e c t r o l y t e .
3 . 5 . 6 . Generalized Eq uiv ale nt C i r c u i t . Nine Elements - 12C0M.
The c i r c u i t in F i g . 3.11 i s a convenient v e r s i o n of more complex
c i r c u i t s 17T0T and 13C0M. I t does not have a c a p a c i t i v e element p a r a l l e l
to r e s i s t o r , Rbu, which i s r e a l i s t i c when the frequency range covered
does not extend high enough to probe the d i e l e c t r i c p o l a r i z a t i o n of the
bulk m a t e r i a l . A r e s i s t o r p a r a l l e l to the balance of the c i r c u i t , p a r a ­
metri ze d by conductance Gle, may provide an e x t r a conduction path such
as s u r f a c e or e l e c t r o n i c . The fo ur -e l em e nt s e c t i o n on the r i g h t of
F i g . 3.11 prov ide s f a i r l y f l e x i b l e model fo r e l e c t r o d e e f f e c t s .
3 . 5 . 7 . G en eralized E qu iv ale nt C i r c u i t with D if f u si o n Element. Twelve
Elements - 18MAC.
The c i r c u i t of F i g . 3.12 i s the l a r g e s t network implemented here
with the e x p e c t a t i o n of pr o vi d in g a complete and f l e x i b l e model of a
s o l i d e l e c t r o l y t e / e l e c t r o d e s system. The two main p a r t s ar e meant to
r e p r e s e n t e f f e c t s in the e l e c t r o l y t e i t s e l f and a t th e e l e c t r o d e s . The
l e f t p a r t c o n t a i n s two s u b s e c t i o n s connected in s e r i e s , each b u i l t of
r e s i s t o r s (Rbu and Rgb) with a p a r a l l e l CPE (and a c a p a c i t o r in case of
Rgb), which may r e p r e s e n t the bulk of e l e c t r o l y t e and the g r a i n bounda­
ry i n t e r f a c e e f f e c t s . The CPE (CPto.ANto) and the c a p a c i t o r CAto con-
75
(l)Rbu
(4)Rgb
rA/Wi
Model 18MAC
(18)CAdl
(7 )CAgb
(15)CFad
(16)ANad
(2)CPbu
(3)ANbu
(17)Rct
5)CPgb
(6 )ANgb
(8)CPto
(9)ANto
(14)Rdi
■AAArLJH
(10)CAto
(ll)YDIO, (12)KDEF
(13)DII£
P i g .3. L2. Equivalent c ir c u it 18MAC with gen eralized , fin it e - le n g t h d i f ­
fusion impedance, twelve elements.
76
s t i t u t e a branch p a r a l l e l to both s u b s e c t i o n s which i s included as a
model of the d i e l e c t r i c response of the m a t e r i a l ; however, t h i s branch
may not be r e l a t e d t o th e t r a n s p o r t of charge c a r r i e r s , i . e . , i t may
r e p r e s e n t the d i p o l a r r e l a x a t i o n in the framework of s o l i d e l e c t r o l y t e .
The la d d er network model of th e e l e c t r o d e p ro c e ss e s a t th e r i g h t of
F i g . 3.12. i s p a t t e r n e d a f t e r th e work of F r a n c e s c h e t t i and Macdonald
[131,135]. I t in c l u d e s a g e n e r a l i z e d d i f f u s i o n impedance, ZD, which in
terms of our p a r a m e t r i z a t i o n , chosen f o r numerical convenience, i s
given by
1
KDEF s in h ( /j u /D I L E ) + /joTcosh(/ju?DILE)
(3.28)
YDIO/ju KDEF cosh(/ju?DILE) + /jw” sinh(/jw/DILE)
where the 3 para mete rs a r e YDIO, KDEF, DILE.
The impedance given by Eq. ( 3 . 2 8 ) i s fo rm ally e q u i v a l e n t to the
impedance of a d i s t r i b u t e d tr a n s m i s s i o n l i n e of c h a r a c t e r i s t i c impedance l / / $ u and a pro p a g a ti o n c o n s t a n t ,/ju" te rm in a te d a t a d i s t a n c e
1/DILE by a load impedance 1/KDEF. The f a c t o r YDIO i s a normal iz ing
c o n s t a n t . The para mete rs used in E q .( 3 .2 8 ) a r e r e l a t e d t o the param­
e t e r s of the s m a l l - s i g n a l a . c . response th e o r y of d i f f u s i o n [131] as
follows:
KDEF = k,/v^T
r
e
DILE = yD” / l e
(3. 29 )
YDIO = / D " k , p/ ( k l h R k , h )
where Dg i s the d i f f u s i o n c o e f f i c i e n t of r e a c t a n t in the e l e c t r o d e ,
kf
th e r a t e of exchange of r e a c t a n t a t th e o u te r s u r f a c e of the e l e c t r o d e
77
a t d i s t a n c e l g from t h e e l e c t r o l y t e , k . ^ , kl b , k ^ a r e r a t e c o n s t a n t s
d e s c r i b i n g the formation of an adsorbed s p e c i e s a t th e e l e c t r o d e - e l e c t r o l y t e i n t e r f a c e and t h e i r e n t r y i n t o th e e l e c t r o d e , and RT i s the
normali zi ng r e a c t i o n r e s i s t a n c e .
The g e n e r a l d i f f u s i o n impedance of Eq .( 3 .2 8 ) reduces to the c l a s ­
s i c a l Warburg [136] i n f i n i t e - l e n g t h , d i f f u s i o n impedance l//juTwhen
DILE-*0.
With KDEF=0, E q .( 3 .2 8 ) d e s c r i b e s the s i t u a t i o n in which ex­
change of a r e a c t a n t i s blocked a t the o u te r s u r f a c e of th e e l e c t r o d e ;
in t h e low-frequency l i m i t (w-»0), ZQ behaves as a c a p a c i t o r . With
KDEF-»oo , r a t e of exchange i s i n f i n i t e l y ra p id and, in the l i m i t w-»0, ZD
behaves as freq ue ncy -i ndepe nd en t r e s i s t a n c e , which i s the s ta n d a rd f i n i t e - l e n g t h Warburg response [94,137].
According to th e the or y of F r a n c e s c h e t t i and Macdonald [131], the
d i f f u s i o n impedance may be observed when d i f f u s i o n of the r e a c t a n t (or
product s p e c i e s ) , to (or from) the e l e c t r o d e / e l e c t r o l y t e i n t e r f a c e i s
the r a t e - c o n t r o l l i n g s t e p o f an e l e c t r o c h e m i c a l r e a c t i o n . The impedance
given by E q .( 3. 28) i s a g e n e r a l i z a t i o n of the c l a s s i c a l Warburg impe­
dance [136] fo r the s i t u a t i o n when the probing frequency i s low enough to
re v e a l t h a t the d i f f u s i o n le n g t h i s f i n i t e . A f i n i t e le ngt h d i f f u s i o n
impedance has been de ri v e d fo r d i f f u s i o n of a n e u t r a l r e a c t a n t through
an e l e c t r o d e , both in the case of a f u l l y - s u p p o r t e d e l e c t r o l y t e and
unsupported bi n a ry e l e c t r o l y t e [131]. I t a l s o appea rs in e xa ct t r e a t m e n t s
of th e s m a l l - s i g n a l a . c . response of an unsupported bin a ry e l e c t r o l y t e s
un le s s one of the charged s p e c i e s i s s t r i c t l y immobile or both s p e c i e s
have i d e n t i c a l m o b i l i t i e s and r e a c t i o n r a t e s a t the e l e c t r o d e s [138,139],
F i n i t e - l e n g t h Marburg e f f e c t s may a l s o a r i s e from p ro c e ss es in the
e l e c t r o l y t e , even when no e l e c t r o d e d i f f u s i o n i s a c t u a l l y p r e s e n t [140].
The c i r c u i t in F i g . 3.12 in c l u d e s a CPE (CPad.ANad), r a t h e r than a
c a p a c i t o r , p a r a l l e l to the d i f f u s i o n impedance, which may account for
the fr equency-dependent d i s p e r s i o n of th e a d s o r p t i o n a t th e e l e c t r o d e .
I t adds f l e x i b i l i t y t o th e model so i t can be u s e f u l even when a
g e n e r a l i z e d d i f f u s i o n impedance i s not observed.
IV. MEASUREMENTS OF THE COMPLEX PERMITTIVITY £T MICROWAVE FREQUENCIES
4.1. Introduction.
The rela tiv e
e =
i s defined
complex p erm ittivity
e '- je " = e'-jo /u e^
= e '(l-jta n 8 )
(4.1)
as the c o n titu tiv e parameter in the Maxwell's equations for
a region f i l l e d with homogeneous, iso trop ic m aterial, written with e^wt
time dependence
V*H = jue eE
(4.2)
0
V*E = -j«U0|iH
10
Q
The complex perm ittivity can be measured at microwave frequencies
11
to 10
Hz by a variety of techniques [141,142], which can be
divided into waveguide and resonant cavity methods. Here the discussion
i s limited to waveguide techniques, based on measurement of scatterin g
of the fundamental waveguide mode by a sample placed inside rectangular
or coaxial guide. Only nonmagnetic materials are considered, n=l.
In the e a r lie s t method, a voltage r e fle c tio n c o e f fic ie n t i s mea­
sured in front of a section of waveguide completely f i l l e d with the
material under study and terminated by a plate short c ir c u it in contact
with the sample [143]. The complex perm ittivity i s evaluated by solving
a complex transcendental equation. Since solution of th is equation is
not unique, measurements on two samples of d ifferen t length are re­
quired for determination of the correct value. A ltern atively, d ifferen t
terminations of the waveguide f i l l e d with material can be used in order
to eliminate the ambiguity. This leads to the variable termination
79
80
method in which a section of empty waveguide with moveable short c i r ­
cu it terminates the section with the sample. S e n s itiv ity of the mea­
surement can be improved by choosing, for a given value of permit­
t i v i t y , a combination of sample length and position of the short c i r ­
cu it which produces a resonant condition characterized by large ab­
sorption in the sample [144,145]. Principles of the method and the
expected accuracy of the measured e are discussed in Section 4.2.
When the r e fle c tio n c o e ffic ie n t i s measured for several positions
of the short c ir c u it , the complex perm ittivity can be determined with
better precision using averaging procedures [141]. A new method for
performing such averaging has been developed here and i s described in
d e ta il in Section 4 .3 . It i s based on a least-squares estimation of
parameters in nonlinear im p licit model [110] (see Appendix D) and
application of s t a t i s t i c a l weights which r e f le c t resolution of d irectly
observed quantities by the measuring equipment. Together with precision
calibration which takes account of attenuation lo sses and d iscon tin u i­
t i e s in the waveguide between the sample and detector, the new method
i s w ell-su ited for measurement of medium lossy m aterials, i . e . where
0.001<tans<10. Its application for measurements of fa st ionic conduc­
tors was hampered by large systematic errors arisin g from lo ss of
e le c t r ic a l contact between the specimen and the waveguide [146] during
thermal cycling (see Section 4 .4 ). This problem prompted development of
a new technique which u t i l i z e s a rectangular waveguide p a r t ia l l y - f i l l e d
by a centrally-placed, rectangular parallelepiped slab. Contact between
the slab and the broader walls of waveguide was maintained by an exter­
81
nal spring loaded clamp.
Several accounts have been published [63,147-151] which report on
the use of a cen trally-located E-plane slab in a rectangular waveguide
for measurement of the complex perm ittivity. Data reduction in these
stu d ies was based on simple approximations which may be valid only for
very thin slabs or for small p erm ittivity. The propagation factor for
the fundamental mode in the p a r t i a l l y - f i l l e d waveguide was deduced from
measurement of scatterin g of the TE^0 mode of an empty waveguide,
assuming that the same f ie ld pattern e x is t s in the homogenous and
partially-loaded guides. Influence of the higher order, evanescent
modes, excited at the in terfa ce, on the scatterin g of the TEl 0 mode was
neglected and the p ossib le existence of more than one propagating mode
in a slab-loaded guide was excluded from their an alysis.
Our method of data reduction, described in Section 4 .5 , employs
accurate calculation of the scatterin g matrix for the TE1(J mode, based
on a complete solu tion of the boundary value problem in terms of modal
expansion, similar to those described by Chang [152] and Liu et al
[153]. Nonlinear least-squares estimation i s used for evaluation of the
complex perm ittivity from variable termination measurements. This
rather complicated scheme of data reduction has been implemented as a
computer program su ita b le for in teractive use on a minicomputer
(PDP11/34) and was routinely used for determination of the complex
p erm ittivity of fa st ion ic conductors.
82
4.2. The Standard Variable Termination Method.
Consider the experimental setup illu s t r a te d schematically in
F ig .4.1. A sample of length L completely f i l l s the cross section of a
single-mode waveguide. A s lo tte d section of the waveguide with a trav­
elin g probe i s connected between the signal generator and the sample
holder. Other instruments could be used for determination of the abso­
lu te value and phase of the voltage r e fle c tio n c o e f fic ie n t Rft at plane
A, but discussion here i s limited to s lo tte d -s e c tio n techniques
[154,155] which have been used throughout t h is work. A variable short
c ir c u it traveling in the waveguide on the other side of the sample i s
used to control the impedance ZB at plane B.
The complex propagation factor
y
of waveguide homogenously-filled
with material i s related to the complex p erm ittivity e
y2 = (2n/Xc ) 2 - k2e
2
2
where k^ = « u e
0
0 0
(4.3)
i s the free-space propagation constant, X i s the
c
c u t-o ff wavelength (x c =« for the TEM mode in the coaxial waveguide;
Xc=2a for the fundamental TE^Q mode in rectangular waveguide; a i s the
broad dimension of the guide).
The voltage r e fle c tio n c o e f fic ie n t
at the front surface of the
sample i s related to the r e la tiv e impedance Zg imposed by the variable
short c ir c u it
R =
*
ZR(1+jgtanhYL)-l+j-tanhrL
------ 5----------------- 2---------- = f(v,L ,Z B)
ZB(l-j|tanhY L)+l+j“tanhYL
(4.4)
where 8 i s the phase constant for the fundamental mode in empty wave­
guide
83
d e te c to r
p la te
sh o rt
c irc u it
g e n e ra to r
s l o t t e d s e c t i o n 2 waveguide
L o c a tin g r e f e r e n c e p o s i t i o n s 0 o f s t a n d i n g wave minimum
g e n e ra to r
v a ria b le
sh ort c irc u it
L o c a tin g r e f e r e n c e p o s i t i o n d0 o f v a r i a b l e s h o r t c i r c u i t
s l o t t e d s e c t i o n 2 waveguide
d e te c to r
,
g en e ra to r
6
i
1
i
A B
,
£
.
I
-*iLd
d
i
i
N
'
S
I
W ;
V
i
^
----------- --- *---------i-- ■
s L o tte d s e c t i o n '
waveguide
w ith sam ple
measurement w ith sam ple in waveguide
'
v a ria b le
short c ir c u it
F ig .4.1. I llu s tr a t io n of the variable termination method for measurement
of the complex p erm ittiv ity .
84
B2 = k2 - (2h/Xc ) 2
(4.5)
Equation (4 .4) i s derived using formulas [156] for transformation of
the impedance by length L of transmission lin e with propagation factor
Y.
The voltage r e fle c tio n c o e f fic ie n t
at plane A i s determined
experimentally by measurement of the voltage-standing-wave-ratio (VSMR)
Vs =Vmax
../Vmin
. =10W/2° and the position of voltage minimum s
r
1 V -1
= ------------- exp[2jB(s-s -L)]
Rn
o Vs +1
(4.6)
where s Q i s the reference position for voltage minimum which i s estab­
lish ed , during calib ration without a sample, by placing a metal plate
short c ir c u it at plane B. Here RQ i s the absolute value of the r e f le c ­
tion c o e f fic ie n t measured in th is configuration, and i s included in
Eq.(4 .6 ) to correct for attenuation in the waveguide between the sample
and probe (for the case, when a waveguide d iscon tin u ity i s present, see
Section 4 .3 .3 ) .
The impedance at plane B, normalized by the ch a ra cteristic imped­
ance of the waveguide, is
Z_cos8 (d-d/. )+jsinB(d-d >
Zn = — ------------ - ------------------------------------------------------------------- (4.7)
cose(d-d O)+jZ_sin
8 (d-d.)
S
0
where dQ i s the reference position of the variable short c ir c u it which,
without a sample, produces a voltage'minimum at s , Z =(1-R_)/(1+R_) i s
O S
s
s
the e f f e c t iv e normalized resistance of the short c ir c u it . The parameter
Rs i s the ratio of the absolute value of r e fle c tio n c o e f fic ie n t mea­
sured with variable short c ir c u it at dA
o to Ro and i s intended to
B5
account f o r l o s s e s i n t h e v a r i a b l e s h o r t c i r c u i t and a t t e n u a t i o n in the
waveguide between t h e p h y s i c a l s h o r t c i r c u i t and plane B.
A f t e r th e v o l t a g e s t a n d i n g wave r a t i o VS and p o s i t i o n of v o l t a g e
minimum s have been measured f o r a c e r t a i n p o s i t i o n d of s h o r t c i r c u i t ,
Ra and ZB a r e c a l c u l a t e d acc ord in g t o E q s . ( 4 . S ) and ( 4 . 7 ) , r e s p e c t i v e ­
l y , and E g . ( 4 . 4 ) i s s o lv ed nu m e ri c al ly fo r
y
.
The Newton's method
[157], modified f o r complex numbers, g i v e s r a p id convergence t o th e
r o o t n e a r e s t t o th e i n i t i a l gu ess . S o l u t i o n of th e t r a n s c e n d e n t a l
e qu at io n ( 4 . 4 ) i s , however, not unique. Ambiquity can be e l i m i n a t e d by
r e p e a t i n g measurements with two or more p o s i t i o n s of th e s h o r t c i r c u i t
and choosing t h e r o o t y which i s approximately unchanged f o r d i f f e r e n t
v a l u e s of d. Complex p e r m i t t i v i t y i s then ob ta in e d from E q . ( 4 . 3 ) .
E r r o r s of measured Zg and RA can s t r o n g l y a f f e c t t h e va lu e of
p ro pa ga tio n f a c t o r
s u s c e p t i b i l i t y of
y
y
c a l c u l a t e d from t h e n o n l i n e a r eq ua ti on ( 4 . 4 ) . The
t o d e v i a t i o n s of th e d i r e c t l y - m e a s u r e d pa rameters
depends on th e combination of t h e sample le n g t h and p o s i t i o n o f s h o r t
c i r c u i t . U n c e r t a in ty in t h e va lu e of e can be e s ti m a te d by i t s v a r i a ­
t i o n caused by v a ry in g t h e d i r e c t l y measured. ob s e r v a b le s by increments
equal t o th e l i m i t s of t h e i r exp er iment al u n c e r t a i n t i e s ( s t a n d i n g wave
ratio
aw
(dB), p o s i t i o n of v o l t a g e minimum
tive reflection coefficient
aR
s
as,
position
Ad
and e f f e c ­
of s h o r t c i r c u i t , sample le n g t h
aL ).
C a l c u l a t i o n can be done e x a c t l y by s o l v i n g E q . ( 4 . 4 ) with d e v i a t e d
v a l u e s of R^, Zg and L, or in l i n e a r approximation us in g d e r i v a t i v e s ;
f o r example
86
de 3y
a e
|
as
(4.8)
AS
dv as
where from E q . ( 4 . 3 )
(4.9)
P a r t i a l d e r i v a t i v e s of th e r i g h t hand s i d e f(Y,L,Zg) of E q . ( 4 . 4 )
t o g e t h e r with p a r t i a l d e r i v a t i v e s of
and ZQ, as give n by E q s . ( 4 . 6 )
and ( 4 . 7 ) , a r e combined in th e methods of i m p l i c i t d i f f e r e n t i a t i o n to
o b t a i n p a r t i a l d e r i v a t i v e s of
y
with r e s p e c t t o d i r e c t l y measured p a ra ­
m e te rs ; fo r example
( 4 .1 0)
( 4 .1 1)
( 4 .1 2 )
Somewhat le ngt hy e x p l i c i t formulas a r e given in Appendix E. To tal mar­
g i n s of e r r o r f o r e ' and o ^ e ^ e 1’ a r e e s ti m a te d by sums of th e a b s o lu t e
v a l u e s of th e f i v e f i n i t e d i f f e r e n c e s , which c orre spo nds to th e worst
combination of e r r o r s .
In ord e r to e l u c i d a t e the e f f e c t s of t h e sample le n g t h and p o s i ­
t i o n of th e s h o r t c i r c u i t on th e ex perimental p r e c i s i o n , model c a l c u l a ­
t i o n s have been made. Some r e p r e s e n t a t i v e r e s u l t s fo r X-band waveguide
(broad dimension a=22.86mm) a t 10 GHz a r e p l o t t e d in F i g s . 4.2 to 4.12
a s fu n c t i o n of t h e reduced d i s t a n c e to th e s h o r t c i r c u i t , (d- d0 ) / x g .
The wavelength in t h e empty waveguide i s Xg=2n/8=39.71mm. Two complex
87
p e r m i t t i v i t i e s a r e examined:
6
^=1 6 - 0 8 [ i . e . o1=0.0417(flcm)“ 1 )] and
e2=20-j40 [ i . e . o2=0.222(acm)- 1 )] which f a l l in th e range of v a l u e s
t y p i c a l f o r a f a s t i o n i c conductor.
The f a c t o r c o r r e c t i n g f o r a t t e n u a t i o n was taken t o be Ro=0.96, and
th e e f f e c t i v e r e f l e c t i o n c o e f f i c i e n t of s h o r t c i r c u i t was ta ken t o be
R =0.97, v a lu e s t y p i c a l f o r th e s t a i n l e s s s t e e l waveguides used in
s
experiment.
The fo ll ow in g r e s o l u t i o n e r r o r s were assumed, based on t h e c h a r a c ­
t e r i s t i c s of t h e a v a i l a b l e measuring equipment and t y p i c a l r e p e a t a b i l i ­
t y of d a t a points.( i ) s t a n d i n g wave r a t i o aW=0.2dB+0.03W, where th e second term r e ­
f l e c t s lower accuracy fo r high VSWR;
( i i ) p o s i t i o n of v o l t a g e minimum
a s = a s ' + a s ,,[ ( | R | + 0 . 0 0 5 ) ‘ 1- l ] X g
( 4.1 3)
with a s ’= 0. 05mm, as"=0.0005. The term dependent on t h e a b s o l u t e
valu e o f measured r e f l e c t i o n c o e f f i c i e n t |R| r e f l e c t s l o s s of
r e s o l u t i o n f o r small VSWR, e . g . , t h e r e s u l t i n g r e s o l u t i o n of
phase of R^ ( 4 . 6 ) i s 0.95deg f o r |R |= 0 . 9 , 1.26deg f o r |R |= 0 . 5 ,
3.97deg f o r |R |= 0 .1 ;
( i i i ) p o s i t i o n o f s h o r t c i r c u i t ad=0.05mm;
( i v ) r e f l e c t i o n c o e f f i c i e n t o f s h o r t c i r c u i t AR =0.01;
s
(v) le n g t h of sample aL=0.01mm.
Changes of e ’ and o=eoe"<i> caused by e r r o r s ( i ) to (v) were c a l c u l a t e d
by s o l v i n g E q . ( 4 . 4 ) with v a r i e d p ara me te rs. R e s u l t s o b ta in e d using
d e r i v a t i v e s agreed w i t h i n ±15%. For comparison, e f f e c t s of e r r o r ( i i )
88
c a l c u l a t e d usi n g E q s . ( 4 . 8 - 4 . 1 0 ) a r e p l o t t e d in F i g . 4 . 9 , which i s t h e
l i n e a r approximation e q u i v a l e n t of F i g . 4 .4 . I t should be noted t h a t th e
s har p d i p s observed in some of th e p l o t s of t h e a b s o l u t e va lu e of
r e l a t i v e e r r o r s correspond to a change of t h e s ig n o f t h e e r r o r (some
of t h e d i p s do not rea ch zero because of th e l i m i t e d r e s o l u t i o n of th e
c a l c u l a t e d c u rv e s , e . g . , in F i g . 4 . 4 ) .
For th e f i r s t p e r m i t t i v i t y , e=16-j8, t h r e e sample l e n g t h s were
i n v e s t i g a t e d Lj=0.92mm, L2=1.84mm, 1.2=3.68mm, which a r e , r e s p e c t i v e l y ,
equal t o 0.125, 0 . 2 5 , 0 .5 of th e wavelength in a sample-loaded guide
Xs =2n/ImY=7.37mm, and a r e a l l s h o r t e r than t h e a t t e n u a t i o n le n g th
1 =l/Re-r=4.85mm. The r e f l e c t i o n c o e f f i c i e n t f o r th e two s h o r t e r samples
ot
goes through a pronounced resonance - low VSWR and r a p i d v a r i a t i o n of
phase with d i s t a n c e t o s h o r t c i r c u i t ( s e e F i g . 4 . 2 ) . For th e l o s s y mate­
r i a l c on si de re d here ( ta n S = 0 .5 ) , th e phase does not go through zero and
v a r i e s only over a l i m i t e d i n t e r v a l around 180deg. I t i s g e n e r a l l y
acc e pt ed [141,142,144,145] t h a t d e te r m in a ti o n of e can be done with
b e s t p r e c i s i o n a t resonance. The range of d g iv in g th e h i g h e s t VSWR
must be avoided, because i t c o i n c i d e s with maximum d e v i a t i o n s of e
caused by each of t h e aforementioned e r r o r s ( i ) t o (v) (see F i g s . 4.3 to
4 . 7 ) . Minima of s u s c e p t i b i l i t y of e ' and o t o e r r o r a r e broad and do
not correspond e x a c t l y t o th e minima of VSWR. P l o t s of the cumulative
e r r o r s (see F i g . 4. 8 ) show t h a t , fo r s h o r t e r samples, measurements made
in th e range of s h o r t c i r c u i t p o s i t i o n s on t h i s s i d e of minimum which
i s c l o s e r to th e maximum of VSWR a r e a l s o s u s c e p t i b l e to e r r o r s . For a
s i n g l e - p o i n t measurement of e, d should be chosen in th e range c o r r e -
— lj =0. 92nmiE:Ks /B
28-
—12 =1. 84mmi=Ks / 4
a^O. 0445
24m
m
"O
UJ
— 13 =3. 68mm”\ s / 2
20-
:*
in
>
12-
8_
220
n
10 GHz
T
T
T
T
200-
07
Q)
TJ
i_i
~ 180
<
a:
07
L
O
160
0.2
-0. 1
0.2
0 .3
F i g . 4 . 2 . Voltage s t a n d i n g wave r a t i o and phase of th e v o l t a g e r e f l e c t i o n
c o e f f i c i e n t a s f u n c t i o n of reduced d i s t a n c e t o s h o r t c i r c u i t ,
c a l c u l a t e d a t 10 GHz in X-band r e c t a n g u l a r waveguide fo r t h r e e
samples of complex p e r m i t t i v i t y e =1 6-j 8; R =0.96, R =0.97.
0
s
1.0
1j *0. 92mm*s\s /8
|A o |/a
0. 8
a=0.0445 fl^cm
12 *1. 84mm*Ks /4
10 GHz
13 “3. 68mm“Ks /2
0. 6
0.4
0.2
0.0
1.0
IAe‘ l / e '
0. 8
0. 6
0. 4
0. 2
3
0.2
0. 1
0.0
0. 1
0.2
0.
Cd-d0 ) / k .
Fig.4.3. Relative errors of the conductivity and of the real part of the
permittivity caused by error of VSWR, aW=0.2dB+0.03W.
C o nd iti on s same as in F i g . 4 .2 .
- 2 j =0. 92tnm“^s / 8
| Act| / ct
0. 8
• 12 “ 1. 84mm=\s / 4
10 GHz
• 13 =3. 68tnm“ \ s / 2
0.6
0. 2
-
0.0
1.0
I Ae 1 I/e *
a=0. 0445 fl^cni
T
T
T
T
T
T
T
0. 6
0. 4 -
0. 2
0. 0 * - 0 .3
-
0.2
0.2
0. 3
F i g . 4 . 4 . R e l a t i v e e r r o r s of t h e c o n d u c t i v i t y and of th e r e a l p a r t of th e
p e r m i t t i v i t y caused by e r r o r of p o s i t i o n of v o l t a g e minimum,
as=0.05mm+0.0005[(|R|+0. 005) “ ^-l ]X g. Co nd iti ons same as in
F i g . 4 .2 .
-1, “0. 92mm*s\s /8
|Act |/a
0. 8
a=0. 0445 O^cm-1
-12 “1. B4mm"Ks /4
10 GHz
•13 "3. 68mm®A.s /2
0.6
0. 4
0.2
IAe * 1/6 ‘
0.0
.20
T
T
T
T
T
T
.0 8
.0 4
'jum
-
0.2
-0. 1
0. 0
(d-d0 ) / \
0. 2
0. 3
Fig.4.5. Relative errors of the conductivity and of the real part of the
permittivity caused by error of reflection coefficient of short
circuit
a R s =o
.01. Conditions same as in Fig.4.2.
0.5
| A ct I / a
0. 4
- 1 1 *0. 92mm“Ks / 8
■18 “ 1. 84mm*\s /4
a=0. 0445
10 GHz
• 13 “ 2 . 6 8 m m " \ s / 2
0 .3
0. 0
1.0
T
T
T
T
T
T
I Ae ' l / e '
0. 8
0.6
0.2
0. 0L- 0 .3
-
0.2
- 0. 1
0. 2
0. 3
Fig.4.6. Relative errors of the conductivity and of the real part of the
permittivity caused by error of position of short circuit,
ad=0.05mm. Conditions same as in Fig.4.2. ■
-1 , “0 .92mmss\s /8
• I z “ 1- 84mme!X.s /4
10 GHz
• 13 "3 . 68mm=K.s / 2
lA ffl/a
cx=0. 0445 n"1cm*1
T
T
T
T
\h e ' |/ e '
T
001-
-0 .3
0.2
0 .3
Fig. 4.7. Relative errors of the conductivity and of the real part of the
permittivity caused by error of sample length Al=0.01mm.
Conditions same as in Fig.4.2.
2.0
a °0 . 0445 0-1cm
\ha\/a
10 GHz
0.8
T
T
T
lA e' 1 / 6 ’
0.0
2.0
0. 0
0. 2
Fig.4.8. Total relative errors of the conductivity and of the real part
of permittivity for the worst combination of measurements'
errors. Conditions same as in Fig.4.2.
M ai/a
- 1 1 ®Q. 92mm=\s /B
ff=Q. 0445
• 12 *1. 84mm*\s / 4
10 GHz
■13 *3. 68mm*!,\ s / 2
0.6
0. 4
0.2
1.0
T
T
T
T
I Ae ' l / e '
0. 8 -
0.6
0.2
~ _
-
t
:
0.2
0.2
0 .3
Fig.4.9. Relative errors of the conductivity and of the real part of the
permittivity caused by error of position of voltage minimum linear approximation equivalent of Fig.4.4.
10 GHz
28-
VSWR
[dB]
2420166 1=20
12-
£"=40
• 12 *®1. 84mm
oa 0. 223
Ke “ 5. 29mm
220
- 1 , “ 0. 92mm
T
T
• 13 =0. 66mm
l a ° 1.35mm
T
T
T
a rg (R A>
Cdeg]
200-
180
160
-
0.2
0.2
0 .3
F ig .4.10. Voltage standing wave ra tio and phase of the voltage r e f l e c ­
tion c o e f f ic ie n t as function of reduced distance to short
c i r c u it , calculated for three samples of complex p erm ittiv ity
e=20-j40. Other cod ition s as in F ig .4.2.
2.0
6 '= 2 0
6"=40
<7=0. 223 fi-1cm
|A o l / f f
K.s =5. 29mm
1. 2
-
1
• 13 =0. 66mm
10 GHz
0.40.0
2.0
T
T
T
IT
T
T
T
|A e ' | / e '
1. 6 -
0. 8 -
0.4
0 . 0L -
- 0 .3
-
0.2
-
0. 1
0.2
0 .3
Fig.4.11. Total relative errors of the conductivity and of the real part
of permittivity for the worst combination of measurements'
errors. Conditions same as in Fig.4.10.
99
sponding to the low values of VSWR on th is sid e of minimum which is
further away from the maximum of VSMR.
Considering the choice of sample length, we note that the shortest
sample, Lj=xs / 8 , behaves w ell, i f the region of high VSWR, where the
errors skyrocket, i s avoided (see F ig .4 . 8 ). In case of the I*2 =xs / 4
sample, when the measurements are made with the short c ir c u it posi­
tioned far outside the range of the maximum s e n s i t iv it y (d*0 ), the
errors are s li g h t l y larger than for the other two samples. Despite the
fa c t, that for the longest sample, I*3 =Xs / 2 , the front-surface r e f le c ­
tion Ra varies l i t t l e with short c ir c u it movement, the best accuracy i s
expected because the resu lts are le a s t susceptible to errors of sample
length and reference positions dQ, s Q.
For the perm ittivity e2=20-j40, the three lengths of samples are
Lj=0.92mm, 1*2=1.84mm, 1. 3 =0 . 66 mm. The f i r s t two lengths are chosen to be
the same as for e^=16-j8 in order to t e s t the f e a s i b i l i t y of doing
measurements on the same sample over a range of temperatures. The wave­
length in a sample-loaded guide i s now Xs =5.29mm, the attenuation
length 1tt =1.35mm. The third length i s LatXs_ / 8 instead of the longest
sample which would be much longer than l a . The VSWR and phase of RA are
plotted in F ig .4.10. The cumulative errors calculated for measurements
errors ( i ) to (v) are plotted in F ig .4.11. The second sample i s now
longer than l ft and RA i s not se n sitiv e to p osition of the short c i r ­
c u it. Errors can, however, be reduced using short c ir c u it positions
resulting in a shallow minimum of the VSWR. The shortest sample can
give the best accuracy i f i t i s measured away from maximum of the VSWR.
100
The sample length Lj=o.92mm s t i l l works w ell, but the uncertainties are
higher than in the case of lower conductivity. Especially large errors
(about 30%) can be expected for the real part of the p erm ittivity.
4.3. Application of Nonlinear Least-Sguares F ittin g for Evaluation of
the Variable Termination Measurements.
4 .3 .1 . Estimation of Scattering Parameters for
a Reciprocal Discon­
tin u ity in a Single-Mode Waveguide.
A reciprocal discon tin u ity structure in a single-mode waveguide
can be treated as a two-port network and described in d iffe r e n t repre­
sentations - impedance, admittance, sca tterin g , tran sfer, special net­
works [158] - by a s e t of three complex, or s ix rea l, parameters. The
scatterin g matrix i s used here because i t i s d ir e c tly related to the
r e fle c tio n c o e f f ic ie n t s . The voltage scatterin g parameters which relate
the incident and reflecte d parts of the t o ta l voltages at reference
planes A and B chosen, outside the d isco n tin u ity, in a uniform wave­
guide (see F ig .4.12) are;
bl = Sl l al + S12a 2
(4>14)
= S21 al + S2 2 a 2
The scatterin g matrix S i s symmetric i f the ch a ra cteristic im­
b2
pedances of waveguides connected to ports A and B are equal. If those
impedances are not equal, input and output voltages can be normalized
to make S symmetric [158] and we take S12 =S21. Relation between the
input r e fle c tio n c o e f fic ie n t R^b^/a^ and the r e fle c tio n c o e f fic ie n t of
the load R0 =a2 /b 2 can be written in the form
101
QI
It
a*!
< —f
■4—>
•b 2
bj • S n Qj + S J2 a*
ba “ S2J Qj + Saa Og
F ig .4.12. S ca tterin g matrix representation for a d isc o n tin u ity in s in g le
mode waveguide.
In p r i n c i p l e , measurements of
w ith th r e e d i f f e r e n t lo ad s R0 a re
2
s u f f i c i e n t to d eterm ine th e th r e e complex p aram eters S ^ , S12 , $ 2 2 in
th e in p u t- o u tp u t r e l a t i o n , E q .( 4 .1 5 ). Accuracy of such a procedure
depends on th e ch o ice of th e th r e e lo a d s and r e s u l t s a r e s tr o n g ly
a f f e c t e d by th e e r r o r s of each measurement. I t has been long understood
t h a t more r e l i a b l e r e s n ' . d can be o b ta in e d by a v eragin g th e r e s u l t s of
a la rg e r
num ber
measurements. Numerous g r a p h ic a l and sem ig rap h ica l
schemes f o r perform ing such av erag in g can be found in th e l i t e r a t u r e
[158]. Some o f th e s e p ro c e d u re s, which r e l y on th e v a r i a b l e te rm in a tio n
measurements, a r e : th e re a c ta n c e tra n s fo rm a tio n diagram and th e ta n g e n t
r e l a t i o n network fo r l o s s l e s s s t r u c t u r e s ; th e c a n o n ic a l network, th e
H e is s flo c h network and th e Deschamps method fo r d i s s i p a t i v e s t r u c t u r e s .
Each of th e s e e l a b o r a t e proced ures employs a s p e c i f i c r e p r e s e n t a t i o n
fo r th e tw o -p o rt netw ork, which le a d s to c o n s tr u c ti o n of a c h a r a c t e r i s ­
t i c curve from th e d a t a p o i n t s , e . g . a c i r c l e in th e Smith c h a r t .
G raphical a n a l y s i s can be re p la c e d by num erical f i t t i n g o f c i r c l e s
[1 59 ], but such an approach r e t a i n s th e a r t i f i c i a l l i m i t a t i o n s of
s p e c i f i c g e o m e tric a l c o n s tr u c ti o n .
K ajfez [160] developed a s t r a ig h tf o r w a r d and g e n e ra l method fo r
th e d e te rm in a tio n of param eters of l i n e a r tw o -p o rts by a l e a s t - s g u a r e s
e s tim a tio n of param eters of th e b i l i n e a r tra n s fo rm a tio n . His method i s
b r i e f l y p re s e n te d h e re in o rd e r t o p in p o in t s i m i l a r i t i e s and d i f f e r ­
ences between h i s and our approach. The r e l a t i o n between th e in p u t w
103
and th e o u tp u t z p aram eters fo r a l i n e a r tw o -p o rt can be w r i t t e n in
form o f a b i l i n e a r tr a n s fo rm a tio n
c , z + c~
w = — -------- CgZ + 1
(4 .1 6 )
Equation (4 .1 5 ) i s of t h i s form when we d e f in e **=RA. z=Rg,
K ajfez e s tim a te s th e param eters c ^ ( c 2> c^ by a l i n e a r l e a s t s g u a re s method form ulated in complex a r i t h m e t i c . The minimized o b je c ­
t i v e fu n c tio n Qg i s a weighted sum of squared a b s o lu t e v a lu e s of e r r o r s
o f th e i m p l i c i t l i n e a r r e l a t i o n
(4 .1 7 )
(4 .1 8 )
Here w^, z k denote th e ex p erim en tal v a lu e s of th e in p u t and o u tp u t
v a r i a b l e s in k - th measurement, m measurements a r e c o n sid e re d . K ajfez
a rg ues t h a t m in im ization of th e e r r o r s e^ of th e c o n s t r a i n t fu n c tio n
E q .(4 .1 7 ) i s e q u iv a le n t to m inim ization o f th e d e v ia t io n s o f v a r i a b l e s
w^, z^ s c a le d by th e in v e rs e squared d i s t a n c e s between th e v a r i a b l e and
th e pole o f th e b i l i n e a r tra n s fo rm a tio n r e l a t i n g one v a r i a b l e w ith th e
o th e r . That i s E q .(4 .1 8 ) can be ap pro xim ately r e w r i t t e n as
(4 .1 9 )
where w£, z£ denote th e " t r u e ” v a lu e s of v a r i a b l e s . Such s c a l i n g of
d e v ia t io n s i s d e s i r a b l e , because i t does n o t a llo w th e la r g e v a lu e s of
v a r i a b l e s near th e po le of tra n s fo rm a tio n to dominate th e o b je c t iv e
4
104
function Eq.(4.18).
The w eights pk a r e o b ta in e d by a p e c u l i a r p ro c e d u re, which s t a r t s
w ith th e requirem ent t h a t th e w eights should be i n v e r s e ly p r o p o r tio n a l
to th e co rrespo n ding v a r ia n c e s [1 1 3 ], b u t, in e f f e c t , th e v a r ia n c e s of
e k a re e stim a te d by unknown v a r ia n c e s of param eters c ^ , c 2 , c^. K ajfez
claim s t h a t th e r e s u l t s o f th e l e a s t - s q u a r e s e s tim a tio n a r e p r a c t i c a l l y
2
2
2
2
independent o f th e r a t i o s o (c 2 ) / o ( c ^ ) , o ( c ^ ) / a ( c
found in h i s
formula f o r pk> Taking th o se v a lu e s t o be u n i t y , r e s u l t s in th e w eights
Pk = [ l + l z k l 2 ( l + | w k | 2 ) ] ~ 1
(4.20)
The r e l a t i o n of w eights given by E q .(4 .2 0 ) to v a r ia n c e s of o b serv ab le s
z k ’ wk *s r a ^*ier ob scure.
Summarizing, in th e method developed by K ajfez [1 6 0 ], th e o b je c ­
t i v e f u n c tio n , E q .( 4 .1 8 ) , i s n o t d i r e c t l y e q u iv a le n t to th e o b je c t iv e
fu n c tio n re q u ire d by s t a t i s t i c a l c o n s id e r a tio n s [113] and th e r e l a t i o n
between th e two in v o lv e s a r b i t r a r y assum ptions. The l e a s t - s q u a r e s
method designed fo r models in which o b s e rv a b le s a r e e x p l i c i t l y ex­
p re sse d by p aram eters i s a p p lie d to system d e s c rib e d by th e i m p l i c i t
r e l a t i o n s E q .( 4 .1 7 ) , which le a d s t o in c o n s is te n c y . I t i s , however,
q u ite l i k e l y t h a t E q .(4 .2 0 ) p ro v id es s a t i s f a c t o r y s c a l i n g fo r d e v ia ­
t i o n s in E q .(4 .1 8 ) and th e whole procedure le a d s to meaningful e s t i ­
mates o f p aram eters.
Our approach h e re i s based on th e method of a l e a s t - s q u a r e s e s t i ­
mation of param eters in th e n o n lin e a r i m p l i c i t models given by B r i t t
and Luecke [1 1 0], which i s reviewed in Appendix D. The o b je c t iv e
fu n c tio n i s form ulated in terms of d e v ia t io n s of th e measured v a r i a b l e s
105
from t h e i r t r u e v a lu e s , which a re weighted by th e in v e rs e of co v arian c e
m a trix . Both th e param eters and th e t r u e v a lu e s of measured v a r i a b l e s ,
which s t r i c t l y obey th e i m p l i c i t model e q u a tio n s , a r e e s tim a te d sim ul­
ta n e o u s ly .
Let us d e f in e th e v e c to r of p aram eters X a s fo llo w s
-
R e(Sj^),
X2 —Ini( Sj^ ),
x3 = Re(S122 ) ,
x4 = Im(S122 ) ,
Xj - Re(S22) ,
Xg ~ Im(S22) ,
(4 .2 1 )
and th e v e c to r o f measurements Y a s fo llo w s
yM4i-3 = Re(RA i).
yM4i-2 = Im(RAi)
yM 4i-l = Re(Rb i }'
yM4i
1 = 1 , 2 , . . .m
(4 .2 2 )
= Im(IW
where RAi i s th e in p u t r e f l e c t i o n c o e f f i c i e n t measured w ith load r e ­
f l e c t i o n c o e f f i c i e n t Rfi. , m measurements w ith d i f f e r e n t Rfij a r e c o n s i­
d ered . The in p u t- o u tp u t r e l a t i o n E q .(4 .1 5 ) i s r e w r i t t e n a s a s e t of
i m p l i c i t model e q u a tio n s
1= 1 , 2 . . .m
(4 .2 3 )
The model fu n c tio n s in E q .(4 .2 3 ) a r e complex. The l e a s t - s q u a r e s
e s tim a tio n fo rm u lated in r e a l numbers can be a p p lie d t o models w r i t t e n
in complex numbers by t r e a t i n g th e r e a l and imaginary p a r t s o f a com­
p lex v a r i a b l e a s two r e a l v a r i a b l e s . Such approach i s more g e n e ra l than
fo rm u latio n o f l e a s t sq u a re s in th e complex a r i t h m e t i c . D if f e r e n t
w eights can be a s s o c i a t e d with r e a l and imaginary p a r t s o f th e d e v ia ­
ti o n of a complex v a r i a b l e and t h e i r mutual c o r r e l a t i o n s can be taken
106
in to a c c o u n t, i . e . a complex d e v ia t io n e= e’+ je " can e n te r th e o b je c t iv e
2
2
fu n c tio n a s g ^ e ' N ^ e " • ^ g ^ e 'e ” ; whereas, in th e l e a s t s q u a re s formu­
l a t e d in complex numbers, a more r e s t r i c t e d form i s used
g e e * = g (e '^ + e " ^ ).
T
The c o v arian c e m a trix o f ex p erim en tal e r r o r s Cy=<(Y-YM )(Y-YM) >
[ f o r th e v e c to r o f measurements YM d e fin e d in E q .( 4 .2 2 )] which i s an
in h e re n t elem ent in th e e s tim a tio n (se e Appendix D), depends on th e
ex p erim ental procedure and i s g e n e r a lly n o t known. I t would be extreme­
ly d i f f i c u l t to g a th e r a la r g e enough p o p u la tio n o f d a ta p o in ts to make
a s t a t i s t i c a l e s tim a te of Cy. For a given te ch n iq u e of measurement, a
sim ple approxim ation of Cy can be c o n s tr u c te d , based on e x perim ental
r e s o l u t i o n e r r o r s of th e d i r e c t l y measured v a r i a b l e s . At t h i s p o i n t , we
s p e c if y th e ex p erim en tal te c h n iq u e . The in p u t r e f l e c t i o n c o e f f i c i e n t s
R^. a r e o b ta in e d by s l o t t e d - s e c t i o n measurements o f VSWR Vj and p o s i­
t i o n of th e v o lta g e minimum s^ .
V1
RAi = - —
e x p [ 2 j0 ( S i - s . ) ]
V.+l
1 0
(4 .2 4 )
where s Q i s th e p o s i t i o n of v o lta g e minimum when a s h o r t c i r c u i t i s
placed a t plane A. The r e f l e c t i o n c o e f f i c i e n t s Rgj of th e load a re
imposed by th e v a r i a b l e s h o r t c i r c u i t a t p o s i t i o n s d^
RBi = -Rs i e x p [-2 jf l(d .-d 0 )]
(4 .2 5 )
where dQ i s th e p o s i t i o n of v a r i a b l e s h o r t c i r c u i t which produces r e a l
r e f l e c t i o n c o e f f i c i e n t RB=-RS a t plane B. The measured o b s e rv a b le s a re
W.=201ogjQV . , S j , d j and Rgj . The e f f e c t i v e r e f l e c t i o n c o e f f i c i e n t of
the short circuit R g is actually measured only during calibration and a
107
constant value Rgi=Rg i s used in Eq.(4.25). The true value of the re­
fle c tio n c o e f fic ie n t Rgi may, however, vary with the p osition of short
c ir c u it because of mechanical imperfections. Adjustment of each of Rg .
may be desirable to account for those imperfections and hence they are
treated as separate observables.
One can assume that measurements with d ifferen t p o sitio n s, d ., of
the short c ir c u it are not correlated. Thus, the elements of the covari­
ance matrix which r ela te measurements with d iffe r e n t i [see Eq.(4.22)]
are zero. The q u an tities Wj, s^, d. and Rg are measured d ir e c t ly , each
in a d iffe r e n t way, and i t seems reasonable to assume that th eir errors
are not correlated. Hence, the elements of the covariance matrix which
r ela te input and load measurements are zero and the only nonzero e l e ­
ments outside the main diagonal are those which express correlations
between the real and the imaginary part of each r e fle c tio n c o e f fic ie n t
[CY 4 i-3 ,4 i-2 for RAi and CY 4 i- l,4 i for RBi' see E<l ( 4 -22)1- From the
ch a ra cteristics of the equipment used in the experiments and typ ical
r e p e a t a b i l i t y o f th e measured q u a n t i t i t e s , one can e s tim a te th e u n cer­
t a i n t i e s to be aW^, ASj, Ad. and ARg , which can be d i f f e r e n t fo r each
measurement. Squares of th e s e v a lu e s a r e used in p la c e of th e v a ria n c e s
of e r r o r s o f th e r e s p e c tiv e v a r i a b l e s . A l i n e a r approxim ation fo r th e
p ro p a g a tio n of random e r r o r s [113,114] i s used to e x p re ss th e elem ents
of th e c o v arian c e m a trix Cy. The only nonzero elem ents of m a trix Cy
are.-
( i ) for each input r e fle c tio n c o e f fic ie n t R^:
CY 4 i-3 ,4 i-3 = <ReRAi>2<4RAi>2 + d m R ^)2^
) 2,
(4.26a)
(ii)
fo r each r e f l e c t i o n c o e f f i c i e n t of th e load Rg.:
CY 4 i-l , 4 i - l = (ReRB i) 2( ^ s / RS)2 + <IraRBi> 2<AV 2 ’
(4.26d)
CY 4 i,4 i
= (ReRB i) 2t A,|‘i ) 2 + (ImRB i) 2(ARs /Rs ) 2 f
(4 .2 6 e )
CY 4 i - l , 4 i
= CY 4 i , 4 i - l = £ tARs / Rs ^ 2"*Al|>i ^ ReRBiI,BRB i;
( 4 .2 6 f )
where
aRa
- = ( 1 - | R ^ | ) 2AW .lnl0/(40|RA i I ) ,
A<D.=2BAsi>Ai»i =2BAd. ,
i = l , 2 , . . . ,m. The m a trix Cy c o n s i s t s of 2m symmetric (2*2) m a tric e s
lo c a te d along th e main d ia g o n a l.
For some o b s e rv a b le s th e v a r ia n c e s could be e stim a te d s t a t i s t i c a l ­
l y ; e . g . th e p o s itio n o f th e v o lta g e minimum, s^ could be measured 10
tim es ( s . j , j = l , 2 . . . 1 0 ) , by av erag in g equal v o lta g e p o s i t i o n s [1 54 ],
2
w ith th e same p o s i t i o n d j o f s h o r t c i r c u i t , s ^ = E s.^ /1 0 , and a - E ( s .
2
2
S j) /1 0 used in s te a d of ( a s j ) in E q .( 4 .2 6 ). E rror of th e p o s i t i o n of
v o lta g e minimum i s u nd erstood a s a d i f f e r e n c e between th e measured
v a lu e Sj and th e t r u e v a lu e which in E q .(4 .2 4 ) g iv e s u n d is to r te d value
o f phase of R ^ .. B esides th e e r r o r of l o c a t i n g th e minimum of v o lta g e
2
on th e d is ta n c e s c a le ( c h a r a c te r iz e d by th e above v a ria n c e o ) , i t
in c lu d e s c o n t r i b u t i o n s a s s o c ia te d w ith im p e rfe c tio n s o f th e s l o t t e d
s e c t i o n , th e frequency i n s t a b i l i t y and o th e r f a c t o r s . The sim ple e s t i 2
mate o does no t measure th e v a r ia n c e o f t o t a l e r r o r o f s .. Use of
r e s o l u t i o n e r r o r s does not have a s t r i c t s t a t i s t i c a l meaning, but i t
p e rm its one to d e c id e how much th e v a r i a b l e s a r e a d ju s te d d u rin g th e
l e a s t - s q u a r e s r e g r e s s i o n , th e a d ju stm en ts bein g p r o p o r tio n a l to th e
assumed e r r o r s . Once th e th e r e g r e s s io n i s com pleted, c o n s is te n c y of
109
th e assumed e r r o r s w ith th e r e s i d u a l d e v ia t io n s can be examined.
Real and imaginary p a r t s of E q .(4 .2 3 ) form a 2m element c o n s t r a i n t
v e c to r P. Each complex c o n s t r a i n t f j r e l a t e s o b s e rv a b le s in i - t h ex­
perim e n t, so th e (4m*2m) m a trix B o f p a r t i a l d e r i v a t i v e s w ith r e s p e c t
t o o b s e rv a b le s Y [se e Eq.(D7)] has only 4*2m nonzero elem en ts. The only
nonzero elem ents of th e (4m*4m) c o v arian c e m a trix Cy , E q .( 4 .2 6 ) , form a
band of (2*2) symmetric m a tric e s along th e d ia g o n a l, t o t a l of 6m
d i f f e r e n t nonzero e lem ents. S im ila r ly in th e (2m*2m) m a trix CB,
Eq.(D8), th e nonzero elem ents form a d ia g o n al band of symmetric
m a tric e s and in v e r s io n o f
can be accomplished by i n v e r t i n g each of
th e (2*2) m a tr ic e s s e p a r a t e l y . The m a trix CA, Eq.(D9), i s a symmetric
(6*6) m a tr ix , whose in v e rs e can be found by a number of num erical
methods [1 6 1 ], th e G auss-Jordan a lg o rith m w ith p iv o tin g i s used in our
program. The t o t a l s i z e of m a tric e s re q u ire d by our im plem entation of
th e a lg o rith m i s 39m+72 f l o a t i n g p o in t numbers, which can be handled by
a microcomputer w ith a re a so n a b le m .
The c o n s t r a i n t f u n c tio n s , E q .( 4 .2 3 ) , a re n o t s tr o n g ly n o n lin e a r in
s c a t t e r i n g p a ra m e te rs , th e only n o n lin e a r term bein g rb ^s x1S2 2‘ Conver"
gence of th e i t e r a t i v e l e a s t - s q u a r e s e s tim a tio n can be expected even
w ith a poor i n i t i a l guess of th e pa ra m ete rs. Taking S22=0, th e r e ­
maining terms in E q .(4 .2 3 ) r e p r e s e n t an e q u atio n of a s t r a i g h t l i n e . A
s ta n d a rd l e a s t - s q u a r e s f i t of a s t r a i g h t l i n e (w ith m o d if ic a tio n fo r
2
complex numbers) i s used to o b ta in e s tim a te s of S ^ ,
, which with
S22=0 or S22=S11 prove<* to be a s a t i s f a c t o r y i n i t i a l guess in a l l case s
2
s tu d ie d . Equation (4 .2 3 ) could be made l i n e a r by choosing s x2
11S22
110
2
a s a param eter in s te a d of S12 • b u t s in c e th e a lg o rith m fo r n o n lin e a r
2
problems was a v a i l a b l e , d i r e c t e s tim a tio n of
and i t s con fidence
l i m i t s was p r e f e r r e d .
Nhen th e d i s c o n t i n u i t y in th e waveguide i s symmetric then S11=s22
and only th e f i r s t fo u r param eters of E g .(4 .2 1 ) a r e r e q u ir e d . The model
f u n c tio n s E g .(4 .2 3 ) ta k e th e form
0 * £! ‘Sn ’S122’lW RBi> ■ Sl l +RBi(S122- Sn 2>+RMRBiSn - RAi
<4-27>
Comparison o f r e s u l t s of e s tim a tio n w ith 4 and 6 s c a t t e r i n g p a ra m ete rs,
in th e case where th e d i s c o n t i n u i t y i s known to be symmetric, s e r v e s as
a t e s t o f d a ta c o n s is te n c y . I f , a s a r e s u l t o f e s tim a tio n w ith 6
p a ra m e te rs ,
i s d i f f e r e n t from S 2 2 by more than th e e stim a te d l i m i t s
of c o n fid e n c e , then measurements of RAi- RBi c o n ta in s y s te m a tic e r r o r s
which make a symmetric d i s c o n t i n u i t y appear t o be n o t symmetric.
For th e sake of comparison our a lg o rith m was t e s t e d on th e example
used by K ajfez [1 6 0 ], th e o r i g i n a l d a ta being taken from r e f e r e n c e 153.
Values o f th e s c a t t e r i n g p aram eters o b ta in e d u sin g our method ag re ed ,
w ith in l e s s than one s ta n d a rd d e v i a t i o n , w ith v a lu e s o b ta in e d in r e f e r ­
ences 21 and 22. For example, assuming ex p erim en tal u n c e r t a i n t i e s to be
ARg=0.001, Ad=0.1mm ( a*=0 . 5 ° ) , AU=0.4dB, As=0.1mm ( ao=0 . 5 ° ) , th e c a lc u ­
l a t e d s c a t t e r i n g param eters w ith t h e i r s ta n d a rd d e v ia t io n s a r e :
Su
= 0 . 5119±0.0012
135.43+0.13°
S22 = 0 . 5717±0.0016
1 4 7 .85±0.20°
S12 = 0.6423+0.0014
29.99±0.07° (modulo 180°)
The roo t-m ean-sq uare r e s i d u a l , Eq.(D16), was 0 .3 2 5 , showing t h a t r e s i d ­
u a l d e v ia t io n s were s m a lle r than th e assumed e r r o r s . When p ro p o rtio n s
Ill
between aR , Ad, aW, as were va ried , estim a tes of the standard dev ia s
t i o n s changed s i g n i f i c a n t l y , bu t th e e s ti m a te s of th e param eters r e ­
mained th e same, a t l e a s t w ith in one s ta n d a rd d e v ia t io n .
4 . 3 . 2 . D ire c t E stim a tio n of th e Prop ag ation C onstant fo r HomogeneouslyF i l l e d Waveguide.
A s e c t i o n o f waveguide f i l l e d w ith th e m a te r ia l of i n t e r e s t can be
t r e a t e d a s a d i s c o n t i n u i t y s t r u c t u r e in tro d u ce d in to th e empty wave­
g u id e. When th e r e f e r e n c e p la n e s A and B a r e chosen a t th e s u r f a c e s of
th e sample (se e F i g . 4 .1 ) and th e m a te r ia l i s nonmagnetic, th e d i s c o n t i ­
n u ity i s r e c ip r o c a l and symmetric. The two s c a t t e r i n g param eters S11
and S^ 2 d e s c r ib in g such d i s c o n t i n u i t y a r e f u n c tio n s of th e complex
p ro p a g a tio n c o n s ta n t
y
[d e fin e d in E g .( 4 . 3 ) ] of th e f i l l e d waveguide
and o f th e le n g th of th e sample L
(§ + g ) s i n j r L
Su ( y .L) = --------- 2-* -------2cosjYL + ( — | ) s i n j r L
(4 .2 8 )
S12( y .L) = ------------------- 1-------------- —
2cosjYL + (2 - ?)sinjYL
(4 .2 9 )
Y
B
Y
Using E q s .(4 .2 8 ) and (4 .2 9 )
y
B
can be w r i t t e n in term s of S ^ , S12
2
[141]
(1 -S U ) 2 - S122
/y
\ 2_
d +sn )‘ - s12‘
yjey
ek02-(2 n/X c ) 2
(4 .3 0 )
The s c a t t e r i n g p aram eters
k02-(2Vxc )2
and S12
2
can be e s tim a te d from v a r i ­
a b le te rm in a tio n measurements, a s d e s c rib e d in th e p re v io u s s e c t i o n ,
and th e complex p e r m i t t i v i t y c a lc u la te d from E q .( 4 .3 0 ). U n fo rtu n a te ly
such procedure i s v ery u n r e l i a b l e fo r th e fo llo w in g re a so n s. The s c a t ­
112
t e r i n g param eters
and S12
a r e e s tim a te d a s fo ur independent r e a l
2
2
p aram eters R e S ^ , ImSj^, ReSj2 , ImSj2 , w ith o u t ta k in g in t o account
t h e i r s p e c i a l f u n c tio n a l form given be E q s .( 4 .2 8 - 4 .2 9 ) . An a d d i t i o n a l
degree of freedom i s th u s allow ed . The exp erim en tal e r r o r s , p a r t i c u l a r ­
ly in a cc u ra c y of c a l i b r a t i o n and im p e rfe c tio n s of waveguide, a f f e c t th e
e stim a te d s c a t t e r i n g p a ra m ete rs, so th e y do not e x a c tly obey E q s .( 4 .2 8 4 .2 9 ). Equation (4 .3 0 ) i s v a l i d only when
and S12 a r e s t r i c t l y in
th e form of E q s .(4 .2 8 ) and ( 4 . 2 9 ) , r e s p e c t i v e l y , and i s extrem ely
s e n s i t i v e to e r r o r s .
Much more r e l i a b l e e s ti m a te s of th e complex p ro p a g a tio n f a c t o r and
th e complex p e r m i t t i v i t y can be o b ta in e d when th e p a r t i c u l a r form of
th e in p u t- o u tp u t r e l a t i o n , E q .( 4 .4 ) , v a l i d fo r a h o m o g e n eo u sly -fille d
s e c t i o n of waveguide i s used d i r e c t l y a s th e i m p l i c i t model fu n c tio n
fo r th e n o n lin e a r l e a s t - s q u a r e s e s ti m a ti o n . The p r i n c i p l e of measure­
ment i s th e same a s d e s c rib e d in S e c .4 .2 . The v o lta g e r e f l e c t i o n c o e f­
f i c i e n t RAi [se e E q .( 4 .6 ) ] i s measured fo r s e v e r a l p o s i t i o n s d j of th e
v a r i a b l e s h o r t c i r c u i t . The v e c to r of th e measurements i s d e fin e d
somewhat d i f f e r e n t th an in E q .( 4 .2 2 ) , v iz :
yM4i-3 = Re(RA i>’
yM4i-2 = Im<RAi*
(4 .3 1 )
yM 4i-l = 8 (d i " do ) *
yM4i = RSi
The c o v arian c e m a trix of e r r o r s Cy i s c o n s tr u c te d in s i m i l a r way
a s in S e c .4 .3 . 1 . The elem ents a s s o c i a t e d w ith in p u t measurements a re
th e same as in E q s . ( 4 . 2 6 a - c ) . The te r m in a tio n i s now re p re s e n te d by
d i f f e r e n t v a r i a b l e s and th e co rresp o n d in g elem ents of Cy a re given by:
2
(4.32a)
Two param eters t o be e s tim a te d a r e :
xx = R e ( y ) ,
x2 = JroCr)
(4 .3 3 )
and th e model f u n c tio n s a r e given by E g .( 4 .4 ) which can be w r i t t e n a s
(ZB4- 1 ) y c o s jrL+(B+Zn . Y2/8 )s in j r L
0 = f i ( v *RA i ’2Bi> = — ------------------------ ^ “3---------------------RAi
(ZB.+l)YCosjYL+(B-ZB.YZ/8)sinjY L
(4 3 4 )
i = 1 , 2 . . .m
( l - Rs i )cosB(di -d ) + j( l+ R g .) s in B ( d i -d )
Zri (dj »Rc i ) _ ” — —— —— i— — —— — — —— — —
(l+Rs . ) c o s 8 ( d . - d 0 ) + j ( l - R s i )s in B ( d .- d 0 )
(4 .3 5 )
where a s b e fo re in E g .(4 .2 5 ) RSj =Rg comes from c a l i b r a t i o n measurement,
b u t th e t r u e v a lu e of Rgj may v a ry . Formulas fo r th e p a r t i a l d e r i v a ­
t i v e s of E g .(4 .3 4 ) giv en in Appendix E.
The model f u n c tio n o f E g .(4 .3 4 ) i s s t r o n g l y n o n lin e a r in
i n i t i a l guess which c l o s e l y approxim ates
y
y.
An
i s n e c e s sa ry to o b ta in con­
v ergence. In most c a s e s , an i n i t i a l guess w ith in ±30% of th e e s tim a te d
v a lu e of
y
proved to be s a t i s f a c t o r y .
Once Y=Xj+jx2 i s e s tim a te d , th e complex p e r m i t t i v i t y e i s c a lc u ­
l a t e d u s in g E g .( 4 .3 ) w ith th e r e a l and imaginary p a r t s given by:
e ' = ( (2 ti/A ) 2- x , 2+x»2 ] /k 2
,
■
e" = 2XjX2/ k Q
(4 .3 6 )
The s ta n d a rd d e v ia t io n s f o r e ' and e ” a r e o b ta in e d from th e e stim a te d
c o v arian c e m a trix of param eters
p ro p a g a tio n o f random e r r o r s [113]
[s e e Eg.(D15)] u sing form ulas fo r
114
(4.3 7b )
c o v ( e ’ ,e ” ) = 2 ^ CX22"CX11^x2 x 1+CX12^x 22"X12 ^ /,,Co2
(4 .3 7 c )
The l e a s t - s q u a r e s e s tim a tio n p ro v id e s th e most l i k e l y e s ti m a te s o£
p aram eters under th e assumption t h a t th e model f i t t e d t o th e d a ta i s
v a l i d . The model f u n c tio n s , given by E q .( 4 .3 4 ) , a r e v a l i d when: ( i ) th e
sample homogeneously f i l l s th e c r o s s - s e c t i o n o f waveguide; (2) i s in
c o n ta c t w ith waveguide w a lls ; and (3) has i t s f r o n t and th e back s u r ­
fa c e s p e rp e n d ic u la r to th e a x i s o f waveguide. I t i s a l s o assumed t h a t
th e p a ra m ete rs, which a r e no t e s tim a te d , a r e known e x a c tly .
The le n g th L o f a w ell-machined s o l i d sample can be measured w ith
good p r e c i s i o n . The phase c o n s ta n t 8 i s determ ined by measurement of
th e wavelength w ith s a t i s f a c t o r y a cc u ra cy , i f th e d is ta n c e s d . - d Q, S js Q (which a re m u l t i p l i e d by 8) a r e s h o r t e r th an th e wavelength. The
re fe re n c e p o s i t i o n s of th e v o lta g e minimum s Q and th e s h o r t c i r c u i t dQ
may be s u f f i c i e n t l y in a c c u r a te to cause la r g e s y s te m a tic d e v ia t io n s of
d a ta from p r e d i c t i o n s of model given by E q .( 4 .3 4 ) . P o s itio n s of th e
r e f e r e n c e p la n e s a r e s h i f t e d when th e back s u r f a c e of th e sample i s not
a lig n e d w ith th e c a l i b r a t i o n plane B (se e F i g . 4 .1 ) . When measurements
a r e made over a wide tem p e ratu re range u s in g an e xp erim en tal s e tu p in
which s e c t i o n s of th e waveguide between th e sample and th e s l o t t e d l i n e
and between th e sample and th e s h o r t c i r c u i t a r e long compared w ith
w avelength, th e p o s i t i o n s s Q, dQ may v ary s i g n i f i c a n t l y w ith tem pera­
t u r e . Even when c a l i b r a t i o n i s re p e a te d fo r d i f f e r e n t te m p e ra tu re s of
th e sample h o ld e r , s h i f t s of th e r e f e r e n c e p la n e s may no t be q u ite
re p ro d u c ib le . Thermal expansion changes no t only th e p h y s ic a l le n g th of
115
th e waveguide b u t, in th e case of r e c ta n g u la r g u id e , a l s o th e guided
wavelength Xg , which i s dependent on th e broad dimension of waveguide.
Thus th e change of e l e c t r i c a l le n g th o f th e waveguide may be la r g e r
th an th e l o n g itu d in a l therm al expansion. Over a d is ta n c e of s e v e r a l
w avelengths, th e phase of th e r e f l e c t i o n c o e f f i c i e n t changes n o tic e a b ly
i f even sm all frequency i n s t a b i l i t y of th e s i g n a l sou rce i s e x p e r i­
enced. The f a c t o r s r e p r e s e n tin g lo s s e s - RQ in E q .( 4 .6 ) and Rg in
E q .(4 .3 5 ) a r e determ ined by measurement of a very high VSMR and may
a ls o be in a c c u r a te . When s u f f i c i e n t number of measurements i s made,
c o r r e c t i o n s of c a l i b r a t i o n p aram eters can be e s tim a te d to g e th e r w ith
y
by th e method of l e a s t s q u a re s .
Let us d e fin e fo u r a d d i t i o n a l param eters:
x^
/ 8
- s h i f t of re f e r e n c e plane A towards s h o r t c i r c u i t ,
x^
/ 8
- s h i f t of re f e r e n c e p lan e B towards s h o r t c i r c u i t ,
x,.
- f a c t o r m u ltip ly in g s h o r t c i r c u i t r e f l e c t i o n Rg ,
Xg
- f a c t o r m u ltip ly in g a l l measured R ^ ..
When th e s e param eters a re in tro d u ce d i n to th e model fu n c tio n E q s .(4 .3 4 4 .3 5 ) , Rft. i s re p la c e d by RAixge x p (2 jx 3 ) , S C d j- d ^ by
8
( d . - d 0 ) - x 4> and
Rgj by XgRgj- I n i t i a l v a lu e s x 3 =x4 =0, Xg=Xg=l correspond t o th e o r i g i ­
n a l c a l i b r a t i o n . The computer program p e rm its refinem ent of a l l four
c a l i b r a t i o n param eters to g e th e r w ith
y,
or refin e m e n t of a s u b s e t of
X3 , x4 , x^, Xg. An e s tim a tio n w ith a c o n s t r a i n t x 3 =x4 i s a ls o p o s s i b l e ,
which a llo w s t o c o r r e c t fo r displacem ent of th e sample in waveguide.
The b a s ic shape of th e model fu n c tio n E q .(4 .3 4 ) i s determ ined by
and th e fix e d param eters
8
y
and 1. Refinement of th e c a l i b r a t i o n parame­
116
t e r s may be t r e a t e d a s an ad ju stm en t o f s c a l e s fo r th e model f u n c tio n .
When th e in p u t r e f l e c t i o n c o e f f i c i e n t
v a r i e s s i g n i f i c a n t l y w ith
p o s i t i o n of th e s h o r t c i r c u i t , d j , c a l i b r a t i o n can be r e f in e d w ith ou t
r i s k of d i s t o r t i n g th e e s tim a te o f
y
and d a ta re d u c tio n can be accom­
,
p lis h e d even when th e re fe re n c e p o s i t i o n s a r e in e r r o r p r i o r to r e f i n e ­
ment. On th e o th e r hand, when th e a t t e n u a t i o n in th e sample i s so la r g e
t h a t th e p o s i t i o n of th e s h o r t c i r c u i t has l i t t l e e f f e c t on R ^., th e
d a ta may not c o n ta in enough in fo rm atio n to a llo w refinem ent o f c a l i b r a ­
t i o n param eters.
Refinement of th e c a l i b r a t i o n param eters should only be made when
i t reduces th e ro ot-m ean -squ are r e s i d u a l [see Eq.(D16)] and th e s t a n ­
dard d e v ia t io n of th e e stim a te d p ro p a g a tio n c o n s ta n t, y. O therw ise, i t
i s no t recommended t o in c lu d e in t o th e e s tim a tio n a d d i t i o n a l param eters
which do not improve th e q u a l i t y o f f i t . The c o r r e l a t i o n m a trix of
p aram eters should be examined fo r la r g e c o r r e l a t i o n s between ReY, Imv
and c a l i b r a t i o n pa ra m ete rs. Refinement of a param eter which i s s tr o n g ly
c o r r e l a t e d w ith
y
may cause d i s t o r t i o n of th e e s tim a te fo r
y
,
and
should be avoided.
A n a ly sis of most o f th e v a r i a b l e te rm in a tio n measurements in t h i s
work included refinem ent of th e p o s i t i o n s of th e re fe re n c e p lanes x3 ,
x4 , which u s u a ll y r e s u l t e d in s i g n i f i c a n t re d u c tio n of th e root-m eansqu are r e s id u a l and u n c e r t a i n t i e s of
y
.
A d d itio n a l refin em ent of th e
lo s s f a c t o r s x^, Xg u s u a ll y brought f u r t h e r sm all re d u c tio n of Rmsr but
in few in s ta n c e s th e e stim a te d s ta n d a rd d e v ia t io n s of e a c t u a l l y i n ­
c re a s e d . Unreasonably la r g e v a lu e s were o b ta in e d fo r c a l i b r a t i o n c o r -
117
r e c t i o n s only in s i t u a t i o n s when th e ex p erim en tal c o n d itio n s d i f f e r e d
from th e assumed c o n f i g u r a t i o n , e . g . Nhen a s l a n t e d sample Nas used in
r e c ta n g u la r waveguide, when l o s s of c o n ta c t o ccu rred between w a lls of
th e c o a x ia l waveguide and th e c o n d uctive sample.
4 . 3 .3 . Scheme fo r Data Reduction and Examples.
Waveguide sample h o ld e rs (used fo r measurements of th e complex
p e r m i t t i v i t y a t e le v a te d te m p e ra tu re s w ith th e sample surrounded by
i n e r t atmosphere) o f te n in tro d u c e im p e rfe c tio n s and o b s ta c le s in to th e
waveguide between th e sample and th e s l o t t e d s e c t i o n . Windows in a
re c ta n g u la r waveguide, c o n n ec to rs in a c o a x ia l g u id e , gas i n l e t s and
s t a i n l e s s s t e e l s e c t i o n s of a waveguide w ith la r g e d i s s i p a t i o n a re en­
co u n tered . E f fe c t of such o b s t a c l e s on th e measured r e f l e c t i o n c o e f­
f i c i e n t can be c o r r e c t e d when th e s c a t t e r i n g m a trix
fo r th e two-
p o r t , which r e p r e s e n ts th e waveguide between th e sample and th e d e te c ­
t o r , i s known.
The load r e f e r e n c e plane B f o r th e s c a t t e r i n g r e p r e s e n t a t i o n (see
F i g . 4 .1 2 ) can be lo c a te d a t th e p o s i t i o n used fo r a lig n in g th e back
s u r f a c e of th e sample and where a p l a t e s h o r t c i r c u i t can be p laced .
The in p u t r e f e r e n c e p lan e A^ should be placed near th e d e t e c t o r , in
f r o n t of th e o b s t a c l e s in th e waveguide. The in p u t re fe re n c e p la n e , A^,
can be c o n v e n ie n tly lo c a te d a t th e p o s i t i o n of th e v o lta g e minimum
o b ta in e d w ith a p e r f e c t s h o r t c i r c u i t a t p lane B. Plane Afl does not
need to be p h y s ic a lly a c c e s s i b l e . When n e i t h e r d i s c o n t i n u i t y nor a t ­
te n u a tio n in th e waveguide i s p r e s e n t th e s c a t t e r i n g param eters a re
2
Sh l l =0, Sh i2
’ s h22=0* when only a t t e n u a t i °n in th e waveguide needs
118
to be taken in t o a c c o u n t, one has Sh i2
2
R0<1, Ro b e in g th e r e f l e c t i o n
c o e f f i c i e n t measured w ith a p e r f e c t s h o r t c i r c u i t lo c a te d a t plane B.
The s c a t t e r i n g p aram eters
a r e e stim a te d u s in g th e l e a s t - s q u a r e s
method d e s c rib e d in S e c .4 .2 . 1 . Measurements o f th e v o lta g e s ta n d in g
wave r a t i o and th e p o s i t i o n of th e v o lta g e minimum a r e made w ith an
empty sample h o ld e r f o r s e v e r a l ( t y p i c a l l y 8) p o s i t i o n s o f th e a d j u s t ­
a b le s h o r t c i r c u i t which a re ap p ro x im a te ly e q u a lly -s p a c e d over a d i s ­
ta n c e of o n e - h a lf of a w avelength.
The r e f e r e n c e p o s i t i o n dQ of th e s h o r t c i r c u i t , which produces a
r e a l r e f l e c t i o n c o e f f i c i e n t , -Rg , a t p lane B, i s app ro xim ately e s t a b ­
l i s h e d a s th e p o s i t i o n which g iv e s a v o lta g e minimum a t th e same p o s i­
t i o n s Q a s th e p l a t e s h o r t c i r c u i t a t plane B. When o b s t a c l e s between
plane B and th e probe a r e s i g n i f i c a n t ( l a r g e
S j ^ ) an^ th e
v a r i a b l e s h o r t c i r c u i t i s d i s s i p a t i v e , t h i s procedure i s not a c c u r a te .
Change of th e a b s o lu t e v a lu e of Rg e f f e c t s both th e a b s o lu te v a lu e and
th e phase of R^ [ Eq. ( 4 . 1 5 ) ] . The v a r i a b l e s h o r t c i r c u i t could be c a l i ­
b r a te d by co n n ectin g th e te r m in a tin g waveguide d i r e c t l y to th e s l o t t e d
s e c t i o n , w ith no in te r v e n in g o b s t a c l e s ; however, s in c e t h i s i s not
always p o s s ib le in an ex p erim en tal s e t u p , a method fo r r e f i n i n g dQ and
Rg d u rin g th e e s tim a tio n of
has been developed.
Measurement w ith th e p l a t e s h o r t c i r c u i t a t p la n e B, which i s
norm ally used to e s t a b l i s h th e r e f e r e n c e p o s i t i o n s Q ( e q u iv a le n t to A^)
and th e f a c t o r Rq which c o r r e c t s fo r l o s s e s , i s now in clu d ed in to th e
v e c to r o f measurements w ith RA2 =-R0 , RB1= - 1 ‘ For t h i s measurement» th e
r e s o l u t i o n e r r o r s in form ulas fo r th e c o v a ria n c e m a trix Cy [see
119
Bq. ( 4 .2 6 ) ] a r e assumed to be much s m a lle r th an fo r measurements w ith
th e v a r i a b l e s h o r t c i r c u i t . In e f f e c t , th e v a lu e s R ^ , Rgl a r e fix e d
and Rb1=-1 p ro v id e s r e f e r e n c e f o r th e phase and th e a b s o lu te v a lu e of
Rg . . The r e f e r e n c e p o s i t i o n dQ o f th e v a r i a b l e s h o r t c i r c u i t and th e
e f f e c t i v e r e f l e c t i o n c o e f f i c i e n t Rg a re added to th e v e c to r of th e
p aram eters [se e E g .( 4 .2 1 ) ] and e s tim a te d to g e th e r w ith th e s c a t t e r i n g
m a tr ix , Sh . The r e f in e d v a lu e s of dQ and Rg were u s u a ll y very c lo s e to
th e approxim ate v a lu e s e s t a b l i s h e d d i r e c t l y .
A ccurate measurement of high VSUR d u rin g c a l i b r a t i o n measurements
may be d i f f i c u l t . Probe p e n e t r a t i o n should be kept s m a ll. S ig n al gen­
e r a t o r should be e l e c t r i c a l l y matched in o rd e r to avoid changes of
s ig n a l le v e l when th e a d j u s t a b l e s h o r t c i r c u i t i s moved. C a l ib r a tio n of
th e d e t e c t o r should be checked u sin g a p r e c i s i o n a t t e n u a t o r and a
matched te rm in a tio n or by comparing ex p erim en tal and t h e o r e t i c a l depen­
dence of standing-w ave v o lta g e on th e probe p o s i t i o n s u sin g a s h o r t
c i r c u i t , as d e s c rib e d in th e l i t e r a t u r e [1 5 4,15 5 ]. Only measurements
around th e v o lta g e minimum (th e double minimum method) should be used
to determ ine th e VSUR i f s a t u r a t i o n of th e d e t e c t o r i s observed a t th e
v o lta g e maximum.
With th e sample in th e waveguide, measurements of th e v o lta g e
r e f l e c t i o n c o e f f i c i e n t a r e made fo r s e v e r a l p o s i t i o n s o f th e s h o r t
c i r c u i t , as d e s c rib e d in S e c .4 .2 ; however E q .(4 .2 4 ) i s used in s te a d of
E q .( 4 .6 ) to c a l c u l a t e R ^ . The r e f l e c t i o n c o e f f i c i e n t Rfth measured t h i s
way r e f e r s to th e in p u t plane A^ of th e tw o -p o rt which r e p r e s e n ts wave­
guide d i s c o n t i n u i t i e s in th e sample h o ld e r. The r e f l e c t i o n c o e f f i c i e n t
120
Rg^ o f th e load a t plane B (which, w ith o u t sample, would produce R ^ a t
A^) i s c a lc u la te d u s in g th e s c a t t e r i n g p aram eters S^. Phase o f t h i s
r e f l e c t i o n c o e f f i c i e n t i s then changed by -2 jBL so i t s re f e r e n c e plane
i s s h i f t e d toward d e t e c t o r by th e d is ta n c e L ( le n g th of th e sample) and
c o in c id e s w ith th e f r o n t s u r f a c e o f sample. The c o r r e c te d r e f l e c t i o n
c o e f f i c i e n t a t th e f r o n t s u r f a c e o f th e sample i s given by
R. = ------------ _ %
Sh l l ------
exp(-2jBL)
(4 .3 8 )
RAhSh 2 2 'Sh l l Sh22+Sh l 2
where
R^h i s c a l c u l a t e d u sin g E q.4.2 4 .
I f s i g n i f i c a n t waveguide im p e rfe c tio n s were p re s e n t between th e
sample and th e a d j u s t a b l e s h o r t c i r c u i t , s i m i l a r tre a tm e n t could be
a p p lie d to c o r r e c t th e r e f l e c t i o n c o e f f i c i e n t of load Rg a t plane B.
Once a d a ta s e t i s o b ta in e d c o n s i s t i n g o f a s e r i e s of th e f r o n t
s u r f a c e r e f l e c t i o n c o e f f i c i e n t s R^. and th e c o rresp on d in g load r e f l e c ­
t i o n c o e f f i c i e n t s Rfi. (or impedances Zfij ) a t th e back of sample, th e
fo llo w in g th r e e d i f f e r e n t approaches can be a p p lie d to e v a lu a te th e
complex p e r m i t t i v i t y , e.
(i)
One can u n d e rta k e a l e a s t - s q u a r e s e s tim a tio n o f th e s c a t ­
t e r i n g param eters f o r th e sample t r e a t e d as a tw o -p o rt and c a l c u l a t e e
from E q .( 4 .3 0 ). An i n i t i a l guess i s not r e q u ir e d . Although th e r e s u l ­
t i n g p e r m i t t i v i t y v a lu e may be i n a c c u r a te , a s d is c u s s e d in S e c .4 . 3 . 2 . ,
i t i s u s u a lly a c c u r a te enough to prov ide a good i n i t i a l guess fo r th e
p ro p a g a tio n f a c t o r y . I t i s notew orthy t h a t in few c ase s i t le d to a
wrong s o l u t i o n of E q . ( 4 . 4 ) . E stim a tio n of th e s c a t t e r i n g param eters i s ,
n e v e r t h e l e s s , d e s i r a b l e a s a check on th e c o n s is te n c y o f th e d a ta .
121
P i t o f a g e n e ra l s c a t t e r i n g m a trix should r e s u l t in a sm all v a lu e
o f th e root-m ean-square r e s i d u a l . With r e s o l u t i o n e r r o r s s i m i l a r to
th o se used in th e example a t th e end o f S e c .4 . 2 . , a Rmsr l e s s than 0 .5
was t y p i c a l l l y o b ta in e d . In g e n e r a l, however, only Rmsr v a lu e s fo r d a ta
o b ta in e d w ith th e same ex p erim en tal s e tu p and f o r e s tim a tio n s with th e
same w eights may be m ean in g fu lly compared. When a Rmsr i s g r e a t e r than
one e x p e c ts , th e d a ta a r e l i k e l y t o c o n ta in some c o arse e r r o r s . O ften,
th e r e i s simply a m istake in one datum, which can be p in p o in te d by
exam ination of r e s i d u a l s and e lim in a te d . Since th e sample r e p r e s e n ts a
symmetric d i s c o n t i n u i t y , S ^ and S2 2 should be equal w ith in th e c o n f i ­
dence l i m i t s . When th e sample i s d is p la c e d r e l a t i v e to th e r e fe re n c e
p o s i t i o n s , th e phases of S ^ and S2 2 a r e d i f f e r e n t bu t th e a b s o lu te
v a lu e s should be e q u a l. The in p u t and o u tp u t re fe re n c e p la n es can be
s h i f t e d to o b ta in equal phases of S ^ , S22 p r i o r to e s tim a tio n of
symmetric s c a t t e r i n g m a trix . When | S ^ | and IS22I
sig n ific a n tly
and Rmsr i s low, one s u s p e c ts some s e r io u s s y s te m a tic e r r o r s and th e
c a l i b r a t i o n should be reexamined.
(ii)
Equation ( 4 .4 ) i s so lved fo r th e complex p ro p ag atio n f a c t o r
r s e p a r a t e l y w ith each p a i r
Rf t i ,
Rfii. Values o f e.. a r e then c a lc u ­
l a t e d from E q .( 4 .3 ) and th e margins of e r r o r a r e e stim a te d as d e s c rib e d
in S e c .4 .2 . An a r i t h m e t i c average e and s ta n d a rd d e v ia t io n s o f th e r e a l
and imaginary p a r t s o f p e r m i t t i v i t y may then be c a lc u l a t e d . Measure­
ments r e s u l t i n g in e. v a lu e s which d i f f e r from th e average by more than
two s ta n d a rd d e v ia t io n s should be excluded from a v erag e , e s p e c i a l l y i f
t h e i r margins of e r r o r s a r e l a r g e . When th e remaining e. v a lu e s do not
122
s c a t t e r by more than th e d e s i r a b l e margin of e r r o r , th e a r i t h m e t i c
average i s a good e s tim a te and th e r e i s no r e a l need fo r a l e a s t sq u a re s e s tim a tio n o f
y.
However, when la r g e d e v ia t io n s of th e i n d i v i d ­
u a l r e s u l t s from th e average a re p r e s e n t , th e a r i t h m e t i c average i s not
a proper way to f in d th e most p robable v a lu e of th e p e r m i t t i v i t y and a
le a s t - s q u a r e s e s tim a tio n i s n e c e s sa ry .
(iii)
The complex p ro p ag atio n f a c t o r
y
i s e stim a te d by th e method
of l e a s t s q u a re s , w ith o p tio n a l refin e m e n t of c a l i b r a t i o n p aram eters as
d e s c rib e d in S e c .4 . 3 . 2 .
Examples
Me s h a l l i l l u s t r a t e th e scheme f o r d a ta re d u c tio n by two examples:
(1) measurement of th e p e r m i t t i v i t y of commercial poly-m ethyl m ethacry­
l a t e ( p l e x i g l a s s ) a t room te m p e ra tu re ; and (2) p e r m i t t i v i t y measurement
on N a jZ ^ S ijP O ^ (NASICON) a t 1 2 3 .5°C. A d e s c r i p t i o n of th e procedure
and a d is c u s s io n of th e r e s u l t s i s complemented w ith th e computer
p r i n t o u t s given in Appendix F.
Measurements were made in X-band r e c ta n g u la r waveguide (a=2.286
cm) a t frequency 8.37 GHz. A Polarad GB-2 g e n e r a to r w ith G-711 r . f .
tu n in g u n i t provided a s t a b l e s ig n a l so u rc e. Output from a P o ly tec h n ic
Research and Development 585B tra n s m is s io n wavemeter, s e t a t th e de­
s i r e d frequency was m onitored on an o s c il lo s c o p e and th e s ig n a l f r e ­
quency was a d ju s te d manually when n e c e s s a ry . A Hewlett Packard s l o t t e d
s e c t i o n X810B and probe c a r r ia g e 809B w ith PRD-250A tu n a b le probe were
used fo r th e standing-w ave measurements (equipped w ith a d i a l i n d i c a t o r
123
f o r p r e c is io n d e te rm in a tio n of probe p o s i t i o n ) . A microwave diode type
1N23-C was s e l e c t e d t o g iv e 40dB dynamic range of n e a r ly sgu are-law
response w ith HP415D s ta n d in g -w a v e -ra tio m eter. A n o n -c o n ta c tin g s h o r t
c i r c u i t was a d ju s te d by means of a micrometer head (U aveline 661). The
waveguide which c o n tain ed th e sample was p r e s s u r iz e d w ith helium. The
s ta n d a rd X-band p r e s s u r i z i n g u n i t (U aveline 667), placed between sample
and s l o t t e d l i n e , was s e p a ra te d from s l o t t e d l i n e by a t h i n (0.001mm)
mylar f o i l . Brass waveguide was used (some p a r t s of which were s i l v e r
p l a t e d ) ex cept fo r a 17.86 cm long s t a i n l e s s s t e e l s e c tio n which was
p o s itio n e d in s id e a t u b u la r fu rn ace f o r measurements a t e le v a te d temp­
e r a t u r e s . The b r a s s waveguides connected to th e two ends of th e s t a i n ­
l e s s s t e e l s e c t i o n were w a te r-c o o le d .
a) c alib ratio n
The wavelength in th e s l o t t e d s e c t i o n was \ gj= 5 8 .156+0.016 mm. In
th e te rm in a tin g waveguide th e wavelength could n o t be measured, because
th e d is ta n c e t r a v e l l e d by th e s h o r t c i r c u i t was lim ite d to 27 mm, but
was e stim a te d t o be Xg2=57.63 mm fo r f=8.37 Ghz and a=22.86 mm. This
e stim a te d v a lu e agreed w ith th e wavelength in th e s l o t t e d s e c t i o n
( a f t e r c o r r e c t i n g fo r e f f e c t s of th e s l o t [154]) and a l s o r e s u l t e d in
th e low est r e s id u a l e r r o r in an e s tim a tio n of th e s c a t t e r i n g parame­
te rs.
At f i r s t , a c a l i b r a t i o n was made w ith o u t th e s t a i n l e s s s t e e l wave­
g u id e , fo llo w ing th e method d e s c rib e d above. A measurement w ith p l a t e
s h o r t c i r c u i t lo c a te d a t plane B (th e fla n g e of th e b ra s s waveguide)
gave VSUR U=40.9dB i . e . RQ=0.982 and s Q=13.63 mm. A fte r co n n ectin g th e
124
te rm in a tin g waveguide, n in e measurements were made w ith th e a d j u s t a b l e
s h o r t c i r c u i t d is p la c e d in increm ents of about 0.055 of th e wavelength
(s e e Appendix P). The ex p erim en tal r e s o l u t i o n e r r o r s were assumed s im i­
l a r t o th o se in th e example in S e c .4 .2 , t h a t i s ,
as=0.03m m +0.0005[(|R |+0.005)"1-l] A g ,
aW=0.2dB+0.03W,
Ad=0.03mm
( au>=0.4°), aRs =0 .0 1 . The r e f e r e n c e p o s i t i o n of th e s h o r t c i r c u i t , i t s
e f f e c t i v e r e f l e c t i o n c o e f f i c i e n t and th e s c a t t e r i n g param eters fo r d i s ­
c o n t i n u i t i e s between probe and plane B were e s tim a te d by th e l e a s t
sq u a re s a lg o rith m to be:
d0 = 19.010 ± 0.002 mm,
ReSh n = 0.0050 ± 0.0 00 3,
Rg = 0.9945 ± 0.0006
ImSh l l ="0 -0126 * 0.0003
ReSh l2 2 = ° " 09 * 0.0005
ImSh l 2 2 =- ° * 0246 * 0.0005
ReSh22 =-0.0059 ± 0.0003
ImSh22 =“0 0 1 2 2 * 0.0003
The roo t-m ean-sq uare r e s i d u a l Rmsr=0.079 was v ery low and th e r e s i d u a l
d e v ia t io n s were an o rd e r o f magnitude s m a lle r th an th e assumed r e s o l u ­
t i o n e r r o r s . The s c a t t e r i n g param eters
were sm all w ith th e e x c e p tio n
2
of R e S ^ . The d i s c o n t i n u i t y produced by th e mylar window and by p r e s ­
s u r i z i n g u n i t tu rn e d ou t to be i n s i g n i f i c a n t and could be n e g le c te d in
t h i s c a s e . The c a l i b r a t i o n param eters Sh , s Q, Rg , dQ were used d i r e c t l y
to e v a lu a te th e p e r m i t t i v i t y of th e p l e x i g l a s s sample, which was placed
w ith in th e b r a s s waveguide.
The s t a i n l e s s s t e e l s e c t i o n of waveguide was connected between
plane B and th e te r m in a tin g waveguide. I t was t r e a t e d as a d is c o n t i n u ­
i t y fo r which s c a t t e r i n g param eters Se were o b ta in e d by th e method
o u tlin e d in S e c .4 . 3 . 1 , u sin g in p u t r e f l e c t i o n c o e f f i c i e n t s R^ c o r r e c t e d
125
by means of th e s c a t t e r i n g param eters Sh [ E q . ( 4 .3 8 ) ] . Here Ss l l = -0 .0 23 j0.024 and Ss2 2=-0 - 0 1 5 -j0 .0 2 7 were n e g le c te d in subsequent a n a l y s i s .
2
(Ss l 2 I=0 . 981 was taken a s a measure o f a t t e n u a t i o n in th e s t a i n l e s s
s t e e l s e c t i o n . For experim ents a t e le v a te d te m p e ratu res samples were
placed in th e middle of th e s t a i n l e s s s t e e l s e c t i o n and h e ld th e r e by
an e x te r n a l clamp which a p p lie d p re s s u re to th e o p p o s ite broad w a lls of
th e waveguide. For each sample, d i s t a n c e s from th e ends of s t a i n l e s s
s t e e l guide to th e sample were measured w ith a depth micrometer - s
s
on
th e s id e of s l o t t e d s e c t i o n and dg on th e s id e of s h o r t c i r c u i t . These
d i s t a n c e s were used t o o b ta in th e proper phases of r e f l e c t i o n c o e f f i ­
c i e n t s a t th e s u r f a c e s of th e sample: dQ b eing re p la c e d by <30-<3s and s_ r e p la c in g L in E q .( 4 .3 8 ) . In o rd e r to c o r r e c t fo r a t t e n u a t i o n in th e
s
s t a i n l e s s s t e e l waveguide,
vided by
|S
j
J
-9905
o b ta in e d by means of E q .(4 .3 8 ) was d i ­
a n d Rg
was m u l t i p l i e d by th e same f a c t o r ,
assuming t h a t th e l o s s i s e q u a lly d iv id e d between two p a r t s of th e
waveguide on o p p o s ite s i d e s of th e sample.
C a l ib r a tio n must ta k e in t o account dependence of th e e l e c t r i c a l
le n g th of th e s t a i n l e s s s t e e l waveguide on te m p e ra tu re . Temperature was
measured by a thermocouple in d i r e c t therm al c o n ta c t w ith th e waveguide
w all half-w ay alon g i t s le n g th . At a given te m p e ra tu re , th e s h o r t c i r ­
c u i t p o s i t i o n s were a d ju s t e d u n t i l th e p o s i t i o n s of th e v o lta g e minima
measured a t room te m p e ratu re were reproduced. Displacem ents of th e
s h o r t c i r c u i t , a t a g iven e le v a te d te m p e ra tu re , from th e c o rresp o nd in g
p o s i t i o n s a t room te m p e ratu re were reco rd ed and t h e i r average was taken
to be th e change of e l e c t r i c a l le n g th aD(T) of th e waveguide upon
126
heating to that temperature. The change of e le c t r ic a l length AD(T) was
found to be an approximately linear function of temperature and could
be e a s ily interpolated for temperatures at which i t had not been meas­
ured. It was divided into two parts ADg and AD^, proportional to the
distances between the sample and the two ends of the waveguide
( aDS /AD<
q=s_/d_); these were then added to s_S and d_.
Q S S
S
The calibration when the sample i s positioned in the middle of the
s ta in le s s s t e e l waveguide i s l e s s precise than the calib ration when the
sample i s aligned with the flange of the waveguide. The former configu­
ration was, however, preferred for highly conductive materials because
with th is configuration i t was possible to maintain mechanical contact
between the broader waveguide walls and the sample over a range of
temperatures.
b) measurement of p lex ig la ss
The sample of length 23.75±0.02 mm, machined to f i t tig h t ly into
the cross section of waveguide, was inserted into the brass waveguide
and the back surface aligned flush with the flange. Measurements were
made for 10 p osition s of the short c i r c u i t , covering the distance of a
half-wavelength but separated by smaller increments around the position
giving the smallest VSWR (see Appendix F). Fit of 6 sca tterin g parame­
ters resulted in low Rmsr=0.197 and low residual deviations, but the
estimated sca tterin g matrix was noticeably nonsymmetric (2.45° d i f f e r ­
ence between the phases of
and S22 was equivalent to s h iftin g the
sample by 0.10 mm). A symmetric scatterin g matrix was f it t e d without
correcting for that s h i f t , so the resu ltin g Rmsr=1.064 was higher. The
127
real part of perm ittivity and the lo ss tangent calculated by mean of
-3
Eq.(4 .3 0 ), were e'=2.92±0.04, tan8=(5.5±16.)xlO . The corresponding
value of the propagation factor v=0.9+j266 was not accurate enough to
serve as an i n i t i a l guess for solution of Eq.(4.4) and had to be
corrected before convergence for a l l data points was obtained. The
average of the individual solu tion s was e ,=2.545±0.016, tan8=(4.9±0.6)xlO“3 .
F ittin g the propagation factor without adjustment of the calib ra­
tion gave e'=2.536±0.003, tan$=(4.96±1.4)xlO~3 and a large root-meansquare residual Rmsr=2.97. Adjustment of the positions of the reference
planes achieved a much lower Rmsr=0.288 with only a small change of
perm ittivity e'=2.506±0.001, tan8=(5.13±0.14)xl0“3 . The estimated
s h i f t s of reference planes [ ( 0 . 35±0.01)mm for plane A, ( - 0 . 15±0.01)mm
for plane B] were higher than the expected u n certainties in estab­
lish in g the reference p osition s. A possible explanation may stem from
the fact that the surfaces of the sample were not exactly perpendicular
to the waveguide a x is.
An estimation, with additional refinement of the lo ss factors,
reduced the root-mean-square residual to Rmsr=0.197, the same value as
was obtained when the f i t t i n g was done with a general scatterin g ma­
tr ix . In t h is case, the estimate of the lo s s tangent was a ffected ,
_3
tan8=(6.09±0.3)xl0 , and i t s uncertainty increased. The factor asso­
ciated with lo ss e s in the short c ir c u it remained unchanged x^=l, while
the factor adjusting attenuation in the input Xg=0.993+0.002 indicated
that the scatterin g parameters Sh overcorrected the lo ss e s .
128
Tables of D ie le c tr ic Materials [162] report d ie le c t r ic constant of
poly-methyl methacrylate e'=2.57 at 10 GHZ, which i s s lig h t l y higher
than the values obtained here. The lo s s tangent i s reported
-3
tanS=4.9xlO . Incid en tally, the r e su lts obtained without refinement of
the calibration were closer to the tabulated data,
c) measurement of NASICON at 123.5°C
A rectangular parallelepiped plate of thickness 0.88±0.01 mm was
cut from a sintered p e lle t of NajZ^SijPOj^ NASICON to f i l l the crosssection of waveguide. The narrow sid es were polished down u n til the
sample could s lid e into the s t a in le s s s t e e l waveguide. Gold electrodes
were sputtered in a d .c . plasma onto the two longer sid es of the
sample. The sample, with i t s large faces perpendicular to the axis of
waveguide, was then positioned in the middle of the s ta in le s s s t e e l
section using a rectangular rod made of p le x ig la ss , whose cro ss-section
matched the internal dimensions of waveguide. A squeezing force was
applied by a spring loaded clamp to the opposite broader walls of
waveguide, at the location of the sample. This secured the sample in
place and forced the walls of the waveguide to contact the gold e le c ­
trodes of the sample. Visual inspection revealed that the contact was
not uniform and small air gaps existed between the sample and the
waveguide. Distances from the ends of waveguide to the sample were
found to be ds =87.6 mm, s s =90.01 mm. These values were used to modify
the calibration as described above. The standing-wave-ratio and the
position of the voltage minimum were measured for 10 position s of the
short c ir c u it distributed about a distance of half-wavelength, but with
129
preference for a narrow range corresponding to lower VSWR (see in
Appendix F). The phase of the input r e fle c tio n c o e f fic ie n t varied only
within the interval 160° to 2 0 0 °.
The resolution errors were assumed the same as for the calib ration
measurements. The r e su lts of data reduction are summarized in Table
4.1. Perm ittivity obtained from the scatterin g parameters did not agree
with the other estim ates, but served well as an i n i t i a l guess. Average
of individual so lu tio n s, after neglecting two obviously bad so lu tio n s,
gave lower p erm ittivity values than the least-squares estim ation, but
the difference was only of the order of one standard deviation. When
correction of calib ration estimated by the least-squares f i t t i n g was
made, the average of the individual solu tion s became indistinguishable
from the resu lts of f i t , thus clea rly demonstrating that the sca tter of
the individual s o lu tio n s, prior to applying corrections, was caused by
systematic errors of the ca lib ration parameters. The perm ittivity ob­
tained here, using a sample f i l l i n g cro ss-sectio n of waveguide, can be
compared with r e s u lts [ e ' =22.65+0.2, o=6.08±0.07 (am)- 1 ] obtained for
the same material, a t the same temperature and frequency, using
cen tra lly -located slab (see S e c .4 .5 ). The small d ifferen ces are lik e ly
attributable to the imperfect contact between the sample and the
broader walls of waveguide (see S e c .4 .4 ).
130
Table 4.1
Results of Data Reduction for Measurement of Perm ittivity of NASICON at
123.6°C, 8.37 GHz, sample f i l l i n g waveguide cross sectio n .
No
Procedure
Rmsr
e*
o (flm) " 1
-
Comments
1
f i t of general
scatterin g matrix
0 .2 0 1
-
2
f i t of symmetric
scatterin g matrix
0.755
45.1±4.
5.3±1.9
3
f i t of symmetric
scatterin g matrix
0.231
3 2 .1±0.8
9.1+0.4
4
average of 10
measurements
-
19.9±2.5
4 . 8 ±1 . 0
5
average of 8
measurements
-
20.7±1.5
4.8+0 . 6
6
f i t of propaga­
tion factor v
2.370
2 0 . 4±0.5
4 . 8 +0 . 2
7
f i t of y with
0.262
adjustment of r e fe ­
rence planes
2 1 . 8 6 +0 . 1 2
5 . 82±0.04
reference s h i f t s
plane A -0.25 mm
plane B 0.10 mm
8
same as 7, factors 0 . 2 0 1
correcting attenu­
ation also adjusted
21.60+0.11
5 . 81±0.05
lo s s factors
load 0.9952
input 0.9839
9
average of 10
measurements
2 1 .65±0.24
5 . 80±0.04
calib ration ad­
justed as in 8
-
estimated s h i f t
of sample -0.083mm
data corrected for
sample s h i f t
estimated in 1
2 o u tlie r s
excluded
131
4.4. The Effect of E lectrica l Contact between the Waveguide and the
Sample on the Measured P e rm ittiv ity .
When the variable termination methods described in Section 4.3
were used for measurement of the complex p erm ittivity of Na^Zr2 S i 2 P0 ^2 >
anomalous behavior was observed at elevated temperatures and frequen­
c ie s around 1 GHz. The measurements were made using the coaxial wave­
guide (General Radio 593, 14.2 mm i . d . , 6.2 mm o .d .) in the frequency
range 0.35 to 7 GHz. The r e s u lts obtained at room temperature appeared
to be reasonable, namely, the real part of perm ittivity was between 15
and 2 0 , the conductivity was higher than the bulk conductivity
obtained from the a .c . impedance measurements below 1 MHz and was
increasing with frequency. As the temperature was increased to 300°C,
the real part of p erm ittivity measured at 0.35 GHz rose from 18 to 65.
The conductivity saturated at a le v e l much lower than the low frequency
bulk conductivity; e . g . , at 300°C the bulk conductivity was 0 ^ 0 .2 1 5 0 “
^cm- *, whereas the real part of conductivity measured at 0.35 GHz was
o' =0.005 fl~*cnf*, at 1.4 GHz o ' =0.04 Q^cm” 1 and at 7 GHz o ' =0.08 fl”
*cnf*. The pronounced drop of the conductivity below the low-frequency
lim it could not be rationalized in terms of the theories of ionic
transport and an experimental error was suspected.
The error was traced to the partial lo s s of e l e c t r i c a l contact
between the sample and the waveguide. Even when the sample was machined
to f i t t ig h t ly into the waveguide at room temperature, the difference
of thermal expansion c o e f f ic ie n t s between the sample and the s t a in le s s
s t e e l waveguide resulted in appearence of s ig n ific a n t gaps at elevated
132
temperatures. As has been demonstrated by Champlin and Glover [146]
lo ss of e le c t r ic a l contact between the waveguide walls and the sample
may a f f e c t the measured complex p erm ittivity. For materials exhibiting
substantial lo ss (e">>l), presence of a gap between the waveguide and
the sample in the area of high normal component of the e l e c t r i c f ie ld
causes s ig n ific a n t d isto r tio n of the measured p erm ittivity. It has been
shown by a simple perturbation theory argument that the e f f e c t of a gap
on the measured perm ittivity can be approximated by an equivalent
c ir c u it con sistin g of a capacitor representing the gap connected in
s e r ie s to the impedance of the sample. An approximate relation between
the measured apparent perm ittivity
and the true p erm ittivity of
material, e, i s
e
where
c---------l+K(e-e 1 ) / e 1
(4•39)
i s the e f f e c t iv e p erm ittivity of the gap and K the ratio of
geometric factors for the gap capacitor and the capacitor formed by the
waveguide w alls. For example, in the case of a coaxial waveguide loaded
with a sample having the outside diameter s lig h t l y smaller than the
inner diameter, b, of the outer conductor, such that a circular gap of
width t i s present, the geometric ra tio i s K=2 t /[ b ln ( b /a ) ] (a i s dia­
meter of the inner conductor). The e f f e c t iv e perm ittivity of the gap
may be d ifferen t than 1 and may be complex when p artial contact resu lts
and when space charge i s accumulated on the surface of sample. Taking,
for sim p lic ity , e^=l, we can rewrite Eq.(4.39) for the real and imagi­
nary parts of perm ittivity
For a material ex h ib itin g large lo ss [ 6 l,soV (u e ) » l ] , even a
small gap (K<<1 ) , gives r is e to a s ig n ific a n t error of the measured
p erm ittivity. For example, taking the real part of p erm ittivity e'=20
and the conductivity o'=0.215a”1cm”1 (equal to the bulk conductivity of
Na3 Zr2 S i 2 P0 12 at 300°C) one gets e"=llOO at 0.35 GHz. With a gap width
of 0.06 mm around the outer conductor of a Gft 593 coaxial waveguide,
the geometric factor i s K=0.01 and we get from Eq.(4.40) e^alOO,
e”=8.9,
om
’=0.0017 fl^cm- *. Thus a small gap between the sample and the
m
waveguide produces a decrease in the measured conductivity of two
orders of magnitude and a large increase of the real part of perm itti­
v it y . Q u alitatively, th is reproduces the anomalous behavior observed
during measurements on NASICON. The e f f e c t s of the lo ss of contact are
smaller at higher frequencies but are not n eg lig ib le for conductive
m aterials. Taking
of
to be complex in Eq.(4.39) and varying the values
and K, i t i s p ossib le to arrive at em values which varies over a
broad range for a given true value of the p erm ittivity, e. In the
actual experiment, the exact values of K and
which arise from a
partial lo ss of contact are not known and hence Eq.(4.39) cannot be
used to correct the experimental data.
The contact problem can be handled by using two approaches: (1)
applying an electromagnetic f ie l d configuration which is not s e n s itiv e
134
to contact between the sample and the waveguide; ( 2 ) assuring good
e l e c t r ic a l contact between the sample and these walls of waveguide on
which the normal component of the e le c t r ic f ie ld i s not zero. A mea­
surement method based on
the f i r s t approach was developed by Champlin
e t . a l. [163] who used the TEQ° mode in the circular waveguide. Since
the e l e c t r i c f ie ld of th is mode has zero normal component on the wall
of waveguide, there i s no current flow between the sample and the wave­
guide, hence a truly con tactless measurement can be made. The TEQ° mode
i s , however, not the fundamental one in the circular waveguide and i t
i s d i f f i c u l t to assure a single-mode operation. The second approach to
the contact problem was adopted here. A new experimental technique was
developed which permitted us to maintain uniform e l e c t r ic a l contact
over a wide range of temperatures (see Section 4 .5 ).
4.5. The Centrallv-Located E-olane Slab in a Rectangular Waveguide.
4 .5 .1 . Experimental setu p .
The measurements were made using a rectangular parallelepiped sam­
ple placed cen trally along the axis of a rectangular waveguide in con­
tact with the broad walls but f i l l i n g only a fraction of waveguide
cross section (usually l e s s than 1/10). A rectangular p lex ig la ss rod
was used for positioning the samples inside the waveguide. The width of
the rod matched the broad dimension of the waveguide. The sample was
placed in a s lo t which was cut cen trally through the end of the rod and
which matched the thickness of the sample. The rod with the sample was
s li d into the waveguide to the desired depth (usually such that the
sample was in the middle of the waveguide s e c tio n ). Centers of the
135
broad walls of waveguide were then squeezed at the position of sample
by a spring-loaded external clamp (see. F ig .4.13). The squeezing force
held the NASICON sample in the waveguide. The p lex ig la ss rod, which was
s lig h t l y thinner than the narrow dimension of waveguide, could be
withdrawn without disturbing the sample. Deformation of the rectangular
cross section of the waveguide was n e g lig ib le when the sample was
machined to the correct s iz e . The thickness of the broad walls of the
waveguide was reduced in order to allow easier compliance to the sample
upon squeezing. The slab was touching the waveguide only over a narrow
s tr ip at the center of the broad wall and the contact pressure was
uniform. Since the squeezing force was controlled by springs located
outside furnace, good e l e c t r ic a l contact was maintained throughout the
temperature range d espite the difference in thermal expansion between
the sample and the waveguide.
The two faces of an ion-conducting sample which were in contact
with the waveguide were polished and coated with sputtered gold or
platinum electrod es. Thus, the e le c t r i c ex cita tio n by the TE^0 mode was
applied to the ionic conductor in a sim ilar way to that used during the
a .c . impedance measurements with ion-blocking electrodes.
This method of holding E-plane slab samples in a rectangular wave­
guide i s not su ita b le for samples of very s o ft or very b r i t t l e mate­
r i a l s , but worked well for NASICON ceramics, s il i c o n and p le x ig la ss.
For a graphite sample, i t was d i f f i c u l t to adjust the squeezing force
which would hold sample in place without breaking i t .
136
screw for adjusting
spring tonsion
7ZZZZZZZZZZZZZZ.
TZZZZZZZZZZZZZZZZs
split-tube
furnace
stainless steal
waveguide
sample
7 ///// / / / zz/ zzzl ' a
stainless steal
clamp
copper
insert
F ig .4.13. Cross section of the waveguide sample holder for measurement
of the p erm ittiv ity using a centered E-plane slab.
137
4 .5 .2 . Computation of the Scattering Matrix.
We consider a rectangular parallelepiped slab of width t , length
L, height b, co n sistin g of material exhibiting the complex p erm ittivity
e. This slab i s placed along the center of a rectangular waveguide of
width a, in contact with the broad walls (see F ig .4 .1 4 ). Limiting the
discussion to the frequency range such that only the fundamental wave­
guide mode, transverse e l e c t r ic TE1Q [156], can propagate in the empty
waveguide, we calcu late the scatterin g matrix for t h is mode with refer­
ence planes located a t the ends of slab (z=0 and z=L). We use the
expansion of the electromagnetic f i e l d in a s e r ie s of the normal modes
in order to express the boundary condition for the transverse com­
ponents of the e l e c t r i c f ie l d , E
y
and magnetic f i e l d , Hv , at the
x
interface between the s l a b - f i l l e d and the empty waveguide [164].
Normal Modes
The only component of the e l e c t r i c f i e l d of the TE^q mode i s E^,
p a ra llel to the y -a x is . Since the faces of the E-plane slab at which
the p erm ittivity of the medium changes from e to 1 are p a ra llel to the
y -a x is , the boundary condition for the Maxwell's equations can be
s a t i s f i e d by the e le c t r i c f ie ld having zero components in the x and z
d irectio n s. In view of the fact that the e le c t r ic f ie ld Ey of the TE1Q
mode i s symmetric about the center of the waveguide, the to ta l e le c t r i c
f ie ld excited due to the presence of the cen tra lly -lo ca ted slab i s also
symmetric with respect to x=a/2. We need to consider only TE modes with
E such that E (x)=E (a-x) and E =0. In the empty guide, the TE modes
y
y
y
«
exhibiting the proper symmetry are TE2m_2 0 ( m=l , 2, 3, . . . ) which have
AN
Fi g. 4. 14. Configuration of a f in i t e - le n g t h , centered E-plane slab in a
rectangular waveguide.
139
the e le c t r ic f ie ld d istrib u tion across the waveguide
Ey(x,y) = fm(x) = / 2 / a s in [ ( 2 m - l)x /a ] ,
and the propagation factors r
0 <x<a
(4.41)
, which determine the exp(±rmz) variation
along the waveguide a x is , are
rm2 = [(2m-l)t»/a] 2 -w2 woeo = [(2m -l)u /a] 2 -k02 ,
r^ je
(4.42)
In the s la b - f il l e d waveguide, the symmetric TE modes have the e le c t r ic
f ie ld distrib u tion
sinknx
gn(x) " Wn <Ancosk^(x-a/2)
sinkn(a-x)
with
for
0 ^x4 ( a - t ) / 2
for
(a -t)/2 ^ x $ (a + t)/2 ,
for
( a + t ) / 2 ^x$a,
(4.43)
An = sin [k n( a - t ) / 2 ]/c o s(k ^ t/ 2 ).
The separation constants kn , k^ are calculated using th eir rela tio n to
the propagation factor Yn :
y
n2 + eko 2 n= k' 2 ,
y
n2 + ok 2 =
n k 2
( 4. 44)
and the boundary condition which requires that Hz and Ey must be co n ti­
nuous at the side surfaces of the slab
kntan[kn ( a - t ) / 2 1 = kncot(knt / 2 )
(4.45)
The normalization constant Wn in Eq.(4.43) i s chosen to s a t i s f y
®
*
/g n(x)gn (x)dx = 1
0
where
*
(4.46)
denotes the complex conjugate. An e x p lic it expression for Wn is
given in Appendix G, Eq.(Gl).
The system of nonlinear E q s.(4 .4 4 )-(4 .4 5 ) must be solved numeri­
c a lly for a s e r ie s of roots ( kn »kn ,Yn^
of the s l a b - f i l l e d
correspond
waveguide. In practice,
to normal modes
weconsider af i n i t e
number
of inodes, say LH. In the lim it e-»l, these modes become the f i r s t LH
TE2n-l 0 modes of the emVty 9 Uide. A good i n i t i a l approximation for the
s e r ie s of propagation factors yr i s required in order to obtain conver­
gence of the numerical algorithm to the desired roots. Approximate
values of the propagation factors vn for a certain number, say LM, low
order modes are obtained by the Rayleigh-Ritz method [165] (usually
LM=5 i s s a tis fa c to r y ). Approximate values of the separation constants
kn> k^ for the higher order modes LM^n^LH are found using an asymptotic
expansion [164]. The d e t a ils of both approximations are given in Appen­
dix G. Elimination of the propagation factor y from E q s.(4 .4 4 )-(4 .4 5 )
y ield s a system of tNo nonlinear equations
h1 (k ,k ') = k, s i n [ k ( a - t ) / 2 ] s i n ( k ' t / 2 ) - k c o s ( k 't / 2 ) c o s [ k ( a - t ) / 2 ] = 0
h2 (k ,k ') = k '2- k2- kQ2 ( e - l ) = 0
(4.47)
which are solved numerically for k, k' by the Newton-Raphson method
[157], using the s e r ie s of LHapproximate k, k* pairs as i n i t i a l values
(see Appendix G for d e t a i l s ) . The
approximate i n i t i a l values for kn , k^
2
2
2
are ordered according to increasing Re(vn ) , Re(Yn+p >Re( Yn )» tsee
Eq.(4.44)] and the same condition should be s a t is f ie d by the corre­
sponding numerical solu tio n s of Eq.(4.47). If th is order i s not pre­
served, one cannot be sure that the Newton-Raphson ite ra tio n s converged
to the roots kn , k^, n = l,2 __ LH,
which represent the desired LM d i f ­
ferent modes. In particular, some of the modes may be obtained twice
while other may be omitted. In such an instance the number of modes
included into the Rayleigh-Ritz approximation i s increased in order to
calcu late more accurate i n i t i a l values and the whole procedure i s re­
141
peated.
Boundary Condition and Scattering Matrix
Since the slab i s symmetric with respect to the z=L/2 plane (see
F ig .4.14) there e x is t so lu tio n s of Maxwell's equations which are symme­
t r i c and asymmetric with respect to z=L/2 [ 6 6 ]. The solu tion with the
e l e c t r ic f ie ld being an odd function with respect to z=L/2, [ i . e . ,
Ey (z)=-Ey (L -z ), E y ( L / 2 ) = 0 ] corresponds to presence of a m etallic wall
(short c ir c u it ) at z=L/2. One can consider only half of the structure
(z^L/2) and ca lcu late the voltage r e fle c tio n c o e f fic ie n t R° defined at
z=0 for TE10 mode incident from z=-«>. Sim ilarly, when the magnetic
f ie ld i s an odd function with respect to z=L/2 [ i . e . , HX (z)=-H X (L - z ) ,
H (L/2)=0] a "magnetic wall" (open c i r c u it ) can be imagined at z= l/2;
X
and, since the e l e c t r i c f ie ld i s now an even function with respect to
z=L/2, we define the voltage r e fle c tio n c o e f fic ie n t Re at z=0. The
scatterin g matrix defined at the reference planes z=0 and z=L i s re­
lated to the voltage r e fle c tio n c o e f f ic ie n t s obtained for the two
symmetries
S11 = S22 = (r6+r0) / 2 ’
s 12
= (Re- R°>/2-
(4.48)
The transverse components of the to ta l e l e c t r i c f ie l d , Ey, and
magnetic f i e l d , Hx , are expressed as linear combinations of the normal
modes. The normal modes of the inhomogeneously-filled waveguide
[Eq.(4.43)] form a complete se t of eigenfunctions and can be used for
expansion of the f ie ld equally well as can the sine functions
[Eq. ( 4 . 4 1 ) j be used for expansion of the f ie ld in the empty waveguide
[167]. The transverse components of electromagnetic f ie ld which corre­
142
spond to the TE^Q mode of amplitude 1 incident from -» are:
( 1 ) in the empty guide, z<0 ,
00
E (x ,z ) ■ [exp(-jBz)+R 1 ex p (jB z)]f 1 (x) + E Rmexp(rmz ) f m(x)
y
m=2
(4.49)
00
ox
( x , z ) = - B [ e x p ( - j B z ) - R , e x p ( j B z ) ] f , ( x ) - j E R r e x p ( r z ) f (x)
l
i
m=2 m m
m
(4.50)
where f m are given by Eq.(4.41), 8 and rm are given by Eq.(4.42);
( 2 ) in the section of waveguide containing the slab , O^z^L/2 ,
00
Ey (x ,z ) = E Tn[exp(-Ynz) + rnexp(vnz ) ]g n(x)
n=l
(4.51)
00
»U0 Hx (x ,z ) = j E ' V n t ^ P C - V 2) " rnexp(Tnz ) l g n(x>
n=l
(4-52)
where 9 n(x) are given by Eq.(4.43) and Yn are found by solving
E q .(4.44 )-(4 .4 5) as described above. The factors rn , which express the
boundary condition at the z=L/ 2 plane
r® = exp(-vnL),
r° = -exp(-YnL) ,
(4.53)
co n stitu te the only differen ce in the form of the equation for the even
and the odd cases.
The boundary conditions at z=0, which require continuity of the
transverse components of e le c t r ic and magnetic f i e l d s , are expressed as
+ m , ! ')
m- 2
( 4 - 5">
n-x
j
- B d - R ^ f ^ x ) - 3 E Rmrmfm(«) = 3 V n ( l - rn)gn(x)
m-Z
n -i
(4.55)
We multiply both sid es of E q s.(4 .5 4 )-(4 .5 5 ) by mode functions 9 n(x) and
143
i n t e g r a t e over th e c r o s s s e c t i o n of t h e waveguide. Using t h e orth o g o ­
nality relation
a
£9n (x)gm(x)dx = 0
( 4 .5 6 )
f o r n?*m
we reduce th e boundary va lu e problem t o a system of a l g e b r a i c equa­
tions.00
(l+Bl )pln
-BCX-R^)Pjn - l n , « , P i n - n n( l - r n)Gn ,
m-z
3
2
( 4 .5 7 )
= V n (Urn)0n • n“1’2’3'
m-z
where Gr = Jgn (x)dx,
n-1,2,3...
( 4 . SB)
lGn l =1» and Pnm den ote i n t e g r a l s
pmn *
<4 -5 s >
An e x p l i c i t e x p r e s s i o n f o r pnm i s given in Appendix F.
E l im in a ti n g p ro d u c ts TnGn from E q s . ( 4 . 5 7 ) - ( 4 . 5 8 ) we o b t a i n an i n f i n i t e
system of l i n e a r a l g e b r a i c e q u a t i o n s f o r t h e r e f l e c t i o n c o e f f i c i e n t s
V
1
y/ , RmP
m“
m=l
l+r_n
l.
1
rm
m
-v_(l-r_)
n
n -i
—
n
pln
jB
1
( 4 .6 0)
1+rn-
n = l , 2 , 3 . ..
He r e p l a c e t h e i n f i n i t e summation over m by a f i n i t e sum of LH terms
and r e t a i n t h e f i r s t LH e q u a t i o n s , n = l , 2 , . . . ,LH. The r e s u l t i n g f i n i t e
system of l i n e a r e q u a t i o n s i s s olv ed n u m e ri c a l ly . Sub ro uti ne CGEFS from
t h e SLATEC l i b r a r y [166] i s used f o r t h a t purpose in our program. The
f i r s t component, R^, of t h e s o l u t i o n v e c t o r [R^.R2 . • • -RLH] i s th e
d e s i r e d v o l t a g e r e f l e c t i o n c o e f f i c i e n t f o r t h e TE^0 mode. The system of
E q .( 4 .6 0 ) i s s olv ed f o r th e even and f o r t h e odd c as e s us in g r® and r °
144
of E g.( 4 . 5 3 ) , r e s p e c t i v e l y . As th e r e s u l t , we o b t a i n t h e even Re and
t h e odd R° r e f l e c t i o n c o e f f i c i e n t s , which a r e used to e x p re ss th e
s c a t t e r i n g m a tr ix v i a E g . ( 4 . 4 8 ) .
Proper c hoice of t h e number of modes used in th e expansion, LH, i s
c r u c i a l f o r a s s u r i n g t h e r e g u i r e d accuracy o f t h e s c a t t e r i n g m a tr ix
with a re a so n a b le computation time and l i m i t e d computer memory. The
fo ll o w i n g c r i t e r i a a r e a p p l i e d f o r d e t e r m i n a t i o n of LH by our a l g o ­
r ithm :
( i ) a l l modes, which a r e no t t o t a l l y a t t e n u a t e d upon t r a v e l i n g from
one end of t h e s l a b t o th e o t h e r , a r e taken i n t o acc oun t. In
p r a c t i c e , t h e modes f o r which | r R| >0.001 [ s e e E g . ( 4 . 5 3 ) ] a r e i n ­
cl ude d.
( i i ) t h e expansion of t h e t r a n s v e r s e f i e l d s a t t h e boundary, z=0, in
t h e s e r i e s of normal modes must be s u f f i c i e n t l y complete. The
completeness of t h e expansion of th e normal modes of th e i n ho m ogen eo u s ly -f il le d waveguide i n t o t h e normal modes of t h e empty
waveguide i s t e s t e d :
LH
#
1 0 1 > E, mn*mn > 0-99
f o r n= l , 2 , . . .LH
( 4.6 1)
m -i
( i i i ) t h e computation of t h e s c a t t e r i n g para mete rs i s c a r r i e d out usi ng
LH and LH-3 modes. The d i f f e r e n c e between th e r e s u l t s s e r v e s as
an e s t i m a t e of t h e numerical e r r o r and g i v e s in fo r m at io n on th e
r a t e of convergence of th e i n f i n i t e s e r i e s .
Mhen t h e c o n d i t i o n s ( i ) and ( i i ) a r e s a t i s f i e d , the numerical e r r o r of
t h e s c a t t e r i n g pa rame te rs e s t i m a t e d in ( i i i ) i s u s u a l l y s m a l le r than
145
0.0001, i . e . a t l e a s t an or de r of magnitude s m a l le r than t h e t y p i c a l
ex perimental p r e c i s i o n .
The procedure desc ib ed above f o r c a l c u l a t i o n of th e s c a t t e r i n g
m a tr ix has been implemented as a s e t of FORTRAN s u b r o u t i n e s . The execu­
t i o n time on PDP11/34 minicomputer i s s e v e r a l seconds. Up t o 29 terms
can be used in th e modal expansion. With t h i s number of modes, s a t i s ­
f a c t o r y accuracy i s ob ta in e d when t h e d i e l e c t r i c l o s s e" of th e s l a b
does not exceed -*500 ( f o r t a a / 1 0 , £**20).
4 . 5 . 3 . Eval ua tio n of th e Complex P e r m i t t i v i t y from t h e V a ri a b le Termina­
t i o n Measurements.
Determination of th e complex p e r m i t t i v i t y , v i a measurements of
‘
s c a t t e r i n g of the TE^Q mode in r e c t a n g u l a r waveguide by a c e n t e r e d , Eplane s l a b , c o n s t i t u t e s t h e in v e r s e problem to th e one solved in the
pre vio us s e c t i o n . The two complementary methods of s o l u t i o n developed
he re a r e based on v a r i a b l e - t e r m i n a t i o n measurements and a n o n li n e a r
least-sguares estimation.
( i ) S c a t t e r i n g Matrix Approach
The s c a t t e r i n g m a t r i x SM r e p r e s e n t i n g t h e exp er iment al s l a b i s
e v a lu a te d from a s e r i e s of v a r i a b l e t e r m i n a t i o n measurements by the
l e a s t - s g u a r e s f i t t i n g a s d e s c r i b e d in Se c ti o n 4 . 3 . 1 . As th e r e s u l t , one
o b t a i n s four r e a l pa ra me ters [y1=Re(SM11) , y2=Im(SM11) , y 3 =Re(SM12),
y4=Im(SM12)] and an e s t i m a t e of t h e i r cov arian ce m a t r i x , Cy . The s c a t ­
t e r i n g m a tri x i s a f u n c t i o n of the p e r m i t t i v i t y of s l a b SC( e ) . The
value of permittivity which corresponds to the measured scattering
146
m a tr ix can be found by s o l v i n g th e fol lo wi ng system of four e qu a ti o n s
with two unknowns x^=e’ and x ^ e ” •
yi = R e f i l l ) = Retsc n ( e ) ]
=f ^ . X j ) .
y2 = l « ( ^ n ) = Im[Sc U ( e ) ]
= f 2 (xl f x2 ) ,
y3 = Re^SM12^ = Re^Sc l 2 ^ e ^
= f 3*xl ’x2 ^ ’
y4 =
= ^4^xl ,x2^ ’
= ^m^cl2^e^
^
^
In orde r t o r e t a i n a l l the inf orm at io n a v a i l a b l e from t h e measure­
ments, the p e r m i t t i v i t y e i s e st i m a te d by a n o n l i n e a r l e a s t - s q u a r e s
r e g r e s s i o n method. The i n v e r s e of th e c ovar ian ce m a tr ix Cy , e st i m a te d
fo r th e s c a t t e r i n g para mete rs S^, i s used as t h e weight m a tr ix in the
objective function
Q = [Y -F (x) ]TCy_1[Y-F(x)]
( 4.6 3)
The a lg o ri th m f o r n o n l i n e a r r e g r e s s i o n which has been developed h e re in
connection with a n a l y s i s of th e complex impedance s p e c t r a (see Appendix
B) i s adopted f o r min imiza tion of Q. The f u n c t i o n Sc ( e ) = [ f l t f 2 , f 3 , f 4 ]
in Eq .( 4 . 6 3 ) i s not ex pre sse d a n a l y t i c a l l y , but i s c a l c u l a t e d by the
computer a lg o ri th m d e s c r i b e d above. The a lg o r it h m f o r n o n l i n e a r r e g r e s ­
si o n uses th e p a r t i a l d e r i v a t i v e s of the model f u n c t i o n s with r e s p e c t
to th e par am ete rs , which must be obta in e d nu m e ri c al ly in t h i s ca s e . An
approximation of the d e r i v a t i v e s by the f i n i t e d i f f e r e n c e s i s used,
e.g .,
3Sc l l = Sc l l (e+Ae)~Sc l l ( e )
ae
(4 64)
Ae
Accuracy of t h i s approximation depends on th e s i z e of th e increment Ae
which i s used in the f i n i t e d i f f e r e n c e . I t i s a d j u s t e d such t h a t the
change of th e s c a t t e r i n g parameters i s l a r g e r than t h e i r numerical
147
p r e c i s i o n , but s m a l l e r than 0.01.
This method of d a t a re d u c t i o n i s Nell s u i t e d f o r a s i t u a t i o n when
an i n i t i a l value of th e p e r m i t t i v i t y cannot be given with a good p r e c i ­
s io n . The a p p l i e d a l g o r i t h m i s very s t a b l e and has a wide range of
convergence. In a l l c a s e s in which th e l e a s t - s q u a r e s e s t i m a t i o n was
s u c c e s s f u l , the s o l u t i o n of Eq .( 4 .6 2 ) di d no t depend on t h e i n i t i a l
guess of e and appeared t o be unique.
( i i ) D i r e c t E s tim a tio n of t h e P e r m i t t i v i t y
The i n p u t - o u t p u t r e l a t i o n f o r th e c e n t e r e d E-plane s l a b i s w r i t t e n
in terms of the s c a t t e r i n g m a tr ix [ s e e E q . ( 4 . 2 7 ) ] . The s c a t t e r i n g p a r a ­
meters a r e , in t u r n , ex pre sse d a s foll ows by f u n c t i o n s of t h e p e r m i t t i ­
v i t y e which a r e c a l c u l a t e d by th e method of Se c ti o n 4 . 5 . 2 :
o ■ £ i ( c . R M .RBi) - Sc U ( e ) ( l +RM RB i )t [Sc J ( e ) - S c l 2 ( e ) ] R Bi-RM ,
(4 .6 5)
where th e r e f e r e n c e p la n es A and B a r e a t z=0 and z=L, r e s p e c t i v e l y .
The p e r m i t t i v i t y i s e s ti m a te d d i r e c t l y from th e v a r i a b l e te rm in a ­
t i o n measurements as a parameter in E q . ( 4 . 6 5 ) . The d a t a re d u c ti o n pro ­
ceeds in a s i m i l a r way t o t h a t used in th e case of e s t i m a t i n g th e
prop ag at ion f a c t o r f o r a ho m og en eou sl y-f ill ed waveguide, as d e s c r i b e d
in Se c ti o n 4 . 3 . 2 . P a r t i a l d e r i v a t i v e s of E q .( 4. 63) with r e s p e c t to e
a r e approximated by f i n i t e d i f f e r e n c e s as in E q . ( 4 . 6 4 ) . The l e a s t s q u ar es e s t i m a t i o n of parameters in a n o n l i n e a r i m p l i c i t model, a p p li e d
h e r e , i s l e s s fo o l p r o o f than i s the case f o r e x p l i c i t model used in
( i ) . Th e re fo re , a more a c c u r a t e i n i t i a l guess fo r th e p e r m i t t i v i t y i s
r e q u i r e d and i t i s d e s i r a b l e to use method ( i ) p r i o r to an a p p l i c a t i o n
148
of method ( i i ) . The v a l u e s of th e p e r m i t t i v i t y ob ta in e d by both methods
a r e i d e n t i c a l (when no adjustm ent of c a l i b r a t i o n parameters i s made),
but th e e s ti m a te d u n c e r t a i n t i e s a r e s i g n i f i c a n t l y l a r g e r in method ( i ) .
Method ( i i ) pe rm it s a d d i t i o n a l e s t i m a t i o n of t h e c a l i b r a t i o n p a r a ­
m e te r s , p o s i t i o n s of t h e r e f e r e n c e pla nes and l o s s f a c t o r s , in a s i m i ­
l a r way to t h a t d e s c r i b e d in Se c tio n 4 . 3 . 2 . The a b i l i t y t o a d j u s t p o s i ­
t i o n s of th e r e f e r e n c e p la n es proved to be ve ry u s e f u l because samples
were pla ced in th e middle of le n g t h of a waveguide s e c t i o n and d i r e c t
c a l i b r a t i o n of the r e f e r e n c e p la n es usi ng a s h o r t c i r c u i t in p la ce of
sample was n ot f e a s i b l e .
4 . 5 . 4 . Numerical and Experimental T e s t s .
Comparison with r e s u l t s r e p o r t e d
in the l i t e r a t u r e
Electromagnetic-wave pro pa ga tio n
in j u n c t i o n s between two symmet­
r i c a l , p a r t i a l l y - f i l l e d waveguides has been i n v e s t i g a t e d by Chang
[152]. Only l o s s l e s s d i e l e c t r i c s were co n si d e re d . The r e s u l t s , ob ta in e d
by a procedure e q u i v a l e n t to t h a t used in Se c ti o n 4 . 5 . 2 , were p r e s e n te d
in th e form of an e q u i v a l e n t c i r c u i t . The r e f l e c t i o n c o e f f i c i e n t for
j u n c t i o n s between an empty and a s l a b - f i l l e d s e m i - i n f i n i t e waveguides
was c a l c u l a t e d by our program [ s e t t i n g r n=0 in E q s . ( 4 . 5 1 ) —( 4 .5 2 ) 3 . Good
agreement was o b ta in e d with th e v a lu e s c a l c u l a t e d us in g Chang's e q u i ­
v a l e n t c i r c u i t and pa rame te rs ob ta in e d from h i s graphs.
A procedure fo r c a l c u l a t i o n of th e r e f l e c t i o n and th e t r a n s m i s s i o n
c o e f f i c i e n t s fo r a f i n i t e l e n g t h , l o s s y d i e l e c t r i c s l a b in a r e c t a n g u ­
l a r waveguide was p r e s e n t e d by Liu e t
take
al
[153]. This procedure does not
advantage of the symmetry of the slab about the z=L/2 plane and
149
e q u a ti o n s s i m i l a r t o E q .( 4 .6 0 ) a r e so lved s im u lt a n e o u s ly f o r both
boundarie s of th e s l a b - f i l l e d s e c t i o n : z=0 and z=L. Sev era l examples of
r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t s were given fo r a s l a b of the
p e r m i t t i v i t y e = 3 0 -j l8 a t d i f f e r e n t f r e q u e n c i e s in t h e X-band waveguide.
He were no t a b l e to e x a c t l y reproduce t h e s e v a l u e s by our procedure.
The disagreement was on th e ord er of 5% of th e c a l c u l a t e d s c a t t e r i n g
parameter. The c a l c u l a t i o n s were re p e a te d fo ll ow in g t h e i d e n t i c a l pro­
cedure of Liu e t a l and t h e r e s u l t s were e x a c t l y t h e same as ob ta in e d
us in g our method. T h e r e f o r e , th e d is c r e p a n c y i s not due t o th e d i f f e r ­
ences in th e fo r m u la t io n of th e boundary va lu e problem and we s u s p e c t
t h a t th e examples given in r e f . 15 a r e n o t a c c u r a t e .
Synthetic data
In or der to e x p lo re th e p r e c i s i o n of th e p e r m i t t i v i t y which can be
expected from th e v a r i a b l e t e r m i n a t i o n measurements of t h e c e n te r e d Eplane s l a b , t e s t s of t h e d a t a re d u c t i o n procedure were made usi ng syn­
t h e t i c d a t a which were contaminated by random e r r o r s . At f i r s t , the
s c a t t e r i n g m a tr ix S ( e ) was c a l c u l a t e d assuming c e r t a i n dimensions and
w
th e p e r m i t t i v i t y of t h e s l a b . A s e r i e s of m p o s i t i o n s d. of the s h o r t
c i r c u i t , e q u a l l y spaced over th e h a l f - w a v e l e n g t h , was used. Computer
ge ner ate d random e r r o r s , which were uniformly d i s t r i b u t e d over an
i n t e r v a l r e p r e s e n t i n g t h e assumed ex per im ent al u n c e r t a i n t y , were then
added to d. and t o t h e r e f l e c t i o n c o e f f i c i e n t s Rg . . The in p u t r e f l e c ­
t i o n c o e f f i c i e n t s were c a l c u l a t e d f o r th e p e r t u r b e d d^ and Rg^ usi ng
th e s c a t t e r i n g m a t r i x S (e ) and ex pre sse d as VSWR, W., and th e s h i f t of
w
th e s t a n d i n g wave minimum, s^ . The v a l u e s of
X
and s^ were randomly
150
v a r i e d w it h in t h e i r i n t e r v a l of u n c e r t a i n t y . The s y n t h e t i c d a t a con­
s i s t e d of t h e e q u a l l y spaced d . , th e c o n s t a n t Rs i =Rg and th e e r r o r contaminated Wj and s ^ . The p e r m i t t i v i t y was e v a lu a te d u s in g methods
given in Se c ti o n 4 . 5 . 3 .
Some r e p r e s e n t a t i v e r e s u l t s a r e given in Table 4 .2 . The e x p e r i ­
mental c o n d i t i o n s , i n c l u d i n g th e u n c e r t a i n t i e s , were th e same as fo r
th e examples given in Se c tio n 4 .2 . A small s l a b (t=0.8mm, L=1.2mm) was
chosen t o allow s u f f i c i e n t tr a n s m i s s i o n f o r c a s e s with high v a lu e s of
th e d i e l e c t r i c l o s s . The r e a l p a r t o f p e r m i t t i v i t y was e'=20 and the
d i e l e c t r i c l o s s v a r i e d from e''=0.1 t o e"=640. The l a r g e e r r o r s of the
e st i m a te d r e a l p a r t of th e p e r m i t t i v i t y , which were found fo r the
h i g h e s t v a l u e s of l o s s ( ta n s > 1 0 ), could be expected because th e r e a l
p a r t of p e r m i t t i v i t y l o s e s i t s p h y s i c a l s i g n i f i c a n c e f o r h ig h l y conduc­
t i v e m a t e r i a l s . The imaginary p a r t of th e p e r m i t t i v i t y ( t h e c o n d u c t i v i ­
t y ) was e v a lu a te d wit h s a t i s f a c t o r y p r e c i s i o n . The va lu e of t h e d i e l e c ­
t r i c l o s s e"=640 was a t th e upper l i m i t of t h e c a l c u l a t i o n procedure,
s i n c e fo r hig he r v a l u e s of e" th e scheme f o r th e g e n e r a t i o n of normal
modes (see Se c ti o n 4 . 5 . 2 ) f a i l e d . The l i m i t e d accuracy f o r low v a lu e s
of l o s s i s due t o t h e small s i z e of th e sample.
Experimental t e s t s
Besides measurements on NASICON compounds, t e s t measurements were
made of p l e x i g l a s s , s i l i c o n and g r a p h i t e .
A s l a b of p l e x i g l a s s (t=1.69mm, 1=22.53mm) was measured a t room
tem pe rat ure in a s t a i n l e s s s t e e l waveguide a t 8.37 GHz. Without a d j u s t ­
ment of th e c a l i b r a t i o n pa ra m et e rs , th e e s ti m a te d p e r m i t t i v i t y was
151
Table 4.2
Examples of th e s c a t t e r i n g parameters c a l c u l a t e d fo r a c e n te r e d E-plane
s l a b in X-band waveguide and e r r o r s of th e p e r m i t t i v i t y e s ti m a te d usi ng
s y n t h e t i c d a t a . The dimensions of s l a b a r e t=0.8mm, L=1.2mm, frequency
10 GHz, th e r e a l p a r t of p e r m i t t i v i t y i s e'= 20, val ue of imaginary p a r t
i s given in column 1. S y n t h e t i c d a t a were contaminated by random e r r o r s
- see t e x t . The number of terms in modal expansion, s e l e c t e d by the
a l g o r i t h , was 24, with th e ex cep tio n of th e two h i g h e s t v a lu e s of e"
where 29 modes were used witho ut s a t i s f y i n g th e c o n d i t i o n fo r complete­
ne ss of expansion [s e e E q . ( 4 . 6 1 ) ] .
s c a t t e r i n g pa rameters
e"
e r r o r s of e s t i m a t i o n
no adju stment
r e f e r e n c e planes
of c a l i b r a t i o n
adjusted
ae'
a e " /e " $
ae* a e ’Ve" %
|Sl l '
argSn
|S i2l
argS12
0.1
0.2667
-116 .8
0.9620
-26.4
-0 .2 2
-18.
-0 .1 3
0.5
0.2650
-11 8 .3
0.9554
-26 .3
-0 .2 2
-3.
- 0 .1 3
- 3 .4
1.
0.2631
-1 20 .0
0.9474
-2 6.2
-0 .2 2
- 1 .1
-0 .1 2
- 1 .5
5.
0.2542
-133.6
0.8888
-2 4.9
-0.21
0.5
-0.1 1
0.2
10.
0.2569
-14 8. 6
0.8276
-2 3 .0
-0 .2 0
0.3
-0.11
0.04
20.
0.2861
-170 .6
0.7371
-1 8 .7
-0 .1 9
0 .6
-0.11
0.3
40.
0.3605
166.9
0.6369
-9.7
-0 .1 4
0.13
-0 .0 7
0.6
80.
0.4639
150.1
0.5757
4.3
0.13
0.01
0.64
-1.0
160.
0.5582
139.6
0.5757
18.0
0.95
0.01
2.72
- 1 .2
320.
0.6223
134.1
0.6027
26.8
4.47
0 .2
13.1
- 1 .2
640.
0.6623
131.9
0.6222
31.4
1.0
59.8
1.1
17.0
-19.
152
e ’= 2 . 6 1 i0 .0 2 , t a n s = ( 6 .± 1 0 .) x l O ” 3 . Agreement with th e r e s u l t ob ta in e d
us in g a sample co mpletely f i l l i n g th e waveguide c r o s s s e c t i o n (see
Sec tio n 4 . 3 . 3 ) i s s a t i s f a c t o r y i f we c o n s id e r t h a t th e measurement here
was made in a l o s s y s t a i n l e s s s t e e l waveguide and the sample was too
small f o r a s e n s i t i v e measurement of t h e d i e l e c t r i c l o s s .
The d . c . c o n d u c t i v i t y of a s i l i c o n wafer (Dow Corning, Type N),
measured a t room te mp erat ure by th e fo ur- pro be te chn iq ue [168], was
o=0.103 o ' ^ c n f 1 . A s l a b of dimensions t=0.43mm, 1=1.72mm, with gold
e l e c t r o d e s s p u t t e r e d on f a c e s c o n t a c t i n g waveguide, was measured in th e
K-band waveguide a t 24GHz. The r e s u l t s were: (1) w ith ou t adju stm ent of
c a l i b r a t i o n e '= 1 1 . 5 ± 0 . 3 , o ’=0.075±0.006 a - 1 cm_ 1 ; (2) with a d j u s t e d
p o s i t i o n s of t h e r e f e r e n c e p la n e s e '= 1 0 .9 ± 0 .2 , o ,=0.078±0.001 Q“ *cm“*.
The low val ue of c o n d u c t i v i t y measured a t 24 GHz can be a t t r i b u t e d to
chi pp in g of th e sample along i t s th e narrow s i d e s which e f f e c t i v e l y
reduced th e a r e a of t h e sample in c o n t a c t with th e waveguide w a l l s .
F i n a l l y , a t h i n s l a b of g r a p h i t e (a s l i c e from a s p e c t r o s c o p i c
e l e c t r o d e ) was measured a t 24 GHz. I t was not expected t h a t the conduc­
t i v i t y of g r a p h i t e ( 0 ^ * 6 5 0 jT ^ c n f 1 usi ng a f ou r -p ro be te ch ni qu e) could
be determined from our measurements, because th e procedure f o r c a l c u l a ­
t i o n of th e s c a t t e r i n g m a t r i x l o s e s p r e c i s i o n f o r e">500. When the
i n i t i a l val ue of th e p e r m i t t i v i t y was given w i t h i n range of the a l g o ­
rit hm ( e . g . e=50 -j4 00), th e i t e r a t i o n s f a i l e d because of th e l o s s of
p r e c i s i o n when e" became high. Thus, th e p e r m i t t i v i t y of a h i g h l y conductive m a t e r i a l l i k e g r a p h i t e cannot be determined by our method,
bu t i t a l s o cannot be mistaken f o r a lower c o n d u c t i v i t y m a t e r i a l .
V. PREPARATION OP TWO NASICON COMPOUNDS
5 .1 . P r e p a r a t i o n of NASICON Ceramics.
Dense, p o l y c r y s t a l l i n e ceramics of nominal compositions
Na-jZ^SijPO.^ and Na2 Zr2SiP 2 0 12 were prepared by mechanical mixing of
r e a g e n t s , followed by c a l c i n a t i o n , m i l l i n g , cold p r e s s i n g compacts and
s i n t e r i n g . In orde r t o optimize th e p r e p a r a t i o n procedure s e v e r a l
b a tc h e s of each composition were prepared usi ng two d i f f e r e n t grades of
S i 0 2 and
and v a ry in g te m p e rt u re s of c a l c i n a t i o n and s i n t e r i n g .
The samples used f o r th e f i n a l e l e c t r i c a l measurements and fo r the
neu tron powder d i f f r a c t i o n s tu dy were, however, c u t from two l a r g e
s i n t e r e d c y l i n d e r s , one of each composition.
D e t a i l s of th e p r e p a r a ­
t i o n of the ceramic samples used in th e p r e s e n t i n v e s t i g a t i o n a r e given
below.
Na3Z r2Si2P012
The s t a r t i n g m a t e r i a l s were Zr02 (A lf a, 99+$, 1-3 urn p a r t i c l e
s i z e ) , S i ( > 2 (A lf a, 99.9$, -325 mesh) and Na^PO^'Hl^O (M a lli nc kr odt AP)
[27]. The p r o p o r t i o n s of s t a r t i n g m a t e r i a l s were purposely made d e f i ­
c i e n t in ZrC^, as compared with the Na3Zr 2 S i 2 P0 ^ 2 composition, in order
to reduce th e amount of f r e e z i r c o n i a o f t e n p r e s e n t as a second phase
in s i n t e r e d NASICON [2 4 , 2 8 , 2 9 ] . Taking i n t o account Z ^
pickup from
th e g r i n d i n g media (e s t i m a t e d t o be l e s s than 0.5$ of t o t a l Z ^
t e n t ) , th e p r o p o r t i o n s of Na^PO^,
con­
and ZrOj were 0 .3 3 1 :0 . 2 2 7 : 0 . 4 4 2 ,
which correspond to t h e formula Na3 2Zri 89S*1 99P1 06°12‘
Na-jPO^HI^O c r y s t a l s were dehydrated a t 120°C under vacuum fo r
153
154
s e v e r a l hours. The weight change i n d i c a t e d t h a t r e s i d u a l water remained
in an amount approximated by formula Na^PO^O.5H20. Dried powders were
d i s p e r s e d in to lu e n e and thoroug hly mixed by v i b r a t o r y m i l l i n g with
dense z i r c o n i a b a l l s . The mixture was poured onto f i l t e r paper and
d r i e d under vacuum. C a l c i n a t i o n , in an open z i r c o n i a c r u c i b l e a t 1100°C
fo r 20 ho ur s, r e s u l t e d in a ha rd , p r e - s i n t e r e d compact. X-ray powder
d i f f r a c t i o n , u si ng Cu-Ka r a d i a t i o n , showed a w e l l - d e f i n e d p a t t e r n of
monoclinic NASICON [ 2 3 ] , t o g e t h e r with some d i f f r a c t i o n peaks which
were i d e n t i f i e d as being due t o un re a ct e d Zr02 . The c a l c i n e d compact
was ground in an alumina mortar t o pa ss a 50-mesh s cr ee n and v i b r a t o r y m i l l e d with to lu e n e f o r 10 hours . P o l y ( v i n y l b u t y r a l ) was added as a
bin d e r (l%wt) and t h e powder was vacuum-dried. C yl in de rs (44 mm diam x
20-30 mm high ) were formed by u n i a x i a l p r e s s i n g in a s t e e l d i e a t 80
MPa, followed by i s o s t a t i c p r e s s i n g a t 380 MPa; th e green d e n s i t y of
t h e s e compacts was about 2.2 g/cm^.
The p e l l e t s were nex t pl ace d in a platinu m c r u c i b l e and covered
with c o a r s e , c a l c i n e d powder of th e same composition. The bin de r was
burned o f f a t 560°C f o r 12 hours and compacts were s i n t e r e d a t 1235°C
f o r 12 hours. The a p p ar en t d e n s i t y of t h e s i n t e r e d m a t e r i a l , measured
by an Archimedes p r i n c i p l e method usi n g i s o b u t y l a l c o h o l , was 3.13
3
g/cm , which i s 96% of th e t h e o r e t i c a l X-ray d e n s i t y .
Na2 Zr2S iP 2°12
The s t a r t i n g m a t e r i a l s f o r t h i s compound were Zr02 ( T ra n s e lc o ,
Ferro C orp or a tio n, 1 wn p a r t i c l e s i z e , 99.7% Zr02 + Hf02 ) , Si02 (A lf a,
99.8%, c a t a l y s t s u p p o r t , amorphous), Na2C03 (M a lli nc kro dt AR) and
155
NH^HjPO^ (A lfa , ACS) [1 0 ]. The z i r c o n i a and s i l i c a chosen fo r t h i s
s y n t h e s i s u e re of s m a lle r p a r t i c l e s i z e than t h a t used fo r th e s y n th e 4
s i s of Na3Zr 2 S i 2 P0 ^ 2 > because d i f f i c u l t i e s were encountered w ith th e
s i n t e r a b i l i t y o f Na2 Zr 2 SiP 2 0 ^ 2 * Due to w ater adsorbed on th e high
s u r f a c e a re a S i 0 2 , th e i n i t i a l S i 0
2
c o n te n t of th e m a te r ia l was about
4$ l e s s than given by formula Na2 Zr 2 SiP 2 0 ^ 2 * The p ro c e ss in g of th e s e
powders in to ceramic c y lin d e r s was s i m i l a r to t h a t d e s c rib e d above. The
m ix ture was c a lc in e d in two s t e p s : f i r s t , a t 220°C fo r 5 hours and then
a t 960°C fo r 12 h o u rs. Free z i r c o n i a was p r e s e n t in th e c a lc in e d pow­
d e r. The p re sse d compacts were s i n t e r e d a t 1205°C fo r 14 hours in a
cover of powder of th e same com position. The s i n t e r i n g te m p e ratu re was
lower than in th e case of Na3Zr 2 S i 2 P0 1 2 , because com positions con­
t a i n i n g more phosphorous (b u t l e s s soda and s i l i c a ) m elt in c o n g ru e n tly
a t lower te m p e ra tu re s [3 0 ]. The measured d e n s it y was 3.07 g/cm3 , which
i s 95% of th e t h e o r e t i c a l X-ray d e n s i t y . When samples were immersed in
is o b u ty l a lc o h o l fo r s e v e r a l m inutes d u rin g th e d e n s it y measurement,
th e weight in c re a s e d so t h a t th e Archimedes d e n s it y was 3.13 g/cm3 .
Such beh avior i n d i c a t e s th e p resen ce of open pores in th e x=l samples.
5.2 X-rav Powder D i f f r a c t i o n . M ic r o s tr u c tu r e C h a r a c te r iz a t io n and
Chemical A n a ly s is .
X-rav Powder D i f f r a c t i o n
An X-ray powder d i f f r a c t i o n method was used to c h a r a c t e r i z e th e
form ation of th e c r y s t a l l i n e compounds a t v a rio u s s ta g e s of th e prepa­
r a t i o n p ro ced u re, th e measurements being made w ith a Rigaku d i f f r a c t o ­
156
meter and Cu Ka r a d i a t i o n .
The s i n t e r e d m a te r ia l was examined by
c o l l e c t i n g X-ray d iff ra c to g r a m s from p o lis h e d s u r f a c e s of th e p e l l e t s
and from s u r f a c e s exposed by s e c tio n in g .
The X-ray powder d iff ra c to g r a m of Na3 Zr 2 S i 2 P01 2 showed a w e lld e fin e d p a t t e r n , in q u a l i t a t i v e agreement w ith th e i n t e n s i t i e s and
p o s i t i o n s c a lc u la te d [169] fo r th e m onoclinic s t r u c t u r e proposed by
Hong [1 0 ].
The two main d i f f r a c t i o n l i n e s of m onoclinic z ir c o n ia
[(110) a t 2e = 28.24° and (111) a t 20 = 31.57°] were b a re ly v i s i b l e in
d iff ra c to g r a m s recorded u sin g a con tin uo us scan (1 /2 ° per m in).
o rd e r to determ ine th e amount of Zr 0
2
In
in th e s i n t e r e d Na3 Zr 2 S i 2 P0 1 2 , an
i n t e r n a l s ta n d a rd method was used in which known amounts of m onoclinic
z ir c o n i a powder (0 .5 to 4 wt$) were in c o rp o ra te d in to f i n e l y ground
samples.
The i n t e g r a t e d i n t e n s i t i e s of th e (111) and (110) d i f f r a c t i o n
l i n e s fo r m onoclinic Zr 0
2
were o b ta in e d u sing s te p scan ( s te p width
0 .0 2 ° , p r e s e t tim e 4 s e c ) fo r th e unknown sample and th e unknown sample
( c o n ta in in g a known a d d i t i o n o f Zr02 ).
l i n e a r r e g r e s s io n .
The r e s u l t s were c o r r e l a t e d by
The amount of z i r c o n i a determ ined based on (110)
i n t e n s i t y was 1% and t h a t based on (111) i n t e n s i t y was ^ 1.6fc.
Me
concluded t h a t th e amount of f r e e m onoclinic z i r c o n i a in th e
Na3 Zr 2 S i 2 P0
^ 2
sample was below
2
$ by w eight.
An X-ray powder p a t t e r n of s i n t e r e d Na2 Zr 2 SiP 2 0 ^ 2 was in q u a l i t a ­
t i v e agreement w ith th e rhombohedral s t r u c t u r e proposed by Hong[10].
The main d i f f r a c t i o n l i n e s fo r Zr0 2 were no t d e te c te d , which, in view
of th e i n t e r n a l s ta n d a rd c a l i b r a t i o n d e s c rib e d above, means t h a t th e
amount of f r e e m onoclinic Zr0 2 in th e sample was below 1$ by w eight.
A
157
p r e c is e c a l c u l a t i o n o f th e l a t t i c e param eters based on X-ray powder
d i f f r a c t i o n was not undertaken because th e s t r u c t u r e of t h i s compound
was s tu d ie d in d e t a i l by neutro n powder d i f f r a c t i o n (see Chapter VI).
M ic r o s tr u c tu r e C h a r a c te r iz a t io n
The sam ples, mounted in a p l a s t i c r e s i n , were c a r e f u l l y p o lis h e d
u sin g a Minimet (B uehler Co.) p o lis h e r w ith diamond p o lis h in g compounds
down to 0.25 urn.
The x=2 compound re q u ire d only medium p re s s u re du rin g
p o lis h in g , whereas th e x=l compound had to be p o lis h e d w ith a low
a p p lie d p re s s u re in o rd e r to p rev en t c r e a t i o n of la r g e
u l a r l y shaped v oid s in th e s u r f a c e .
( > 2 0
urn), i r r e g ­
The p o lis h e d s u r f a c e s were exam­
ined w ith an o p t i c a l microscope u sing r e f l e c t e d , p o la r iz e d l i g h t .
The p o lis h e d s u r f a c e of th e x=2 compound (see P i g . 5 . l a ) e x h ib ite d
rounded v o id s , s e v e r a l m icrom eters in s i z e .
The p e rc en ta g e of th e a re a
occupied by v o id s appeared t o be dependent on th e p o lis h in g procedure
and was always l a r g e r than should be expected on th e b a s i s of th e
measured v a lu e s of th e r e l a t i v e d e n s it y .
I t was concluded t h a t th e
m a jo rity of v o id s was c r e a te d d u rin g p o lis h in g and was no t r e p r e s e n t a ­
t i v e of th e i n t e r n a l p o r o s ity of th e ceramic sample.
The s m a ll, w hite
s p o ts v i s i b l e in F i g . 5 . l a a r e in c lu s io n s of a second phase (most l i k e l y
f r e e z i r c o n i a ) . In th e case of th e x=l compound, th e p r e f e r e n t i a l r e ­
moval of g r a in s from th e s u rfa c e du rin g p o lis h in g was even more s e r i ­
ous, and only a f t e r s e v e r a l hours of p o lis h in g a t very low p re s su re
could a s u rfa c e l i k e in F i g . 5 . 2a be produced.
ly shaped and have sh arp c o rn e rs .
The v o id s a re i r r e g u l a r ­
a)
b)
F i g . 5 .1 . O p tic a l micrographs of th e sample of Na3 Zr 2 S i 2 P01 2
a) s u r f a c e p o lis h e d u sin g 1/4 urn diamond compound,
b) etch ed in d i l u t e d HF fo r 4 sec .
Bars denote 10 urn.
.5 .2 . O p tic a l m icrographs of th e sample of Na2 Zr 2 SiP 2 0 ^ 2
a) s u rfa c e p o lis h e d u sing 0.25 urn diamond compound,
b) etched in d i l u t e d HP fo r 90 s e c . to re v e a l g r a in b o u n d a rie s.
Bars denote 10 urn.
160
The p o lish e d s u r f a c e s of th e samples were etch ed w ith d i l u t e d HF
(•''1%) and examined under th e microscope.
In th e case of th e x=2 com­
pound, a f t e r e tc h in g fo r only 4 seconds,
a f i n e mosaic o f g r a in s was
re v e a le d (see F i g . 5 . l b ) .
in s i z e .
The g r a in s a r e i r r e g u l a r l y - s h a p e d and 2-4 um
Sample of th e x=l compound had to be etched lo n g er (1 .5 min)
to re v e a l t h e i r g r a in bou nd aries (see F i g . 5 . 2b ).
The g r a in s of t h i s
compound a re l a r g e r (5-10 tun) than th o se of th e x=2 compound and have
shapes s i m i l a r to th e v o id s developed d u rin g p o lis h in g . The g r a in s
appear to be weakly bonded a s evidenced by t h e i r p r e f e r e n t i a l removal
d u rin g p o lis h in g .
The m icrohardness was measured u sing a L e itz M iniload Hardness
T e s te r .
The t r a c e s o f in d e n ta tio n s on th e x=2 sample had r e g u la r d i a ­
mond shapes and th e V ickers hard ness was 288±4 w ith 1 kg lo a d .
In th e
case of th e x=l sample some c ra c k s o ccurred in th e immediate a re a
around th e in d e n ta tio n w ith th e r e s u l t t h a t th e s i z e of th e in d e n ta ­
t i o n s could only be measured w ith low p r e c i s i o n .
The a p p a re n t V ickers
hardn ess of t h i s compound was 160+25 w ith a 300g load.
Chemical A n a ly sis
The bulk chemical a n a l y s i s was done by an o u ts id e l a b o r a to r y
(S h arp-S chu rtz Co., G r a n v ille , Ohio).
An X-ray flu o r e s c e n c e tech niqu e
was a p p lie d , using samples fused w ith a lith iu m b o ra te f lu x .
mated p r e c i s i o n s
P2 O5 , 0.4$ fo r
(1
The e s t i ­
s ta n d a rd d e v ia t io n ) were 0.2$ f o r Na2 0 , S i 20 and
The a n a l y s i s y ie ld e d com positions which can be
exp ressed by fo llo w in g form ulas: fo r x= 2 compound
N a 3.31Zr1.91S2.01P0.99°11.97* for X=1 C0">P°und
161
N a 2
0 2 Z r 2
0 2 S i 0
97P2
0 1
°1 2 '
R e s u lts *or both compounds a re in good
agreement w ith th e s t a r t i n g co m positions.
A sm all d i f f e r e n c e between
th e s t a r t i n g com position (se e S e c tio n 5 .1 ) and th e r e s u l t i n g chemical
a n a l y s i s fo r th e x= 2 compound can be accounted fo r assuming a l o s s of
P2 0,j d u rin g f i r i n g a t 1235°C and an u n c e r t a i n t y in th e i n i t i a l Na^PO^
c o n te n t due to p a r t i a l d e h y d ra tio n of th e Na3 P04 «12H2 0.
The aforemen­
tio n e d chemical formula f o r th e x= 2 compound i s w r i t t e n in th e form
proposed by C l e a r f i e l d e t a l [3 3 ].
Chemical m ic ro a n aly se s of both com positions were done u sin g th e
energy d i s p e r s i v e X-ray s p e c tro m e te r, EDS, (N uclear Diodes, Edax I n t . )
in th e scanning e l e c t r o n microscope (S te re o sc a n S4, Cambridge
S c ie n tific I n s tr .) .
The a n a l y s i s confirmed u n ifo r m ity of th e composi2
t i o n averaged over a r e a s
400 urn . When th e e l e c t r o n beam was concen­
t r a t e d on s m a lle r a r e a s (beam c u r r e n t 180 uA a t 10 KV), a ra p id i n ­
c re a s e in th e i n t e n s i t y of th e Na Ka c h a r a c t e r i s t i c r a d i a t i o n was
observed and v i s i b l e r a d i a t i o n damage o c cu rred ; t h i s e f f e c t i s a s s o ­
c i a t e d w ith th e high m o b ility of Na* ions in NASICON.
The mobile Na*
ions a r e a t t r a c t e d by th e n e g a tiv e charge of stopped e l e c t r o n s to th e
volume producing X -rays.
Using th ic k g r a p h ite c o a tin g of th e sam ples,
i t was p o s s ib le to o b ta in s te a d y X-ray i n t e n s i t i e s , b u t only when th e
2
e l e c t r o n beam was scanning a r e a s of th e o rd e r of 400 urn or l a r g e r .
A q u a n t i t a t i v e m ic ro a n a ly s is by an energy d i s p e r s i v e X-ray method
was a tte m p te d , b u t th e r e s u l t s were u n r e l i a b l e d e s p it e th e use of s t a n ­
dard s mounted and p o lis h e d to g e th e r w ith th e NASICON samples and a p p l i ­
c a tio n of ZAF method (atom ic number Z, a b s o r p tio n A and flu o r e s e n c e F
162
c o r r e c t i o n s ) f o r c a l c u l a t i o n o f com position [170].
The s ta n d a rd s were
z irc o n (ZrSiO^), a l b i t e (NaAlSigOg)f a p a t i t e [ (CaP)Ca^(P04 ) 3 ] , z i r c o n i a
(Zr02 ) (98* p a r t of a c r u c i b l e ) and q u a rtz (S i0 2 ) .
The chemical compo­
s i t i o n s o f a p a t i t e and a l b i t e were known from independent a n a ly s e s .
The d i f f i c u l t y in a q u a n t i t a t i v e a n a l y s i s of NASICON by th e energy
d i s p e r s i v e microprobe i s caused by o v e rla p between Zr La and P K c h a ra
a c te r is tic lin e s.
T his n e c e s s i t a t e s use of Zr Ka l i n e fo r de term in a -
t i o n of zirconium and s u b t r a c t i o n o f th e c a l c u l a t e d Zr Lcl3 i n t e n s i t y
from th e combined Zr L , P K peak.
ol
cl
U n fo rtu n a te ly , th e r e was no a g re e -
ment between th e expected com positions and th e measured i n t e n s i t y
r a t i o s f o r Zr Ka l i n e r a d i a t e d by th e two s ta n d a rd s (Zr0 2 and ZrSiO^)
and th e two NASICON compounds, th e d isagreem ent being l a r g e r than 20*.
This d is c re p a n c y may be caused by th e poor q u a l i t y of s ta n d a rd s or
by th e in a cc u ra c y of th e a p p lie d ZAF c o r r e c t i o n fo r a heavy elem ent in
a m a trix c o n s i s t i n g o f l i g h t elem ents.
The samples were a l s o examined by a Scanning Auger Microprobe
(P h y sic a l E l e c tr o n ic s I n d u s t r i e s , I n c . ) .
The Auger s p ec tro sc o p y was
performed on p o lis h e d s u r f a c e s of th e sample.
The e f f e c t s of high
sodium m o b ility in th e sample were ag ain a p p a re n t. The d e r i v a t i v e of
th e Na (986V) peak d e crea se d w ith tim e, a dim inution by a f a c t o r of
o n e -h a lf of th e i n i t i a l i n t e n s i t y being observed a f t e r about a 4 minute
exposure to th e e l e c t r o n beam. A fte r s e v e r a l m inutes th e sample ap­
peared to c o n ta in no sodium. In o rd er to prove t h a t sodium was not
e v a p o ra tin g from th e sample due to lo c a l h e a tin g by th e e l e c t r o n beam,
a mass s p e c tro m e te r was used, w ith th e r e s u l t t h a t no sodium was de-
163
t e c t e d to le av e th e sample when th e h ig h e s t a v a i l a b l e e l e c t r o n beam
c u r r e n t s were employed.
I t was concluded t h a t th e d e c re a se in sodium c o n te n t in th e >'>30 &
th ic k s u rfa c e la y e r which e m itte d Auger e l e c t r o n s was caused by Na+
ions m ig ra tio n from th e s u r f a c e in to th e i n t e r i o r of th e sample where
th e n e g a tiv e charge o f th e stopped e l e c t r o n s had accum ulated.
Determi­
n a tio n of th e sodium c o n te n t by th e Auger e l e c t r o n s p ec tro sc o p y was,
t h e r e f o r e , n o t p o s s ib le . The p h o s p h o r o u s - to - s ilic o n - to - z ir c o n iu m r a t i o
e stim a te d from th e i n t e n s i t i e s of th e Auger peaks ( c o r r e c te d by s e n s i ­
t i v i t y f a c t o r s ) a g re e d , w ith in th e e x p erim en tal u n c e r t a i n t y ( 2 0 %) w ith
th e bulk chemical com position fo r both th e x=l and th e x= 2 compounds.
V I. TIME-OF-FLIGHT NEUTRON POWDER DIFFRACTION AND STRUCTURE REFINEMENT
BY THE RIETVELD METHOD.
6 .1 . Experim ental P ro c e d u re .
Neutron powder d i f f r a c t i o n d a ta were c o l l e c t e d by th e t i m e - o f - f l i g h t
techn iq ue [171] on th e General Purpose Powder D iffra c to m e te r (GPPD) a t
th e In te n s e Pulsed Neutron Source (IPNS) a t Argonne N atio n al L aborato­
ry . The IPNS i s a pro ton a c c e l e r a t o r pulsed s p a l l a t i o n neu tro n so u rc e.
High-energy n e u tro n s , g e n erate d in cou rse of i n t e r a c t i o n between th e
proton beam and a heavy-m etal t a r g e t , a re moderated to e p ith e rm a l and
therm al e n e r g ie s by a p o ly e th e le n e m oderator. In th e t i m e - o f - f l i g h t
tech n iq u e fo r powder n eu tro n d i f f r a c t i o n , th e d i f f r a c t e d n e u tro n s a re
c o l l e c t e d a t fix e d a n g le s as a fu n c tio n of th e time of f l i g h t of
n eutron from th e pu lsed so urce to th e d e t e c t o r . A powder d i f f r a c to m e te r
c o n s i s t s o f m e ch an ically tim e-focu sed d e te c to r a r r a y s in r i g h t - and
le f t- h a n d m irro r image p o s i t i o n s a t nominal s c a t t e r i n g a n g le s (2e=151°
and 2e=90° in th e case of GPPD).
The ceramic samples of Na2 Zr 2 SiP 2 0 j 2 and Na3 Zr 2 S i 2 P01 2 were in
form of c y l i n d e r s , 12 mm in d ia m e ter. Two or th r e e c y l i n d e r s , c u t from
th e same s i n t e r e d p e l l e t , were sta c k e d to g e th e r to th e t o t a l le n g th of
about 70 mm. The samples were mounted in a h o ld e r , which was equipped
with h e a tin g elem ents and therm al s h i e l d s . The h o ld e r assembly was
p o s itio n e d in to th e neu tro n beam and th e d i f f r a c to m e te r chamber was
evacuated. The n eu tro n d i f f r a c t i o n p r o f i l e s were c o l l e c t e d fo r each
sample a t room tem p e ratu re (25±5°C) and a t 300+10°.
164
The u n c e r ta in t y of
165
th e tem perature ta k e s in to account d is ta n c e between th e sample and th e
measuring thermocouple. Data were c o l l e c t e d fo r approxim ately 12 hours
a t each tem p eratu re.
An o p e ra tin g e r r o r a t th e a c c e l e r a t o r , which p ro v id es th e protons
fo r h i t t i n g th e s p a l l a t i o n source which produces th e n e u tro n s , gave fo r
a l l of our runs a s p u r io u s , a d d i t i o n a l background which o r i g i n a l l y
caused much tr o u b l e in r e f i n i n g th e d i f f r a c t i o n d a ta . Subsequently, the
primary d i f f r a c t i o n d a ta were e m p ir ic a lly c o r r e c te d fo r t h i s a d d itio n a l
background. A fte r t h i s c o r r e c t io n was made, th e r e s u l t s became s e l f
c o n s i s t e n t between th e refin e m e n ts o f th e d a ta c o l l e c t e d in back r e ­
f l e c t i o n (around 151° in 2e) and a t r i g h t a n g le s (around 90° in 2e) in
th e v a rio u s d e te c to r a r r a y s . The re fin e m e n ts of th e back r e f l e c t i o n
d a ta provided th e b e s t r e s o lu tio n and was used as a b a s i s fo r s t r u c t u r e
d e te rm in a tio n .
A ll c a l c u l a t i o n s were based on R i e t v e l d 's method fo r c r y s t a l
s t r u c t u r e refinem ent u sin g powder p r o f i l e i n t e n s i t i e s [172]. Computer
programs developed f o r th e t i m e - o f - f l i g h t tech n iq u e [173,174] were
used. C a l ib r a tio n p aram eters fo r th e d i f f r a c t o m e t e r , based on r e f i n e ­
ment of S i ( > 2 s t r u c t u r e , were provided by th e IPNS s t a f f . The s t a r t i n g
param eters of th e c r y s t a l s t r u c t u r e s fo r th e refin e m e n ts were taken
from Hong [10 ]. The number of p a ra m ete rs, e stim a te d by th e l e a s t sq u ares f i t t i n g o f th e experim ental d i f f r a c t i o n p r o f i l e s , was in c re ase d
step w ise in th e c o n se c u tiv e c y c le s of re fin e m e n t. The f i n a l refin e m e n ts
were performed w ith a n i s o t r o p i c tem perature f a c t o r s fo r a l l th e atoms
( in th e Tables 6.2 and 6.3 th e corresponding e q u iv a le n t i s o t r o p i c tern-
166
p e r a tu r e f a c t o r s , B, a r e r e p o r t e d ) . For th e rhombohedral c a s e s , th e
number of r e f in e d p aram eters was 48 (10 of which were param eters de­
s c r i b i n g th e background and th e d i f f r a c t i o n l i n e p r o f i l e ) . For th e
m onoclinic c a s e , th e number o f param eters was 114 (9 of which were
p r o f i l e p a ra m e te rs ). The number of deg rees of freedom ranged from 3106
to 3187, th e range o f d -v a lu e s of r e f l e c t i o n s in clu d ed in th e f i n a l
refin em en t was from 0 .6 7 t o 2.29 8. This corresponded to about 1220
d i f f e r e n t observed Bragg r e f l e c t i o n s ,
fo r th e rhombohedral s t r u c ­
t u r e s in space group R32c, and to 2026 Bragg r e f l e c t i o n s ,
fo r
th e m onoclinic s t r u c t u r e in space group C2/c. A comparison of th e
o bserved, th e c a l c u l a t e d and th e d i f f e r e n c e d i f f r a c t i o n p a t t e r n s i s
given in F i g . 6.1 fo r th e most complex case of th e m o n o c lin ic , roomte m p e ratu re phase of Na.jZr2S i 2P0^2 .
6 .2 . R e s u lts and D is c u s s io n .
The c e l l c o n s ta n ts and th e R f a c t o r s a r e given in Table 6 .1 . The
s t r u c t u r a l p a ra m e te rs , in c lu d in g th e p o s i t i o n a l c o o rd in a te s and th e
observed occupancy f a c t o r s , a r e given in Tables 6.2 and 6 .3 fo r th e
rhombohedral and m onoclinic p h a se s, r e s p e c t i v e l y . S e le c te d in te ra to m ic
d i s t a n c e s , c a l c u l a t e d fo r th e r e f in e d s t r u c t u r e s , a r e l i s t e d in Tables
6.4 and 6 .5 .
The rhombohedral c r y s t a l s t r u c t u r e s a r e e s s e n t i a l l y a s d e s c rib e d
by Hong [1 0 ], and th e m onoclinic s t r u c t u r e of Na.jZr2S i 2POj2 i s very
c lo s e to th e p ro p o sa l made by him. The arrangem ent of th e Zr c o o rd in a ­
t i o n o ctah e d ra and th e S i,P c o o rd in a tio n t e t r a h e d r a i s shown in F ig s .
167
F i g . 6.1. R ie tv e ld refin e m e n t p r o f i l e f o r th e room -tem perature phase of
N a ^ Z ^ S ijP O ^ . Data (denoted by p lu s marks) were c o l l e c t e d in
back r e f l e c t i o n (2e=151°). The s o l i d l i n e i s th e b e s t - f i t
p r o f i l e c a l c u l a t e d w ith a n i s o t r o p i c te m p e ratu re f a c t o r s . Tick
marks below th e p r o f i l e i n d i c a t e th e p o s i t i o n s of a l l allowed
Bragg r e f l e c t i o n s . A d i f f e r e n c e curve (observed minus c a lc u ­
l a t e d ) appear a t th e bottom. The background was f i t as p a r t of
re fin e m e n t, b u t has been s u b s tr a c te d b e fo re p l o t t i n g .
NASIC0N,X=2,RT,ANIS0
COUNTS.
(/i
0.0
1000.0
Ro ™
§'
CJ
2000.0
3000.0
4000.0
NASICON,X=2.RT,ANIS0
ill niiiiiirjisiiiimiiiiiiaiii
0.665
0.718
0.771
0.824
0.877
1.036
0.930
-
1.064
1089
1.117
ittiimtitaeiniiiiiiHitiiHiiiiii i
1.170
1223
1276
mi m i i
in iiiiiiiiiiin i
1.382
1.435
U SB
d-SPACING (4)
d-SPACING (/0
NASICON,X=2,RT,ANIS0
§§
O o8
o
1000.0
—J
2000.0
COUNTS
3000.0
4000.0
NASIC0N,X=2,RT,ANIS0
o
o
1.516
1569
1.622
1675
d-SPACING (X)
1.728
1.781
1.B34
1.8B7
1916
1.969
2.022
2.075
d-SPACING (/S)
2.128
2.181
2.234
16B
1463
2.287
Table 6.1
C ell c o n s ta n ts , c e l l volumes and r e s i d u a l s fo r Na2Zr2SiP20 j 2 (*=1) and Na3Zr2S i2P012 (x=2)
a t room tem perature and a t 300°C. The space group i s R32/c fo r a l l cases except x=2 a t RT,
where i t i s C2/c. In t h i s case th e second l i n e g iv e s th e c e l l c o n s ta n ts transform ed to a
pseudorhombohedral s e t t i n g (by 0 .5 -0 .5 0 / 0 1 0 / 1 0
3 ). The R v a lu e s a re d e fin e d a s:
R»p ■ t S t ^ V o i - ^ c i ^ W o l ' 0 ' 5
a
b
c
a
B
1579.7
8.9348(2)
8.9348(2)
22.8486(3)
90
1
300°C 1588.7
8.9309(2)
8.9309(2)
22.9998(3)
90
2
Room
X
T
1
Room
2
V
Y
RI
R.,_
wp
90
120
3.50
3.28
90
120
4.10
3.28
90
2.15
2.94
4.64
3.27
1086.5 15.6513(17)
9.0550(3)
9.2198(11)
90
1629.8
9.0410(12)
9.0550(3)
23.0019(26)
90
89.378(6) 120.049(6)
9.0580(4)
9.0580(4)
23.0705(7)
90
90
300°C 1639.3
123.742(5)
120
Table 6.2
S t r u c t u r a l param eters fo r th e rhombohedral phases: Na2Zr2SiP2012 a t room tem perature ( f i r s t
l i n e ) ; Na2Zr2SiP2012 a t 300°C (second l i n e ) ; Na3Zr2S i2P012 a t
Atom
S it e
Symmetry
Max.
Occ.
Obs.
Occ.
3 0 0
°C ( t h i r d l i n e ) .
X
y
z
0
0
0
0
0
0
0
0
0
6 .7 (7 )
11(1)
12(4)
0
0
0
1/4
1/4
1/4
3 .4 (9 )
5 .5 (5 )
8(1)
0
0
0
0.1471(1)
0.1473(1)
0.1483(1)
0 .8 3 (3 )
1 .1 0 (5 )
0 .8 3 (4 )
0
0
0
1/4
1/4
1/4
1 .10 (5 )
1 .45 (7 )
1 .41 (8 )
Beq
N a(l)
6(b)
3
1/6
0.0 96 (7 )
0.0 82 (9 )
0.066(13)
Na(2)
18(e)
2
1/2
0.1 16 (8 )
0 .114(11)
0.282(16)
Zr
12(c)
3
1/3
1/3
P,Si
P,Si
Si ,P
18(e)
2
1/2
1/2
0.2912(2)
0.2903(2)
0.2904(3)
0(1)
3 6 (f)
1
1
1
0.1941(2)
0.1954(2)
0.1952(3)
0.1709(2)
0.1713(2)
0.1705(3)
0.0895(1)
0.0906(1)
0.0922(1)
1 .8 (1 )
2 .4 (1 )
2 .2 (1 )
0(2)
3 6 (f)
1
1
1
0.1787(2)
0.1761(2)
0.1712(3)
0.9745(2)
0.9714(2)
0.9644(3)
0.1949(1)
0.1954(1)
0.1946(1)
2 .6 (1 )
4 .0 (1 )
2 .9 (1 )
0.637(1)
0.635(2)
0.6 37 (3 )
0
0
0
Table 6.3
S t r u c t u r a l param eters fo r th e m onoclinic Na2 Zr 2 S i 2 POj2 phase.
Atom
S ite
Symmetry
Obs.
Occ.
X
y
z
N a(l)
4(d)
I
1/2
0.2 2 (3 )
1/4
1/4
1/2
Na(2)
4 (e)
2
1/2
0 .5 5 (3 )
1/2
0.891(1)
1/4
Na(3)
8(f)
1
1
0 .3 2 (4 )
0.836(2)
0.079(4)
0.842(5)
8(1)
Zr
8(f)
1
1
1
0.1015(2)
0.2472(7)
0.0539(4)
1 .1 (1 )
S i,P (1 )
4(e)
1/2
1/2
S i,P ( 2 )
8(f)
1
1
0.3569(4)
0.1117(7)
0.2586(8)
1 .5 (2 )
0(1)
8(f)
1
1
1
0.1466(4)
0.4366(6)
0.2226(6)
1 .8 (2 )
0(2)
8(f)
1
1
1
0.4374(4)
0.4439(6)
0.0821(6)
2 .6 (2 )
0(3)
8(f)
1
1
1
0.2527(4)
0.1812(6)
0.2028(8)
2 .4 (3 )
0(4)
8(f)
1
1
1
0.3812(4)
0.1337(6)
0.1116(6)
2 .1 (2 )
0(5)
8(f)
1
1
1
0.4491(4)
0.1809(7)
0.4369(7)
2 .7 (2 )
0(6)
8(f)
1
1
1
0.0806(4)
0.1475(6)
0.2444(6)
2 .3 (2 )
Max.
Occ.
0
0.0392(9)
1/4
B
eq
21(2)
3 .6 (4 )
1 .5 (2 )
Table 6.4
In te ra to m ic d is ta n c e s (£) fo r th e rhombohedral s t r u c t u r e s of Na2Zr2SiP20 j 2 a t room tempera­
t u r e (1RT), a t 300°C (1HT) and fo r N a ^ Z r ^ ^ P O ^ a t 300°C (2HT).
1RT
6 x N a (l)-0 ( l)
1HT
2.622(2) 2.657(2)
2HT
2.703(2)
1HT
2HT
3 x Z r-0 (l)
2.103(2) 2.102(2) 2.111(3)
3xZr-0(2)
2.039(2) 2.041(2) 2.037(2)
2 x N a(2 )-0 (l)
2.438(2) 2.436(4)
2 xN a(2)-0(l)
2.533(8) 2.525(15) 2.584(8)
2xNa(2)-0(2)
2.791(3) 2.816(4)
2.823(3)
2 x P ,S i- 0 ( l)
1.549(2) 1.548(2) 1.577(3)
2xNa(2)-0(2)
2.900(7) 2.890(14) 2.870(7)
2 x P ,S i-0 (2 )
1.555(2) 1.556(2) 1 .598(3)
means
2.666(5) 2.671(9)
2.465(3)
1RT
2.686(5)
means
means
2.071
2.072
2.074
1.552(2) 1.552(2) 1.588(3)
Table 6.5
In te ra to m ic d i s t a n c e s (8) fo r fo r th e m onoclinic room tem perature s t r u c t u r e of N a ^ Z ^ S ijP O ^
2 x N a (l)-0 ( l)
2.550(4)
2x N a(2 )-0 (l)
2.478(4)
Zr-O (l)
2.154(7)
2 x N a (l)-0 (3 )
2.723(5)
2xNa(2)-0(4)
2.698(9)
Z r-0 (2)
2.022(8)
2 x N a(l)-0(6 )
2.833(5)
2xNa(2)-0(5)
2.615(6)
Z r-0(3 )
2.057(5)
2.702(5)
2xNa(2)-0(6)
2.555(9)
Zr-0(4)
2.006(6)
2.587(14)
Z r-0(5 )
2.102(5)
mean
mean
N a(3)-0(1)
2 .5 3 (2 )
2 r-0 (6 )
2.155(6)
N a(3)-0(2)
2 .2 4 (4 )
mean
2.083(6)
N a(3)-0(3)
2.4 9 (4 )
S i,P ( 2 ) - 0 ( 1 )
1.599(8)
N a(3)-0(3)
2 .6 5 (3 )
S i,P ( 2 ) - 0 ( 3 )
1.545(5)
N a(3)-0(5)
2 .6 3 (3 )
S i,P ( 2 ) - 0 ( 4 )
1.611(6)
S i,P (l)-0 (2 )
1.554(7)
N a(3)-0(6)
2 .7 8 (3 )
S i ,P ( 2 ) - 0 ( 5 )
1.594(8)
S i,P (l)-0 (6 )
1.621(6)
mean
2 .5 5 (3 )
mean
1.587(7)
mean
1.588(7)
174
6.2 and 6.3 fo r th e 300°C phase o f Na^Zr2Si2P0^2- The two kinds o f Na
s i t e s a re d is p la y e d a s c i r c l e s . The c o o rd in a tio n of th e N a (l) s i t e by
th e s i x Na(2) s i t e s i s v i s u a l i z e d in P i g . 6 .3 . The m onoclinic d i s t o r ­
t i o n , which o ccurs in t h i s compound a t room te m p e ra tu re , i s sm all and
would not be n o tic e a b le in th e s e draw ings.
In o rd e r to check th e c o n s is te n c y of s t r u c t u r e re fin e m e n ts , th e
c a lc u la te d in te r a to m ic d i s t a n c e s a r e compared w ith th e corresponding
v a lu e s in s i m i l a r s t r u c t u r e s . The mean bond le n g th Zr-0 fo r th e four
s t r u c t u r e d e te r m in a tio n s (th e x=l and th e x-2 compounds a t r . t . and a t
300°C) i s 2.075 8 , t h a t i s s l i g h t l y s h o r t e r than th e 2.08 8 (o r 2.10 8
i f we assume th e mean c o o rd in a tio n of th e oxygen atoms to be 4) which
i s th e sum of th e io n ic r a d i i o f Zr and 0 [175]. The mean t e t r a h e d r a l
bond le n g th s : ( P , S i ) - 0 equal 1.552 8 in N a jZ ^ S iP jO j^ and ( S i , P ) - 0
equal 1.588 8 in N a ^ Z ^ S ijP O ^ a r e much too s h o r t when compared w ith
r e l i a b l e v a lu e s from th e l i t e r a t u r e . The mean in te r a to m ic d is ta n c e S i-0
fo r 50 o r t h o s i l i c a t e g rou ps, from p r e c i s e l y determ ined c r y s t a l s t r u c ­
t u r e s , i s 1.636 8 , w ith a v a r i a t i o n range from 1.622 to 1.654 8 [176].
The mean in te ra to m ic d is ta n c e fo r 64 ortho ph osph ate groups i s 1.536 A°,
w ith th e v a r i a t i o n range from 1.516 to 1.548 8 [177].
I f we ta k e th e s e
v a lu e s as a measure t o c a l c u l a t e by i n t e r p o l a t i o n th e P:Si r a t i o in
Na2Zr2Si?20^2 we o b ta in 2 .5 2 :0 .4 8 in s te a d o f 2 .1 , whereas fo r
Na3Zr2S i 2P012 we g e t 1 .7 4 :1 .2 6 in s te a d of 1 :2 . However, th e chemical
a n a ly s e s of th e samples (see S e c tio n 5 .2 ) i n d i c a t e th e sim ple composi­
t i o n r a t i o s . A s i n g l e c r y s t a l s t r u c t u r e d e te rm in a tio n of N a ^ Z ^ S i^ O ^
y ie ld e d a mean S i- 0 d is ta n c e of 1.625 8 [2 2 ] , whereas fo r Na^ZrP^O^
175
F i g . 6 .2 . View, in th e d i r e c t i o n approxim ately p a r a l l e l to [0001], of the
h ig h -te m p e ra tu re phase of Na^ZRjSijPOjj (300°C). Small c i r c l e s
denote th e N a (l) s i t e s ; la r g e c i r c l e s denote th e Na(2) s i t e s ;
c o o rd in a tio n o c tah e d ra around Zr and c o o rd in a tio n te t r a h e d r a
around S i,P .
F i g . 6 .3 . View, in th e d i r e c t i o n [1010], of th e h ig h -te m p e ra tu re phase of
Na3 Zr 2 S i 2 POj2 - For e x p la n a tio n s see F i g . 6 .2 .
In th e upper
r i g h t , an N a(l) atom can be seen between two Zr c o o rd in a tio n
o ctah e d ra and surrounded o c ta h e d r a lly by th e s ix Na(2) atoms to
which th e wide conduction p a th s a re a v a i l a b l e .
177
s i n g l e c r y s t a l s , which have NASICON s t r u c t u r e , th e mean P-0 d is ta n c e i s
1.515 & [2 5 ]. Nhen th e s e v a lu e s a r e used as s t a r t i n g p o in ts fo r an
i n t e r p o l a t i o n , th e mean observed v a lu e s of 1.552 X fo r th e ( P , S i ) - 0
bond le n g th and 1.588 X fo r th e ( S i , P ) - 0 bond le n g th correspond e x a c tly
to th e P:Si r a t i o s of 2:1 and 1 :2 , r e s p e c t i v e l y .
Occurence o f s h o r t e r than normal t e t r a h e d r a l bond le n g th s in
framework s t r u c t u r e s may i n d i c a t e pseudosymmetry, a s has been p o in te d
ou t by Meier [178] f o r a l u m i n o s i l i c a t e frameworks. The pseudosymmetry,
a s opposed to a mere s t a t i s t i c a l d i s o r d e r , im p lie s t h a t th e t r u e sym­
metry of c r y s t a l s i s lower th an assumed in cou rse of s t r u c t u r e r e f i n e ­
ment. I f th e s h o r t mean ( P , S i ) - 0 and ( S i , P ) - 0 in te r a to m ic d is t a n c e s in
our NASICON c r y s t a l s were observed only in th e rhombohedral p h a se s, one
could assume t h a t th e y a r e composed of pseudosymmetric in te rg ro w th s of
m onoclinic p hases. However, even th e m onoclinic phase has too s h o r t
d i s t a n c e s . I f our NASICON samples a r e indeed pseudosymmetric, th ey have
t o c o n s i s t of in te rg ro w th s of even lower space group symmetry than th e
m onoclinic C2/c. The maximal non-isom orphic subgroups of C2/c a re C2,
Cc, C l, P 2 j/c or P2/c [179], I t should be i n v e s t i g a t e d whether f u r t h e r
i n d i c a t i o n s of lower symmetry in NASICON-type compounds can be found.
Among NASICON-type compounds th e only one w ith normal P-0 bond le n g th s
(mean P-0 d is ta n c e 1.536 X ) a p p ears to be N a Z ^ P ^ O ^ *
and i t i s u n l i k e ­
ly to be pseudosymmetric.
Hong [10] had proposed t h a t in Na3Zr2Si2P0^2 th e Si and th e P
atoms a re o rd ered over th e two ( S i , P) p o s i t i o n s 4 (e ) and 8 ( f ) of th e
C2/c space group. Incomplete o rd e rin g was e x p e rim e n ta lly found fo r
178
com positions w ith xs 1 .6 and 2 .0 [2 6 ]. In our c r y s t a l s , th e average
( S i , P ) - 0 bond le n g th s f o r th e s e two p o s i t i o n s a r e equal (see Table 6 .4 )
and th e r e i s no i n d i c a t i o n whatsoever fo r o r d e rin g to o ccur.
We v a r ie d th e occupancy f a c t o r s of th e Zr atoms, b u t th e y stay e d
a t u n i t y , or in c re a s e d s l i g h t l y . Thus, we cannot confirm th e fin d in g s
concerning zirconium d e f ic ie n c y in NASICON compounds, which were r e ­
p o rte d fo r Na5ZrP3012 t 25l and
N a 3
i Zri
7 8
S i l 24P1 76°12 t* 1 ]- E ith e r
th e s e compounds a r e r e a l l y d i f f e r e n t in t h i s r e s p e c t or e l s e th e X-ray
re fin e m e n ts a r e a f f e c t e d by th e use of atom ic s c a t t e r i n g f a c t o r s fo r Zr
in s te a d o f th e proper io n ic s c a t t e r i n g f a c t o r s , a s has been sug g ested
[2 5 ].
The t o t a l occu pancies of th e Na atom p o s i t i o n s , o b ta in e d by un­
c o n s tr a in e d re fin e m e n t, a r e about two t h i r d s of what would be expected
from th e chemical com position of th e two compounds (see T ables 6.2 and
6 . 3 ) . U nderoccupations of Na p o s i t i o n s were r e p o rte d even fo r a s i n g l e
c r y s t a l s t r u c t u r e d e te rm in a tio n of N a ^ Z rjS ijO ^ [180]. A p p aren tly , th e
h ig h ly a n i s o t r o p i c therm al motions of th e Na atoms make i t d i f f i c u l t to
g e t r e l i a b l e v a lu e s f o r t h e i r o c cu p a n c ies, even when anharmonic temper­
a t u r e s f a c t o r s a r e used [1 2 ]. The q u a l i t y of our powder d a ta does not
w a rra n t c a l c u l a t i o n s in v o lv in g anharmonic te m p e ratu re f a c t o r s .
We p e r­
formed c o n s tr a in e d re fin e m e n ts in which th e Na c o n te n t was forced to be
a t th e s to i c h i o m e t r i c v a lu e s . The r e s u l t s a r e very s i m i l a r to th o se
re p o rte d h ere fo r th e u n c o n stra in e d re fin e m e n ts and th e r e s i d u a l s a re
only m a rg in a lly h ig h e r. E s s e n t i a l l y , th e same r e s u l t s a re o b ta in e d by
s c a l i n g th e occupancies up to th e t o t a l sodium c o n te n t corresp o nd in g to
th e s to i c h i o m e t r i c form u la, a s i s done in Table 6 .6 .
For Na2Zr2SiP201 2 , th e r e i s very l i t t l e d i f f e r e n c e in th e occupan­
cy f a c t o r s of th e N a (l) and th e Na(2) s i t e s between room tem p e ratu re
and 300°C. The t o t a l amount o f sodium i s s l i g h t l y h ig h e r in th e th r e e
a v a i l a b l e Na(2) s i t e s than in th e N a (l) s i t e , but th e r e l a t i v e occupan­
cy (as a f r a c t i o n of th e number of atoms t h a t can be accomodated in th e
s i t e ) i s 80 to 90% f o r N a (l) and only about 38fc f o r Na(2). Obviously
th e N a (l) s i t e i s e n e r g e t i c a l l y more fa v o ra b le fo r sodium in th e x=l
compound. In th e x=2 compound a t room te m p e ra tu re , th e r e l a t i v e occu­
pancy o f th e N a(l) s i t e i s s i m i l a r to th o se of th e x=l compound. At
300°C, however, th e occupancy of th e N a(l) s i t e i s s m a lle r and th e
Na(2) s i t e occupancy i s in c re a s e d . In th e h ig h -te m p e ra tu re , high io n ic
c o n d u c tiv ity phase o f Na3Zr2S i 2POj2 , a sm all s h i f t of sodium from N a(l)
to Na(2) i s observ ed, a s compared w ith th e room -tem perature phase. In
f a c t , th e h ig h -te m p e ra tu re phase o f th e x=2 compound i s th e only one of
th e phases s tu d ie d h e r e , which e x h i b i t s a h ig h e r r e l a t i v e occupancy fo r
th e rhombohedral Na(2) s i t e , than fo r th e N a (l) s i t e . In th e m onoclinic
phase , th e Na(2) s i t e i s f u l l y occupied and, t h e r e f o r e , te n d s to
i n h i b i t io n ic movement. A p p aren tly , in th e h ig h -te m p e ra tu re
Na3Zr2S i 2P012 th e Na(2) s i t e i s e n e r g e t i c a l l y more fa v o ra b le than th e
N a (l) s i t e . The f in d i n g s re g a rd in g Na o ccupancies have to be seen in
term s o f th e l i m i t a t i o n s of th e ex p erim en tal p r e c is io n of th e p re s e n t
s tu d y . Namely, th e t o t a l amount of sodium d i r e c t l y o b ta in e d from s t r u c ­
t u r e refin e m e n t i s l e s s than re q u ire d by th e chemical com position. In
a d d i t i o n , th e s tu d ie d s t r u c t u r e s appear to be pseudosymmetric.
Table 6.6
Observed and s c a le d up occupancy f a c t o r s and r e l a t i v e occupancies fo r
each s i t e of p o s i t i o n s N a (l) and Na(2) in Na2Zr2SiP2012 ( 1RT a t room
te m p e ra tu re ; 1HT a t 300°C and in Na3Zr2Si2P012 (2RT, m onoclinic a t room
te m p e ra tu re ; 2HT a t 300°C).
In 2RT th e Na(2) p o s i t i o n i s s p l i t in to
Na(2) and Na(3). R e l a tiv e occupancy i s given in p e r c e n t. The m u l t i p l i c ­
i t y of th e N a(l) s i t e allo w s i t to c o n ta in up to one sodium atom; th e
Na(2) [o r th e combined Na(2) and Na(3) s i t e s in th e m onoclinic case ]
can accomodate up to th r e e sodium atoms.
N a(l) occupancy
Na(2) occupancy
phase
observed
s c a le d
%
observed
s c a le d
%
1RT
0.58
0 .9 1 (7 )
91
0.7 0
1 .0 9 (8 )
36
1HT
0.49
0 .8 4 (9 )
84
0.68
1 .16 (11 )
39
2RT
0.44
0 .8 1 (1 1 )
81
1.64
2 .1 9 (2 1 )
73
[Na2:
1.00
1 .0 0 (6 )
100]
[Na3:
0.64
1 .19 (15 )
60]
1.69
2 .4 3 (14 )
81
2HT
0.4 0
0 .5 7 (1 1 )
57
181
When Tranqui e t a l [22] determ ined th e c r y s t a l s t r u c t u r e of
N a^Z ^S i^O ^* they proposed t h a t th e conduction p a th s from one Na(2)
p o s i t i o n to n eig h b o rin g Na(2) s i t e s a re more im po rtant than th e pathway
between N a(l) and Na(2) s i t e s , because a wider channel opening was
observed fo r th e former in case of t h e i r co m position, x=3. Kohler a t a l
[11] found, fo r t h e i r c r y s t a l of com position Na3 ^Zrj 70S i 1 2 4 Pi 76°12’
t h a t th e N a(l) to Na(2) pathway i s e n e r g e t i c a l l y more fa v o r a b le , in
agreement w ith th e pro p o sal made by Hong [1 0 ]. Me f in d t h a t th e N a(l)
to Na(2) channel in th e h ig h -te m p e ra tu re Na3Zr2S i 2P0^2 i s about 5% more
open than i t i s in Na4Zr2S i 30 12< i t i s a l s o about 10$ wider th an in
e i t h e r of th e phases w ith x = l. This a g re e s w ith th e d i f f e r e n c e s found
in th e io n ic c o n d u c t i v i t i e s of th e two p hases. The openings between th e
n e ig h b o rin g Na(2) s i t e s a r e s l i g h t l y s m a lle r and have ap prox im ately th e
same dimensions in a l l of th e phases s tu d ie d h e re a s they have in
Na4Zr2S i 301 2.
The N a (l) to Na(2) opening i s so wide because th e mean
N a ( l) - 0 ( 1 ) d is ta n c e i s very l a r g e , a c t u a l l y much l a r g e r th an th e mean
d is ta n c e N a(2)-0, where Na(2) i s e ig h t c o o rd in a te d , w hile N a(l) i s only
s i x c o o rd in a te d . The sum of th e e f f e c t i v e io n ic r a d i i [175] fo r s i x
c o o rd in a te d Na i s 2 .40 ft, fo r e ig h t c o o rd in a te d Na i t i s 2.56 ft. The
mean N a ( l) - 0 d is ta n c e in th e 300°C phase of th e x=2 compound i s 2.70 ft,
t h a t i s , 0 .3 ft lo ng er th an th e sum of th e io n ic r a d i i .
In th e same
phase, th e mean N a(2)-0 d is ta n c e i s 0.13 ft lo n g er than th e sum of th e
correspo nd in g io n ic r a d i i . These long mean in te ra to m ic d is ta n c e s give
s u f f i c i e n t space to th e sodium atoms to move from one s i t e to ano ther
and to be s t a t i s t i c a l l y d i s t r i b u t e d w ith in th e space of th e c o o rd in a -
182
fcion po lyhedra. The la r g e therm al p aram eters o f th e sodium atoms c o r r e ­
l a t e w ell w ith th e s i z e s o f th e c o o rd in a tio n polyhedra.
To conclude, th e p re s e n t study of th e c r y s t a l s t r u c t u r e on s i n ­
te r e d samples of Na2Zr 2 SiP 2 0 12 and Na3Zr2S i2P012 su g g e s ts t h a t th e high
io n ic c o n d u c tiv ity of th e l a t t e r , a t te m p e ra tu re s around 300°C, can be
c o r r e l a t e d w ith th e wide openings between th e N a(l) and th e Na(2)
s i t e s , and w ith th e high r e l a t i v e occupancy f a c t o r s of th e Na(2) p o s i­
t i o n s o c c u rrin g only fo r th e x=2 com position a t e le v a te d te m p e ra tu re s .
V II. ELECTRICAL PROPERTIES QF NASICON-TYPE SOLID ELECTROLYTES
7 .1 . Complex Impedance Measurements and A n a ly s is .
The a . c . impedance measurements Here made u sin g r e c ta n g u la r p a r a l ­
le le p ip e d samples o f t y p i c a l s i z e 4x4x8 mm which Here cu t from s i n t e r e d
c y l i n d e r s . The two s m a lle r opposing fa c e s were f i r s t diamond p o lis h e d
and co ated w ith Pt or Au e le c tr o d e s d e p o s ite d by s p u t t e r i n g in d .c .
plasma (a Polaron SEM Coating Unit E5000 was u s e d ). Each e le c tr o d e was
f i r s t s p u t t e r e d fo r 15 m inutes a t *4 mA c u r r e n t and then fo r 30 m inutes
a t *18 mA. The s u r f a c e r e s i s t a n c e of th e e l e c t r o d e s was t e s t e d u sing a
d . c ohmmeter. The samples were placed in t o a h o ld e r between two s p r in g loaded platinu m p l a t e s . The impedance measurements were made over th e
te m p e ratu re range 20 to 450°C w ith th e sample under vacuum or in a dry
argon atmosphere. The complex impedance was measured in th e frequency
R
range 1 t o 7x10"' Hz u sin g a c o m p u te r-c o n tro lle d impedance measuring
system (see S e c tio n 3 .2 ) . In a few ^ a s e s , th e frequency range was ex7
tended to 10 Hz u s in g a v e c to r impedance meter (HP-4815A). The imped­
ance of th e sample w ith io n -b lo c k in g e l e c t r o d e s was o b ta in e d from th e
measured impedance c o r r e c t e d fo r s t r a y impedances. In th e case of
measurements a t f r e q u e n c ie s above 1 MHz, such c o r r e c t i o n s were la r g e
and th e impedance of sample was determ ined w ith c o rre sp o n d in g ly lower
p re c isio n .
The complex impedance s p e c tr a were analyzed u sing th e n o n lin e a r
l e a s t - s q u a r e s e s tim a tio n o f param eters in g e n e r a liz e d e q u iv a le n t c i r ­
c u i t s d e s c rib e d in Chapter 3. The ex p erim en tal s p e c t r a were a c c u r a te ly
183
184
modelled by c i r c u i t s c o n s i s t i n g o£ r e s i s t o r s and c o n s ta n t phase e l e ­
ments, CPE [see Eg.3 .2 2 ) ] . The impedance s p e c t r a of both Na2Zr2SiP20^2
and N a ^ Z ^ S ijP O ^ e x h ib ite d a d i s p e r s i o n c h a r a c t e r i s t i c of io n -b lo c k in g
e l e c t r o d e s and g ra in -bo un dary p o l a r i z a t i o n fo r a p o l y c r y s t a l l i n e mate­
r i a l . The s p e c tr a o f th e x=2 compound were more c o m p licated , e s p e c i a l l y
in th e range of te m p e ra tu re s from 120 to 200°C, where th e phase t r a n s i ­
t i o n ta k e s p la c e . I t was not always p o s s ib le to a s s o c i a t e a l l f e a tu r e s
observed in th e s p e c t r a w ith w e ll- d e f in e d p h y s ic a l phenomena, d e s p it e
th e f a c t t h a t we were a b le to reproduce th e s p e c t r a u sin g e q u iv a le n t
c i r c u i t s . In n e a r ly a l l c a s e s , however, th e r e s i s t a n c e r e p r e s e n tin g th e
bulk io n ic c o n d u c tiv ity could be c l e a r l y s e p a ra te d from th e frequencydependent elem ents which were used to model th e p o l a r i z a t i o n of g ra in
b o un daries and e l e c t r o d e s . The bulk c o n d u c ti v ity was a s s o c ia te d w ith a
broad p la te a u in th e s p e c t r a l p l o t s o f th e r e a l p a r t of th e a d m ittan ce
a t fr e q u e n c ie s above th e re g io n of d i s p e r s i o n . At tem p eratu re above
*>80°C, th e bulk c o n d u c tiv ity appeared to be freq uency -in d ep en d en t up to
10 MHz and i t i s b e lie v e d t h a t t h i s re p re s e n te d th e s t a t i c i n t r a g r a i n
io n ic c o n d u c tiv ity . At lower te m p e ra tu re s , th e o n s e t of an a d d i t i o n a l
d i s p e r s i o n was observed around 1 MHz.
P r io r to d is c u s s in g th e d e t a i l s of th e complex impedance s p e c tr a
fo r samples of each compound, l e t us compare t h e i r bulk io n ic conduc­
t i v i t i e s , which a re shown in th e form of A rrh en iu s p l o t s F i g . 7 .1 .
For Na3Zr2Si2P01 2» an exam ination of th e A rrheniu s p l o t o f conduc­
t i v i t y shows t h a t th e c o n d u c tiv ity undergoes th r e e d i s t i n c t k ind s of
beh av ior as th e te m p e ratu re i s r a i s e d from 20°C to 450°C; th e ap p aren t
Temperature
[°C]
300
100
3
20
2
'e 1 0
1
r-1
G
i- 10 0
l
J Q ""H
1.2
t
♦
N a o Z r^ S i^ P O
q
N a ^ Z ro S lP o O
l
i
1.6
t
i
I
i
I
2.0
2.4
2.8
1000/T
[K_1]
i
!
3.2
1
3.6
Fig.7.1. The bulk ionic conductivity of Na3Zr2Si2 P012 and Na2 Zr2 SiP20 1 2 .
186
a c t i v a t i o n e n e r g ie s and p re e x p o n e n tia l f a c t o r s a r e given in Table 7 .1 .
In th e high te m p e ratu re rang e, 180 t o 450°C, th e average a c t i v a t i o n
energy i s 0.21 eV; however w ith a s l i g h t change of s lo p e , v i s i b l e in
P i g . 7 .1 , a b e t t e r f i t of th e d a ta i s o b ta in e d u sin g two s t r a i g h t - l i n e
segments (see Table 7 . 1 ) . In th e in te rm e d ia te tem p e ratu re range (110 to
180°C) where a phase t r a n s i t i o n o c c u rs , th e a c t i v a t i o n energy was
e s tim a te d w ith lower p r e c i s i o n . We no te t h a t a t 20°C, th e io n ic conduc­
t i v i t y was 5.6x10“ * fl“ *cm“ * and a t 300°C i t was 0.215 fl“ *cm“ *, both
v a lu e s ly in g in th e range of v a lu e s t y p i c a l l y re p o rte d fo r t h i s NASICON
com position [ 9 ,3 3 ,3 8 ,3 9 ,4 0 ] , The bulk c o n d u c tiv ity was not a f f e c t e d by
therm al c y c lin g , d e s p i t e th e f a c t t h a t some d i s c o l o r a t i o n of th e sample
s u r f a c e was observed a f t e r prolonged h e a tin g a t te m p e ra tu re s above
350°C.
For th e second com p o sitio n , Na2Zr2S iP 2 0j2 . th e bulk c o n d u c tiv ity
a t 20°C was 6.6*10“ ^ a~1cra“ 1’, and a t 300°C
0.028 fl“ 1cm“ 1 , v a lu e s
ap prox im ately an o rd e r of magnitude lower than th o se found fo r th e
compound NagZr2S i 2POj 2 - The average a c t i v a t i o n energy e stim a te d fo r a
te m p e ratu re range 20 t o 460°C was 0.335+0.01 eV, but a d e c re a se of
a c t i v a t i o n energy w ith in c r e a s in g te m p e ratu re was ap p aren t in th e
A rrh en ius p l o t and a b e t t e r f i t of th e d a ta could be o b ta in e d u sin g
fou r s t r a i g h t - l i n e segments. The a c t i v a t i o n e n e r g ie s and p re e x p o n e n tia l
f a c t o r s fo r such a f i t a re l i s t e d in Table 7 .1 . I t i s noteworthy t h a t
th e a c t i v a t i o n e n e r g ie s fo r both compounds a r e ap pro x im ately equal in
th e low -tem p erature range and t h a t changes of a c t i v a t i o n e n e r g ie s ta k e
p la c e a t s i m i l a r te m p e ra tu re s fo r samples of both com positions.
187
Table 7.1
Apparent a c t i v a t i o n e n e r g ie s and p re e x p o n e n tia l f a c t o r s fo r th e bulk
io n ic c o n d u c ti v ity . The numbers in p a re n th e s e s a re e stim a te d sta n d a rd
d e v ia t io n s of th e l e a s t s i g n i f i c a n t f ig u r e .
Temperature
A c tiv a tio n
P ree x p o n e n tial
range [°C]
energy [eVJ
f a c t o r [fi~'1‘cm_1K3
Na3Zr2S i 2P012
20 - 110
0 .3 6 2(4 )
2 . 6 (3 ) x l0 5
110 - 180
0 .4 3 (2 )
2 .0 ( 6 ) x l 0 6
180 - 315
0 .2 3 5(5)
1 . 4 (2 ) x l0 4
315 - 430
0 .19 5 (4 )
6 .5 ( 3 ) x l 0 3
Na2Zr2SiP2Oj2
20 - 100
0 .3 68 (8)
4 .2 ( 4 ) x l 0 4
100 - 195
0 .3 4 0 (3 )
1 . 7 5 (1 3 )x l0 4
195 - 325
0 .31 5(1 )
9 .5 ( 3 ) x l 0 3
325 - 460
0 .28 0(2 )
4 . 8 ( 2 ) x l0 3
188
Ma2Zr2SiP2°12
The c o n d u c tiv ity of N a jZ ^ S iP jO jj was found to in c re a s e slow ly i f
th e c o n d u c tiv ity sample was m aintain ed under vacuum or an i n e r t gas
atmosphere a t te m p e ratu res in excess of 350°C. An in c re a s e of about 10%
was observed over a p e rio d of 30 h o u rs, fo llo w in g which th e conduc­
t i v i t y appeared to s t a b i l i z e . Although th e e x a c t cause of t h i s behavior
i s no t known, i t seems p l a u s i b l e t h a t a s l i g h t re d u c tio n of th e sample
occurs a t th e low oxygen p a r t i a l p r e s s u r e s used. The measurements
r e p o rte d here were tak en du rin g a c o o lin g c y c le , a f t e r a n n e a lin g th e
sample w ith th e s p u t t e r e d gold e l e c t r o d e s fo r 2 days a t 460°C under
vacuum.
The exp erim ental impedance s p e c t r a were f i t t e d by th e impedance
c a l c u l a t e d acco rd ing to th e e q u iv a le n t c i r c u i t d is p la y e d in F i g . 7 .2 .
c
With a lim it e d frequency span 1 to 7x10 Hz, n o t a l l elem ents of th e
e q u iv a le n t c i r c u i t were observed over th e e n t i r e tem p erature range.
R e p r e s e n ta tiv e examples of th e e x perim ental s p e c t r a a re given in F igs.
7 . 3 - 7 .6 , to g e th e r w ith th e f i t t e d c u rv e s , th e e q u iv a le n t c i r c u i t s and
th e e stim a te d v a lu e s of param eters.
The v a lu e s of c i r c u i t p a ra m ete rs, a s e stim a te d from th e measured
s p e c t r a , a re given w ith o u t ta k in g i n to account th e dimensions of th e
sample. In ord er to c a l c u l a t e v a lu e s of param eters independent of
sample s i z e , th e c i r c u i t elem ents must be i d e n t i f i e d a s being a s s o ­
c i a t e d w ith th e volume of th e sample or w ith th e a re a of th e e l e c ­
tr o d e s . The c a p a c ita n c e s (or magnitudes of th e c o n s ta n t phase elem ents)
189
CPgb
ANqb
CPbu
ANbu
r lF i’
Ro
a~JW v ~
“V W
Rbu
W V v
Rgb
CPel
ANel
P i g . 7 .2 . E q u iv ale n t c i r c u i t used to model th e complex impedance spectrum
o f th e sample o f Na2 Zr 2 $ i P 2 0 i 2 w ith io n -b lo c k in g gold
elec tro d e s.
190
a s s o c ia te d w ith th e e le c tr o d e s a re p r o p o r tio n a l to th e a r e a , A, of
e l e c t r o d e . Since th e two i d e n t i c a l e l e c t r o d e s on th e o p p o s ite s id e s of
th e sample a r e re p re s e n te d by a s i n g l e c i r c u i t , th e c a p a c ita n c e per
u n i t a r e a , expressed by th e c i r c u i t c a p a c ita n c e , Cdl, i s 2Cdl/A. The
impedances a s s o c ia te d w ith th e volume of sample should s c a le as the
r a t i o of sample le n g th to th e e le c tr o d e a re a L/A, e . g . , th e r e s i s t a n c e
Rb r e p r e s e n tin g th e bulk io n ic c o n d u c ti v ity , ob , i s obviously
Rb=L/(obA), th e magnitude o f th e CPE r e p r e s e n tin g th e bulk d i e l e c t r i c
response i s p r o p o r tio n a l to A/L. For th e g r a in b o u n d a rie s, th e t r u e
g e o m etric al f a c t o r i s unknown, b u t th e v a lu e s of th e corresponding
impedances should s c a l e a s L/A when th e g r a in bo un daries a re uniform ly
d i s t r i b u t e d th roughout th e volume of th e sample. The convention used
h e re i s to e x p re ss th e g ra in -bo un dary c o n d u c tiv ity as ogb=A/(RbL). For
th e sample used in t h i s experim ent, th e a r e a of each e le c tr o d e was
2
A=0.13 cm and th e r a t i o of sample le n g th , L, t o th e e le c tr o d e a r e a , A,
was 5.77 cm- 1 .
The d i e l e c t r i c d is p e r s io n a s s o c ia te d w ith th e bulk o f th e mate­
r i a l , which i s re p re s e n te d by th e c o n s ta n t phase element (CPbu,ANbu) in
th e c i r c u i t of F i g . 7 .2 , was c l e a r l y observed a t te m p e ratu res below
</>80°C in form of a h ig h -fre q u en c y spur in th e adm ittan ce p l o t s (see fo r
example, th e impedance spectrum measured a t 54.5°C i s d is p la y e d in
P i g . 7 .4 ) . At room te m p e ra tu re , th e h ig h -fre q u en c y spur can be i d e n t i ­
f i e d a s a p a r t of an a d d i t i o n a l a rc in th e adm ittan ce p l o t (see
F i g . 7 . 3 ) . The h ig h -fre q u en c y i n t e r c e p t of t h i s a r c w ith th e r e a l a x is
i s given by 1/Ro, where Ro i s th e r e s i s t a n c e added in s e r i e s to th e
F l l a APMV02. Spac AP3C19. T - 19 .5°C . 51 fp a q a . Lav 2 .
Modal C7QT. f , - 9 . 0QE-H3Q t o f SI - 7 . 00Ei-Q5 Hz.
6 p a r. Rm sr- 2 .1 0 2
191
1 .4
1.2
J f 1.0
0 .8
a . 0 .6
01 0 . 4
0.2
0 .2
0 .4
0 .6
0 .8
1 .0
Conductance
1 .2
1 .4
CIO*3 fl-1 J
1.6
1.8
T
T
T
T
T
T
T
T
T
2 .4
3 .0
3 .6
4 .2
4 .8
2.0
S
>
g » - 5 .6
-
6 .0
-6 .4
- 4 .8
- 5 .2
-
6.0
0 .6
1.2
1.8
5 .4
6.0
log (f)
F i g . 7 .3 . Complex ad m itta n ce of th e Na2Zr2SiP20^2 sample w ith gold e l e c ­
tr o d e s a t 19.5°C. The e stim a te d v a lu e s of p aram eters a r e :
Ro = 1 . 8 ( 2 ) x l0 4 Q,
ANbu = 0 .2 7 ( 2 ) ,
'Rbu = 7 . 0 ( 2 ) x l 0 4 a,
CPbu = 4 .2 ( 9 ) x l 0 “ 10 a-1 ,
CPel = 4 . 4 ( l ) x l O “ 8 F,
ANel = 0 .1 6 8 (4 )
F l l a APLVOS. Spac AP3C1S. 7 - 5 4 .5 °C . 56 f r a q s .
Modal C70T, f | "2 . 49£*QQ to fjg "7 . Q0E*Q5 Hz.
Lav 2.
8 p a r. Rm sr- 0 .6 0 7
192
4 .2
CPgb
[ANgb
C Pbu
ANbu
3 .6
CPal
ANal
3 .0
e
2 .4
0 .6
I
2 .4
3 .0
Conductance
3 .6
4 .2
4 .6
S. 4
6.0
CIO"3 fl"1 1
- 4 .5
-S.0
-
6 .0
-
T
T
T
T
T
T
- 4 .3
-5 .0
01- 5.5
-
6 .0
-6 .5 -7 .
0 .6
1.2
2 .4
3 .6
4 .2
4 .8
£ .4
6 .0
F i g . 7 . 4 . Complex a d m ittan ce of th e Na^ZrjSiPjOj^ sample w ith gold e l e c ­
tr o d e s a t 54.5°C. The e s tim a te d v a lu e s o f p aram eters a r e :
Rbu = 2 .0 9 4 ( 5 ) x l0 4 Q.
CPbu = 4 .7 ( 4 ) x l 0 ‘ 10 a-1 . ANbu = 0 .2 8 3 (6 ) ,
Rgb = 8 .1 ( 1 2 ) x l0 4 Q,
CPgb = 4 . 2 ( 2 ) x l 0 “ 7 a-1 ,
CPel = 6 .9 ( l) x lO ~ 8 a"1, ANel = 0 .1 2 3 (6 ) .
ANgb = 0 .2 7 1 (8 ) ,
F i l s A P L V J9. S p e c A P 3 C 1 9 . T - 1 9 5 .3 ° C . 6 0 f r e q s .
M o d e l CCOM. f j ” 1 . 0 0 £ * 0 0 t o f e g “ 7 . OOE^OS H z.
L ev 2 .
7 p o r,
R m s r« 0 . 6 7 2
193
10
Cbu
ta
ao
c
*oa.
»
ao
3ul
U
0 .2
0.6
0 .4
0.B
C o n d u c ta n c e
- 3 .0
T
T
T
1. 0
1 .2
tl D " 3 I T 3 3
T
- 4 .0
at
- 5 .0
-
6 .0
T
T
T
T
T
- 3 .0
- 4 .0
-
6 .0
0 .6
.2
8
2 .4
3 .0
lo g (f)
3 .6
4 .2
4. B
5 .4
6 .0
F i g . 7 .5 . Complex a d m ittan ce of th e Na2Zr2SiP20^2 sample w ith gold e l e c ­
tr o d e s a t 195°C. The e stim a te d v a lu e s of param eters a r e :
Rbu = 714(2) 0 , (
Cbu = 7 . 9(15)xlO "12 F,
CPgb = 3 .4 ( 4 ) x l 0 -6 a"1,
ANel = 0 .1 3 6 (1 ) .
ANgb = 0 .3 6 6 (1 2 ),
Rgb = 903(41) R,
CPel = 2 . 7 0 ( l ) x l 0 * 7 a"1,
F i l e APLV41, Spec AP2C19. T -4 0 8 . 5°C. 60 f r a q s .
Model CCOR. f } - 1 . 0 0 E -00 to f M - 6 . 97E+05 Hz.
Lav 2.
7 p o r.
Rm er- 0. 468
C Pgb
ANgb
•m
t
cs
o
Rgb
0
u
sa.
ao
<0
3
C/1
CIO"3 D~J J
C o n d u c ta n c e
-2.0
T
T
T
T
T
T
T
T
T
T
1 .8
2 .4
3 .6
4 .2
T
T
4 .8
5 .4
-3 .0
- 5 .0
-
2 .0
- 3 .0
- 5 .0
-6.
0 .6
3 .0
lo g (f)
6 .0
P i g . 7 .6 . Complex a d m ittan ce of th e Na2Zr2SiP2012 sample w ith gold e l e c ­
tr o d e s a t 4 0 8 .5°C. The e stim a te d v a lu e s of p aram eters a r e :
Rbu = 9 6 .5 (5 ) 0,
ANgb = 0 .4 9 ( 2 ) ,
ANel = 0 .2 1 8(3) .
Rgb = 80(11)0,
CPgb = 7 .0 (1 7 ) x l0 ~ 50_ 1 ,
Cdl = B.B(5)xlO_8F,
CPel = 1 .0 9 4 (3 )x l0 ‘ 6 fl"1 ,
195
r e s t of th e c i r c u i t (s e e F i g . 7 .2 ) .
I t i s proposed h e re t h a t th e h ig h -fre q u en c y d is p e r s io n observed a t
low te m p e ra tu re s in th e impedance s p e c t r a i s a s s o c ia te d w ith th e f r e ­
quency dependence of hopping c o n d u c ti v ity . I f t h i s were th e c a s e , th e
r e s i s t a n c e Ro would correspond to th e h ig h -fre q u en c y l i m i t fo r hopping
c o n d u c tiv ity o(»)=L/(RoA). I f Ro^O i s e stim a te d from th e d a t a , then th e
bulk c o n d u c tiv ity in th e normal sense of th e low -frequency l i m i t i s
given by ob=(Rb+Ro)_1L/A. U n fo rtu n a te ly , th e r e s i s t a n c e Ro cannot be
p r e c i s e l y e stim a te d from th e d a ta in F i g . 7 .3 because o f th e lim it e d
frequency span and th e reduced p r e c is io n of th e measuring equipment a t
high v a lu e s of r e s i s t a n c e . With an in c re a s e of te m p e ra tu re , th e o n set
o f th e d is p e r s io n s h i f t s to h ig h e r f r e q u e n c ie s , beyond th e a v a i l a b l e
frequency range. At te m p e ra tu re s above -10OoC, th e p aram eters of th e
c o n s ta n t phase elem ent (CPbu,ANbu) cannot be e s tim a te d from th e spec­
tra .
The d is p e r s io n a s s o c i a t e d w ith th e g r a in bou nd aries i s not c l e a r l y
v i s i b l e in th e complex a d m ittan ce p l o t s due to sev e re o v e rla p of th e
two s e m i c ir c le s which correspond to th e p o l a r i z a t i o n of th e e le c tr o d e s
and th e g r a in b o u n d a rie s. The corresp o n d in g elem ents of th e e q u iv a le n t
c i r c u i t , t h a t i s , th e r e s i s t a n c e Rgb and th e c o n ta n t phase element
(CPgb,ANgb) can, however, be e stim a te d by th e l e a s t - s q u a r e s f i t t i n g for
th e tem p eratu re range 55 to 410°C. For example, in th e complex adm it­
ta n c e p l o t s of th e s p e c t r a measured a t 1 9 5 .3°C and a t 408°C (see
F i g s . 7 . 5 - 7 . 6 ) , th e e stim a te d v a lu e l/(Rgb+Rbu) i s marked by an arrow.
I f th e s e m i c ir c le s had no t ov erlap p ed , t h i s would be an e x tr a p o la t e d
196
i n t e r c e p t of th e two s e m i c ir c le s w ith th e r e a l a x i s . The exponent of
th e c o n s ta n t phase elem ent, which r e p r e s e n ts th e g r a in boundary p o l a r i ­
z a t i o n , in c re a s e s w ith tem p erature from 0.28 a t 55°C to 0 .4 9 a t 409°C.
This can be i n t e r p r e t e d a s a change of th e e f f e c t of th e g r a in bounda­
r i e s on io n ic t r a n s p o r t from m ostly blo c k in g ( p i l e - u p o f charged p a r­
t i c l e s ) a t low te m p e ra tu re s t o d i f f u s i o n - l i k e charge t r a n s f e r a t high
te m p e ratu re. Below 55°C, p o l a r i z a t i o n due to th e g r a in bo un daries
cannot be s e p a ra te d from t h a t due to th e e l e c t r o d e s even by th e l e a s t sq u ares a n a l y s i s . Both e f f e c t s a r e w ell modelled by a s i n g l e c o n s ta n t
phase element and th e high v alu e of th e Rgb cannot be e x tr a c te d from
th e d a ta . At te m p e ra tu re s above 410°C, th e g r a in boundary r e s i s t a n c e
Rgb i s h ig h ly c o r r e l a t e d w ith th e p a r a l l e l CPE (CPbu,ANbu) which has
la r g e exponent (ANbu&0.5) and th e e s tim a te o f Rgb i s not r e l i a b l e . The
tem p e ratu re dependence of th e g r a in boundary c o n d u c tiv ity i s d is p la y e d
in form of A rrhenius p l o t in F i g . 7 .7 . Three d i s t i n c t tem peratu re ranges
a r e a p p aren t w ith th e e stim a te d a c t i v a t i o n e n e r g ie s : (1) 0.34±0.03eV
below 120°, (2) 0.64±0.03eV between 120 and 200°C, and (3) 0.32±0.02eV
above 200°C.
In th e e q u iv a le n t c i r c u i t used to f i t d a ta a t 408°C (se e F i g . 7 . 6 ) ,
th e e le c tr o d e i s re p re s e n te d by a p a r a l l e l combination of a c a p a c ito r
Cdl and a c o n s ta n t phase element (CPel,ANel). When th e CPE i s used
a lo n e , th e q u a l i t y of f i t d e t e r i o r a t e s and th e r e s i d u a l s a t low f r e ­
qu encies in c r e a s e . The c a p a c ita n c e Cdl in c r e a s e s w ith tem p e ratu re from
380 nF/cm2 a t 250°C to 1610 nF/cm2 a t 436°C. At lower te m p e ra tu re s , th e
c a p a c ita n c e s Cdl i s e s tim a te d to be z e ro , which means t h a t th e e l e c -
450
300
[®C3
100
50
20
10 D
ffT
Cn^cm^K]
10 z
T em p e ra tu re
200
__i__
__i__
2. 4
2. B
1000/T
CK"1]
3 .2
3 .6
F i g . 7 .7 . Temperature dependence o f th e g ra in -b o u n d a ry c o n d u c ti v ity of
198
tro d e p o l a r i z a t i o n i s a d eq u a te ly re p re s e n te d by a s i n g l e CPE fo r th e
frequency range above 1 Hz. The exponent, ANel, in c r e a s e s from 0.11 a t
38°C t o 0.23 a t 436°C and th e magnitude CPel r i s e s from 0.75x10“ ® to
20xl0“ 6 Q_1/cm2 .
Na3Zr2S i 2P012
The sample of N a jZ rjS ijP O ^ Mas equipped w ith platinum e l e c t r o d e s .
The a r e a of each e le c tr o d e was 0.1 9 cm and th e r a t i o of th e sample
l e n g t h - t o - t h e a re a o f th e e le c tr o d e was 4.14 cm“ *. Measurements of th e
5
complex ad m ittance in th e frequency range from 1 to 7x10 Hz were made
d u rin g a c o o lin g c y c le under dry argon atmosphere.
The e q u iv a le n t c i r c u i t used to model th e ex perim en tal adm ittan ce
s p e c t r a of N a jZ rjS ijP O jj w ith platinum e l e c t r o d e s i s shown in F i g . 7 .8 .
The Warburg elem ents (denoted by W in th e schem atic drawing) a r e used
in two segments o f th e c i r c u i t . The Warburg elem ents were in tro d u ce d
when i t was e s t a b l i s h e d t h a t , in cou rse o f f i t t i n g a c o n s ta n t phase
element w ith an a d j u s t a b l e exponent [ E q . ( 3 . 2 2 ) ] , th e value of th e
exponent was e stim a te d t o be ap pro xim ately equal 0 .5 ( a t l e a s t w ith in
two s ta n d a rd d e v i a t i o n s ) . I f th e exponent of th e CPE remained n e a r ly
equal 0 .5 fo r a range of te m p e ra tu re , th e v a lu e of exponent was fix e d
to be 0 .5 , and th e f i t s were re p e a te d . In e f f e c t , an e q u iv a le n t c i r c u i t
w ith a lower number o f a d j u s t a b l e param eters was e s t a b l i s h e d . Thus, th e
use of Warburg elem ents was not an a r b i t r a r y assum ption, b u t a r e s u l t
of n o n lin e a r l e a s t - s q u a r e s a n a l y s i s of th e ex p erim en tal s p e c t r a . The
magnitude of th e Warburg adm ittan ce i s denoted W ( i . e . , Yw = w «/ju).
199
Wgb
CPbu
Cgb
M AAr
Rbu
IWgh
r ifl
l/\
Ro
I
Cgh
■A/VV ^ A /V
Rgb
Rgh
F ig .7 .8 . Equivalent c ir c u it used to model the complex admittance spec­
trum of the sample of NajZ^SijPOj^ with ion-blocking platinum
electro d es.
200
The c i r c u i t o f P i g . 7.8 c o n ta in s fo u r r e s i s t a n c e s in s e r i e s , b u t,
a t th e most, th r e e of them were used to f i t th e impedance spectrum a t
any given te m p e ra tu re . The two p o l a r i z a t i o n p ro c e s s e s , re p re s e n te d by
segments denoted gb ang gh were dominant a t d i f f e r e n t tem p eratu re
ra n g e s. The therm al e v o lu tio n of th e a d m ittan ce spectrum fo r
N a ^ Z ^ S ijP O ^ i s d e s c rib e d below. R e p r e s e n ta tiv e example of th e adm it­
ta n c e s p e c t r a a r e p l o t t e d in F ig s. 7 .9 -7 .1 2 in two forms: complex plane
p l o t s and lo g a rith m ic p l o t s of adm itta n ce v s . frequency. Continuous
l i n e s r e p r e s e n t th e f i t t e d s p e c tr a which were c a lc u la te d a ccordin g to
th e e q u iv a le n t c i r c u i t s given in th e i n s e r t s of complex plane p l o t s .
At te m p e ratu res above ^300°C, two g r e a t l y overlapped s e m i c ir c le s
appeared in th e complex plane p l o t s of adm ittance (see F i g . 7 . 9 ) . The
e x perim en tal s p e c tr a were f i t t e d by a c i r c u i t c o n s is tin g of a r e s i s t o r
r e p r e s e n tin g th e bulk r e s i s t a n c e , Rbu, a c o n s ta n t phase element r e p r e ­
s e n ti n g th e e le c tr o d e (CPel, ANel) and a segment c o n ta in in g th e Warburg
elem ent. The p a r a l l e l combination of r e s i s t a n c e , Rgh, c a p a c ita n c e , Cgh,
and Warburg a d m itta n c e , Wgh, probably r e p r e s e n ts a p o l a r i z a t i o n p ro cess
a s s o c i a t e d w ith th e inhomogeneity o f th e p o l y c r y s t a l l i n e NASICON sam­
p le . I t co u ld , however, be a ls o a s s o c ia te d w ith th e c h a r g e - t r a n s f e r
p ro c e ss through th e e l e c t r o d e / e l e c t r o l y t e i n t e r f a c e , which becomes
r a t e - l i m i t i n g a t high tem p e ratu res when th e r e s i s t a n c e of th e s o l i d
e l e c t r o l y t e i s low. No convincing e v id ence, confirm ing one of th e two
p o s s ib le p h y s ic a l meanings of th e gh s e c t i o n , i s p r e s e n tly a v a i l a b l e .
At 408°C, th e r e s i s t a n c e Rgh i s of a s i m i l a r o rder of magnitude as th e
bulk r e s i s t a n c e (Rgh at 7xRbu). With d e c re a s in g te m p e ra tu re , th e r e s i s t ­
201
ance Rgh in c r e a s e s much more r a p id ly th an th e bulk r e s i s t a n c e , Rbu. At
te m p e ratu res below *1800C, th e r e s i s t a n c e Rgh can no lo ng er be e s t i ­
mated from th e a d m ittan ce spectrum , because th e a b s o lu te v a lu e of
impedance of th e p a r a l l e l Warburg element becomes s m a lle r than Rgh in
th e e n t i r e frequency range.
At te m p e ratu res around 300°C, an a d d i t i o n a l d is p e r s io n becomes
v i s i b l e in th e complex adm ittan ce diagrams in th e form of a d i s t o r t i o n
of th e h ig h -fre q u en c y p a r t of th e a r c . At te m p e ratu res below 250°C,
t h i s d i s t o r t i o n was e x h ib ite d a s a h ig h ly overlapped and dep ressed
sm all s e m i c ir le (see F i g . 7 .1 0 ). In th e e q u iv a le n t c i r c u i t , t h i s semi­
c i r c l e was re p re s e n te d by an a d d i t i o n a l segment c o n s i s t i n g of a r e s i s t ­
a n ce, Rgb, and a Warburg elem ent, Wgb, which a r e connected in p a r a l l e l .
The r e s i s t a n c e Rgb i s i n i t i a l l y s m a lle r than th e bulk r e s i s t a n c e , Rbu,
and in c r e a s e s more r a p i d l y w ith d e c re a sin g te m p e ratu re. This a d d itio n a l
d is p e r s io n c auses some d i f f i c u l t i e s in d e term in in g th e bulk r e s i s t a n c e
a t te m p e ra tu re s , where th e d i s t o r t i o n of th e complex plane diagram i s
a lre a d y a p p a r e n t, b u t n o t s u f f i c i e n t l y re s o lv e d to allow th e l e a s t sq u ares e s tim a tio n o f th e r e s i s t a n c e Rgb and th e Warburg adm ittance
Wgb. The bulk r e s i s t a n c e , e stim a te d w ith o u t ta k in g i n to account th e gb
segment of th e e q u iv a le n t c i r c u i t , i s b ia se d toward th e sum o f the tr u e
Rbu and Rgb. T h e re fo re , th e e stim a te d v a lu e of bulk c o n d u c tiv ity i s
lower than th e tr u e v a lu e . F o r tu n a te ly , Rgb i s much sm a lle r than Rbu in
th e tem perature range where proper e s tim a tio n of Rbu i s not f e a s i b l e .
The segment c o n s i s t i n g of Rgb and Wgb i s b e lie v e d to r e p r e s e n t the
g rain -b o u n d ary p o l a r i z a t i o n in N a jZ rjS ijP O ^ . At tem p e ratu res below
F l l a N2EA41. Spac NN1142. T -4 0 8 .7 °C . 59 f r a q s . Lav 2.
Modal CTOT. f , -i.O O E-O O t o f S9 -5 .5 7 E * 0 5 Hz.
7 p a r. Rm ar- 0 .5 4 8
.0 6
Cbu
.05
rUFi
________|Cgh
AA/V AAAr
Rbu
Rgh
.01
.00
.01
.02
.0 3
. 04
. 05
. 06
C a n d u e ta n e a
Cfl-1 1
T
-1 .5
T
T
3 .0
3 .6
.0 7
.0 6
.0 9
-2.0
L-2.5
( P - 3 .0
- 3 .5
- 4 .0
-1.0
-2.0
- 4 .0
0.6
1.2
2 .4
4 .2
4 .8
5 .4
6.0
log (f)
F i g . 7 .9 . Complex a d m ittan ce of th e Na3Zr2S i 2P012 sample w ith platinum
e le c tr o d e s a t 4 0 8 .5°C. The e stim a te d v a lu e s of
Rbu = 11.92 (2) Q,
Cbu = 6 .0 ( 8 ) x l0 -10 F,
Rgh = 75(3) a.
Wgh = 4.5 4 (4)x lO “ 4 a*1,
CPel = 3 .3 9 (l)x lO ~ 5 a- 1 ,
ANel = 0 .1 8 0 (1 ) .
param eters a r e
Cgh =1 .4 7 (3 )x lO " 6
F,
F l l a N 2E A 20. S p a c N N 1 1 4 2 . T - 2 0 2 .9 ° C . 5 9 f r a q s .
M o d a l CTOT, f j "l.O O E + O O t o fgg “ 5 . 5 7 E + 0 5 H z,
L av 2 .
p ar,
8
203
R m sr“ 0 .5 0 1
CPel
ANal
Q.
in
0 .4
0.2
0 .2
0 .4
0 .6
-2.0
2.0
0 .6
1 .0
1 .2
1 .4
1.
C o n d u c ta n c a
CIO-2 fl-1 3
2.2
2.4
-2 .5
- 4 .0 -4 .5
T
T
T
0.6
2 .4
3 .0
l o g Cf>
T
T
T
4 .2
4 .8
5 .4
-2.0
-3 .0
>
Wcn
- 4 .0
-5 .
3 .6
6 .0
F i g . 7.10 . Complex a d m ittan ce of th e Na2Zr2Si2P0^2 sample w ith platinum
e l e c t r o d e s a t 203°C. The e stim a te d v a lu e s of param eters a r e :
Rbu = 4 0 .2 (4 ) fl,
Rgb = 1 2 .1 (3 ) ft,
Rgh = 1 .0 8 ( 7 ) x l0 4 ft,
Wgh = 3 .3 2 (3 )x l0 ~ 5 ft"1 ,
CPel = 1 . 0 4 ( l ) x l 0 -5 ft"1 ,
ANel = 0 .1 6 5 (2 ) .
Wgb = 8 .3 ( 8 ) x l 0 " 5 fl"1
Cgh = 2 . 3 4 ( 6 ) x l 0 '6 F
F i l s N2EA13. Spac NN1142. T -1 3 4 .3 °C . 59 f r a q a .
Modal CTOT. f j " 1 .0 0 E + 0 0 t o f jg ” 5 . 57E+05 Hz.
Lav 2.
9 p a r, Rmsr" 0 .5 0 2
204
4 .2
3.6
C P al
A N al
2 .4
0 .6
0.6
I
2 .4
3 .0
3 .6
4 .2
C o n d u c ta n c e
CIO"3 0 " 1 1
4 .6
5 .4
6 .0
- 2 .5
-3 .0
>-- 3 . 5
- 4 .5 -5 .0
-
2 .0
-5 .0
0.0
0 .6
1.2
1.8
2 .4
3 .0
lo g (f)
3 .6
4 .2
4 .8
5 .4
6 .0
F i g . 7.11. Complex ad m itta n ce of th e Na.jZr2Si2P0^2 sample w ith platinum
e l e c t r o d e s a t 1 3 4 .5°C. The e stim ate d v a lu e s of param eters a re
Rbu = 179 .3(5) ft,
Cbu = 4 .6 (6 )x lO " U F,
Wgb = 4 .3 3 ( 6 ) x lO '5ft- 1 ,
Cgb = 4 .0 ( 3 ) x l 0 " 8 F,
Cgh = 1 .7 3 (5 )x lO "6F,
CPel = 6 .9 (2 )x lO -6 ft"1 ,
Rgb = 220(8) ft
Wgh = 7 .4 (2 )x lO " 6 ft-1
ANel = 0 .1 22 (8) .
3.6
F l l a N2EA0S. Spac NN1142. T - 4 9 . 8°C. 59 f r a q a .
Modal CTQT, f j • ! . 00E+0Q to fsa “ 5. 57E+05 Hz.
Lav 2.
9 pap. Rmsr- 0 .5 7 1
205
3 .2
C Pal
A N al
2 .9
—O
C 2 .0
1 .6
0 .6
0 .4
0.4
0. 6
1 .6
2 .0
2 .4
C o n d u c ta n c e
2 .6
3 .2
CIO"* fl" 1 1
3 .6
4 .0
4 .4
4 .6
-3 .5
-4 .0
- 5 .0 -5 .5
-3 .0
- 3 .5
- 4 .0
- 5 .0 -5 .5
-6.
0.6
2 .4
3 .0
lo g (f)
3 .6
4 .2
4 .6
5 .4
6 .0
F i g . 7 .1 2 . Complex a d m ittan ce of th e Na^Zr2Si2P0^2 sample w ith platinum
e le c tr o d e s a t 50°C. The e s tim a te d v a lu e s of param eters a r e :
Ro = 1 .0 0 (6 ) ka,
ANbu = 0 .1 0 5 (1 7 ) ,
Cgb = 4.7 1 (5 )x lO -8 F,
Rbu = 1 .3 4 (7 ) ka,
Rgb = 59(2) ka
CPbu = 7 . 3(12)xlO “ 10 a - 1 ,
Wgb = 3 .3 4 ( 2 ) x l0 “ 6 a - 1 ,
CPel = 1 .2 5 ( l ) x l 0 ~ 6 a ' 1 ,
ANel = 0 .0 4 9 (7 )
206
160°C, where th e g ra in -b ou nd ary p o l a r i z a t i o n i s w e ll- r e s o lv e d in th e
adm ittan ce spectrum , a p a r a l l e l c a p a c i t o r , Cgb, i s added to t h i s seg ­
ment in o rd e r to improve th e q u a l i t y o f f i t (s e e F i g . 7 .1 1 ). In th e same
tem peratu re range, th e segment gh of th e e q u iv a le n t c i r c u i t i s no
longer n e ce ssa ry to f i t th e e x perim en tal a d m ittan ce s p e c t r a . A b e t t e r
f i t i s o b ta in e d when a c a p a c i t o r , Cdl, i s i n s e r t e d in p a r a l l e l w ith the
CPE r e p r e s e n tin g th e e l e c t r o d e s .
A h ig h -fre q u en c y spur becomes v i s i b l e in th e complex plane p l o t s
of ad m ittan ce a t te m p e ra tu re s below -‘100°C, and below •'>60°C i t develops
in t o a p a r t of an a r c (see F i g . 7 .1 2 ) . I t i s re p re s e n te d in th e equ iva­
l e n t c i r c u i t by a c o n s ta n t phase element (CPbu, ANbu) and a r e s i s t a n c e ,
Ro, which co rresp on ds to th e h ig h -fre q u e n c y , i n t e r p o l a t e d i n t e r c e p t
w ith th e r e a l a x i s . The proposed i n t e r p r e t a t i o n o f th e h ig h -fre q u en c y
d is p e r s io n observed a t low te m p e ra tu re s fo r N ajZrjSijPO j^ i s a ls o in
terms of th e frequency-dependence o f hopping c o n d u c ti v ity , a s was th e
case fo r Na2 Zr 2 SiP 2 0 ^ 2 (see above).
7 .2 . R e s u lts of Microwave Measurements.
The microwave measurements re p o rte d h e re were made u sin g th e
c o n f ig u r a tio n of a c e n te re d E-plane s la b in a r e c ta n g u la r waveguide
(se e S e ctio n 4 .5 ) and were t h e r e f o r e f r e e o f th e c o n ta c t problem, which
a f f e c t e d most o f th e e a r l i e r measurements in course o f t h i s work. The
two fa c e s of th e sample which were in c o n ta c t w ith waveguide were po­
li s h e d and covered w ith s p u tte r e d Au or Pt e l e c t r o d e s . In e f f e c t , the
e l e c t r i c e x c i t a t i o n was a p p lie d to th e io n ic conductor in s i m i l a r way
207
to t h a t used in th e a . c . impedance measurements a t lower fre q u e n c ie s .
Experim ental s e tu p s were used in X-, K- and Ka-bands, and measure­
ments were made a t one frequency in each band (8 .3 7 , 2 4.0 and 38.9 GHz)
between 20 and 450°C. The v o lta g e r e f l e c t i o n c o e f f i c i e n t was measured
u sin g s l o t t e d - s e c t i o n te c h n iq u e . The te rm in a tin g impedance was con­
t r o l l e d by v a r i a b l e s h o r t c i r c u i t . The sample was held in a s t a i n l e s s
s t e e l s e c tio n of th e waveguide which was en closed by a s h o r t s p l i t - t u b e
fu rn ac e and p r e s s u r iz e d w ith dry helium gas. The tem p eratu re was s t a b i ­
l i z e d to ±0.5°C fo r each measurement.
The tem p eratu re dependence o f th e r e a l p a r t o f the microwave
c o n d u c ti v ity , o ' , i s d is p la y e d in form o f A rrhenius p l o t s in F igs. 7.13
and 7.14 fo r N a ^ ^ S i P j O ^ and Na3Zr2Si2P012i r e s p e c t i v e l y . The the
bulk c o n d u c ti v ity , o^, o b ta in e d from a n a l y s i s of th e complex impedance
s p e c t r a , i s included fo r comparison. The s t r a i g h t - l i n e segments which
connect d a ta p o in ts were o b ta in e d by f i t t i n g th e e x p re ssio n
OT = 0Qexp(-Ea /kT)
(7 .1 )
to th e c o n d u c tiv ity d a ta . The e x te n t o f a tem p eratu re regio n a s s o c ia te d
w ith a s i n g l e a c t i v a t i o n energy was determ ined by th e q u a l i t y of th e
l e a s t - s q u a r e s f i t . Although th e A rrh en ius p l o t s of microwave con­
d u c t i v i t y change s lo p e in th e low -tem perature re g io n , t h e i r approxima­
t i o n by s t r a i g h t - l i n e segments r e p r e s e n tin g E q .( 7 .1 ) i s u s e f u l a t l e a s t
fo r comparison. The e s tim a te d a c t i v a t i o n e n e r g ie s and p re e x p o n e n tia l
f a c t o r s fo r th e microwave c o n d u c ti v ity , o ' , a r e l i s t e d in Table 7 .2 ,
th e same d a ta fo r th e bulk io n ic c o n d u c ti v ity , o^, a re given Table 7 .1 .
For both compounds, th e a c t i v a t i o n e n e r g ie s f o r th e microwave conduC-
Temperature
[°C]
300
100
10
208
20
Na2Zr2S iP 2^12
1 0
[Q
cm
K]
1 0
0
1 0
a
1 0
38. 9 GHz
□ 24.0 GHz
-l
x 8.37 GHz
* bulk below 1 MHz
1 0 " 2 I------- 1------- 1------- L
1.2
F i g . 7.1 3 .
1.6
J
I
L
2.0
2.4
2.8
1000/T
D C 1]
J
I
3.2
L
3.
6
Temperature dependence o f th e c o n d u c ti v ity of Na2Zr2SiP2012
a t d i f f e r e n t fr e q u e n c ie s . E r ro rs a re s m a lle r than th e s i z e
o f p o in ts u n le s s marked by e r r o r b a r s .
209
Temperature
C°C]
300
100
10
20
T
T
T
Na3Zr3S i2^0 12
[ft
cm
K]
1 0
1 0
a
10
0
38.9 GHz
□ 24.0 GHz
x 8. 37 GHz
* bulk below 1 MHz
J
10'
1.2
F i g . 7 .14 .
1.6
2.0
2.4
2.8
1000/T
[K-1]
Temperature dependence of th e c o n d u c ti v ity of
1
3.2
L
^.
Na3Zr2S i 2P012
a t d i f f e r e n t f r e q u e n c ie s . E r ro rs a r e s m a lle r than the s i z e
o f p o in ts u n le s s marked by e r r o r b a rs .
6
Table 7.2
Estim ated a c t i v a t i o n e n e r g ie s E.a [eV] and p re e x p o n e n tia l f a c t o r s o O
. [n- 1 cm- 1 K] in v a rio u s ternp e r a tu r e re g io n s fo r th e microwave c o n d u c tiv ity of Na3Zr2S i 2P012 (x=2) and
Na2Zr2SiP20^2 ( x = l) .
Numbers in p a r e n th e s is a r e e stim a te d s ta n d a rd d e v ia t io n s of th e l e a s t s i g n i f i c a n t f ig u r e .
8.37 GHz
Temp.
Ea
38.9 GHz
24.0 GHz
*0
Ea
°o
Ea
°o
20 - 70
0 .1 4 8 (5 )
1 .4 ( 2 ) x l0 3
0 .1 18 (2)
1 .0 5 ( 9 ) x l0 3
0.1 00 (3 )
8 .5 (9 ) xlO2
70 - 120
0 ;192(4)
6 .4 ( 6 ) x l0 3
0 .1 6 2(2 )
4 .5 (4 ) XlO3
0 .1 3 5(4 )
2 .6 (3 ) xlO3
120 - 180
0 .2 9 0 (7 )
1 .2 ( 2 ) x l0 5
0 .2 2 0(9 )
2 .5 (6 ) xlO4
0.1 88 (4 )
1 .2 (2 ) xlO4
180 - 450
0 .16 3 (1 )
4 .9 (l)x l0 3
0 .14 8 (2 )
3 .9 (1 ) xlO3
0 .14 3 (1 )
3 . 9 0 (5 )x l0 3
20 - 100
0 .1 6 5 (3 )
1 .2 0 ( 8 ) x l0 3
0 .14 5 (5 )
1 .0 (l)x l0 3
0 .13 8 (1 )
1 .1 5 (3 ) x l0 3
100 - 210
0 .2 00 (3 )
3 .4 (2 ) xlO3
0 .1 6 9 (3 )
2 . 4 ( 2 ) x l0 3
0 .1 68 (1)
2 .8 (1 ) xlO3
210 - 450
0 .21 9 (2 )
5 .4 (1 ) xlO3
0.20 0(4)
4 . 7 ( 3 ) x l0 3
0 .1 84 (1)
4 .1 (1 ) xlO3
x=2
x=l
211
t i v i t y a r e lower than fo r th e c o rresp on d in g bulk io n ic c o n d u c ti v ity and
d e c re a se w ith in c r e a s in g frequency.
The tem p e ratu re dependence of th e r e a l p a r t of p e r m i t t i v i t y , e ' ,
i s d is p la y e d in F ig s. 7.15 and 7.16 f o r Na2Zr2SiP20 12 and
N a g Z ^ S ijP O ^ . r e s p e c t i v e l y . The in v e rs e te m p e ratu re s c a l e i s used to
f a c i l i t a t e comparison w ith th e A rrheniu s p l o t s of c o n d u c ti v ity . For th e
x=2 compound, a f t e r an i n i t i a l in c r e a s e w ith te m p e ra tu re , th e r e a l p a r t
of p e r m i t t i v i t y , e ' , d e c re a s e s w ith tem p e ratu re above ^150°C. The
maximum o f e ' c o in c id e s w ith th e te m p e ratu re re g io n of th e phase t r a n ­
s i t i o n in N a g Z ^ S ijP O ^ . For th e x=l compound, e ' in c r e a s e s w ith tem­
p e r a tu r e and l e v e l s o f f a t high te m p e ra tu re s . S a tu r a tio n of e ' i s f i r s t
seen a t h ig h e r fr e q u e n c ie s . For both compounds, th e r e a l p a r t of th e
p e r m i t t i v i t y d e c re a s e s w ith frequency.
The frequency dependence of th e r e a l p a r t o f th e microwave conduc­
t i v i t y , o ' , of Na2Zr2SiP2012 and Na3 Zr2S i 2P0 12 i s p l o t t e d fo r s e v e r a l
te m p e ra tu re s in F ig s . 7.17 and 7 .1 8 , r e s p e c t i v e l y . The bulk io n ic con­
d u c t i v i t y , o^, i s a l s o d is p la y e d fo r comparison p u rp o se s. Both th e r e a l
and th e imaginary p a r t s of th e ex p erim en tal complex c o n d u c tiv ity were
f i t t e d by a phenomenological formula
o(w) = o ' ( u ) + jo"(w ) = ob + A (j» )n + j we0e f ,
( 7 .2 )
u sin g th e n o n lin e a r l e a s t - s q u a r e s f i t t i n g d e s c rib e d in Chapter I I I . The
v a lu e o f th e bulk c o n d u c t i v i t y , cr^, was o b ta in e d from th e complex
impedance s p e c tr a and was t r e a t e d a s a c o n s ta n t d u rin g f i t t i n g . The
r e s u l t s o f th e f i t a r e re p re s e n te d by con tinu ou s l i n e s in F ig s. 7.17
212
Temperature
200
[°C]
100
50
20
38.9 GHz
XX
24.0 GHz
8.37 GHz
1.2
F i g . 7.1 5 .
1. 6
2. 4
2. 8
1000 /T CK_13
2.0
3. 6
3.2
Temperature dependence of th e r e a l p a r t of p e r m i t t i v i t y of
Na2Zr2SiP20j2 a t microwave f r e q u e n c ie s .
The e r r o r
b a rs
g iven fo r s e v e r a l p o in ts r e p r e s e n t t y p i c a l u n c e r t a i n t y .
213
TempQrature
C°C]
50
100
200
22
16AA
38.9 GHz
24.0 GHz
8.37-GHz
1.2
1.6
2.0
2. 4
’ 1000/T
F i g . 7.1 6 .
2. 8
3. 6
3 .2
CK"1]’
Temperature dependence of th e r e a l p a r t o f p e r m i t t i v i t y of
Na3Zr2Si2P0^2 a t microwave fr e q u e n c ie s .
The e r r o r
b a rs
given fo r s e v e r a l p o in ts r e p r e s e n t t y p i c a l u n c e r t a i n t y .
214
and 7 .1 8 . The e s tim a te d v a lu e s of th e exponent n a r e given in th e
f i g u r e s to th e r i g h t o f th e d a ta p o in ts . The framework d i e l e c t r i c
c o n s ta n t, e f , was found to be independent of te m p e ra tu re , th e v a r i a t i o n
of th e e stim a te d v a lu e s of
w ith te m p e ratu re being s m a lle r than th e
e stim a te d s ta n d a rd d e v i a t i o n s . During th e f i n a l f i t t i n g , an average
v a lu e of ef , o b ta in e d f o r a range o f te m p e ratu re f o r th e given com­
pound, was used a s a c o n s ta n t. The only e x c e p tio n to t h a t was found fo r
th e x=2 compound, a t te m p e ra tu re s below 140°C, where th e framework
d i e l e c t r i c c o n s ta n t, e^, could n o t be e s tim a te d . This occured because,
in t h a t re g io n , th e c a p a c i t i v e term j e Qe f was h ig h ly c o r r e l a t e d w ith
th e f r a c t i o n a l power-law term , A (j» )n , which d is p la y e d la r g e v a lu e s of
th e exponent. The exponent n de crea se d w ith te m p e ratu re fo r both com­
pounds, but v a r ie d over a wider range fo r th e x=2 compound.
7 .3 . D is c u s s io n .
The c o n d u c ti v ity o f a c l a s s i c a l ion hopping system in c r e a s e s w ith
frequency when th e s t a t i c e l e c t r i c f i e l d d i s t o r t s th e e q u ilib riu m d i s ­
t r i b u t i o n fu n c tio n [7 ,7 0 - 7 3 ]. The h ig h -fre q u en c y c o n d u c ti v ity , o ( « ) ,
depends on th e change o f t r a n s f e r r a t e s produced by th e s t a t i c e l e c t r i c
f i e l d . The s t a t i c c o n d u c ti v ity , o ( 0 ) , i s reduced by th e c u r r e n t - c u r r e n t
c o r r e l a t i o n s . The o v e r a l l v a r i a t i o n i s given by th e charge c o r r e l a t i o n
fa c to r, f
w
f c = o (0 )/o (« )
( 7 .3 )
which may become sm all a t low te m p e ra tu re s when th e l a t t i c e s i t e s a re
n ot e q u iv a le n t [73] or when d i f f e r e n t hopping r a t e s a re p re s e n t [73,75]
and fo r ordered system s of i n t e r a c t i n g c a r r i e r s [181]. In th e s im p le s t
215
[°C ]
0 . 20
0 . 31
300
0 . 441
0 . 55 :
180
E
*
0. 6 2 !
100
O
-
440
35
Q (go)
9
(iw)
10
Frequency
1
[Hz]
F i g . 7 .1 7 . Frequency dependence of th e r e a l p a r t of microwave c o n d u c tiv i­
t y of Na2Zr2SiP20j2 a t s e v e r a l te m p e ra tu re s . The p o in ts on th e
l e f t of th e f i g u r e i n d i c a t e th e bulk io n ic c o n d u c ti v ity ,
and have no co n n ectio n w ith th e frequency s c a l e . The s t r a i g h t
l i n e s r e p r e s e n t f i t s o f E q . ( 7 .2 ) , and th e e stim a te d v a lu e s of
th e exponent, n, a re in d ic a te d to th e r i g h t of d a ta p o i n t s .
216
10 0
»
I
i
i
t
i 11
i—i—
rrrrr
n
C°C]
440
— X o. 15
0. 24
X-
X
300
Y
f 1 0 '1b* a
0. 30
200
0. 42
150
0. 6 4 - |
E
0. 85
0
N
1
G
i i
100
40
-
b
10"2 r z
3 ( u ) =CTb+ i i a 6 Qe f -*-A (ioo) n
6f-1 3 .2
CT>140°C)
Na3Zr2S i2P012
2
0
^
B
10 9
J I______ 11____ 11___ tI
I___ L...
1 _ I_ L , l I___________
I___
1J->____ ■
L ,..1
101Q
FrequGncy
II II II Ii
1011
[Hz]
F i g . 7 .1 8 . Frequency dependence of th e r e a l p a r t of microwave c o n d u c tiv i­
ty of Na3z r 2Si2P0^2 a t s e v e r a l te m p e ra tu re s . The p o in ts on th e
l e f t of th e f i g u r e i n d i c a t e th e bulk io n ic c o n d u c ti v ity , o^,
and have no co n n ectio n w ith th e frequency s c a l e . The s t r a i g h t
l i n e s r e p r e s e n t f i t s of E q . ( 7 .2 ) , and th e e s tim a te d v a lu e s of
th e exponent, n, a re in d ic a te d to th e r i g h t of d a ta p o in ts .
217
approxim ation th e frequency-dependent c o n d u c ti v ity i s given by E q .( 2 .3 )
which can be r e w r i t t e n as
0 (» ) = o (0 ) + [ o ( « ) - o ( 0 ) ]
j« T /(l+ j» T )
+
( 7 .4 )
j»eoef ,
where th e t h i r d term d e s c r ib e s th e d i e l e c t r i c p o l a r i z a t i o n of th e r i g i d
framework of th e io n ic con d ucto r. The c h a r a c t e r i s t i c tim e,
t
,
may be
much s h o r t e r than th e in v e rs e of th e hopping r a t e , r , deduced from th e
d . c . c o n d u c ti v ity [7 3 ,7 5 ].
In th e case o f NASICON, u s in g a sim ple formula fo r th e c o n d u c tiv 9
^
+
i t y o=pe a r/3kT and assuming t h a t a l l Na io n s a r e mobile and t h a t
jumps occur between N a(l) and Na(2) s i t e s (a&3.5&), we g e t th e f o l ­
lowing low -frequency bounds fo r th e d i s p e r s i o n of c o n d u c tiv ity : fo r th e
x=l compound (p*7.6xlO
21
-3
cm ) r 2Qoc /2iiatO.5 MHz,
th e x=2 compound ( p a l . 1x10
22
.5 GHz; fo r
-3
cm ) r2 0oc /2 n a 3 .2 MHz, r 300oc /2 u * 2 .4 GHz.
At high te m p e ra tu re s , th e in c re a s e of c o n d u c ti v ity w ith frequency i s
expected in th e microwave re g io n . Near room te m p e ra tu re , th e e stim a te d
bounds f a l l in th e range covered by th e a . c . impedance sp ec tro sc o p y .
Thus, th e o n s e t o f th e c o n d u c ti v ity d i s p e r s i o n , t y p i c a l l y observed
above alOOkHz a t low te m p e ra tu re s (an a d d i t i o n a l a rc or an i n c lin e d
spur in th e complex adm itta n ce diagram s, see S e c tio n 7 . 1 ) , could be a
m a n if e s t a tio n of freq uency-dependent c o n d u c ti v ity .
Our d a ta do n o t show s a t u r a t i o n o f th e c o n d u c tiv ity a t high f r e ­
q u e n c ie s. Although o (» ) i s n o t known, th e r a t i o o ^ / o '( u ) may se rv e a s a
rough e s tim a te of th e charge c o r r e l a t i o n f a c t o r f c
For th e x=2 com-
pound, th e r a t i o 0b/ o '( w ) i s l a r g e r than t h a t fo r th e x=l compound. The
o b s e rv a tio n t h a t th e h ig h -te m p e ra tu re phase o f Na3Zr 2 S i 2 P0
1 2
e x h ib its
218
th e h ig h e s t v a lu e s o f o ^ / o 'f u ) c o r r e l a t e s w ell w ith th e fin d in g s of th e
s t r u c t u r e stud y o f th e s e compounds (se e S e c tio n 6 . 2 ) . Only in t h a t
phase do th e N a(l) and Na(2) s i t e s show s i m i l a r r e l a t i v e o ccu p an cies,
i n d i c a t i n g t h a t they a re n e a r ly e n e r g e t i c a l l y e q u iv a l e n t. Thus, th e
e f f e c t of th e in e q u iv a le n c e of l a t t i c e s i t e s on th e c o n d u c ti v ity , as
measured by f , i s s m a lle r than in th e case o f th e f u l l occupancy of
v
N a(l) s i t e s found f o r Na2Zr2SiP20 j 2 .
The ex p erim en tal frequency dependence of th e complex c o n d u c ti v ity ,
o ( u ) , cannot be reproduced by E q .( 7 . 4 ) , but i t i s w e l l - f i t t e d by
E q . ( 7 . 2 ) , which has been proposed by Jo nscher [78] as th e u n iv e r s a l
response fo r hopping system s. In a s e r i e s of p apers on Na 8-alum ina and
io n ic -c o n d u c tin g g l a s s e s [6 8 ,6 9 ] , Almond and West proposed a procedure
fo r th e d e te rm in a tio n of th e hopping r a t e and th e charge c a r r i e r con­
c e n t r a t i o n from th e frequency-dependent c o n d u c ti v ity e xp ressed as in
E q .( 7 .3 ) . The b a s i s fo r t h e i r p ro p o sal i s t h a t th e same th e rm a lly a c t i ­
v a ted hopping r a t e determ in e s th e d . c . c o n d u c ti v ity and th e frequencydependent term. As a d i r e c t consequence o f t h i s assum ption, th e a c t i v a ­
t i o n e n e r g ie s fo r th e frequency independent c o n d u c ti v ity , o^, and fo r
th e magnitude o f th e power-law term , A, should be r e l a t e d through th e
v a lu e of th e exponent n. The bulk c o n d u c tiv ity and th e microwave con­
d u c t i v i t y observed f o r both NASICON compounds do not e x h i b i t th e tem­
p e r a tu r e dependence governed by th e r e l a t i o n proposed by Almond and
West. T h e re fo re , a p p l i c a t i o n of t h e i r procedure fo r e x t r a c t i n g th e
hopping r a t e and c a r r i e r c o n c e n tr a tio n does n o t appear to be j u s t i f i e d
in th e case o f NASICON. The fr a c tio n a l-p o w e r frequency dependence of
219
th e c o n d u c tiv ity has been t h e o r e t i c a l l y o b ta in e d fo r hopping in d i s o r ­
dered systems [6 6 ], where th e exponent a l s o decrea se d w ith tem p e ratu re
[se e E g .( 2 .8 ) ]
The la r g e in c re a s e of c o n d u c ti v ity , observed a t room te m p e ra tu re ,
from
a t low fr e q u e n c ie s to o ' ( » ) a t microwave fre q u e n c ie s ,
( ° b/0(24GHz)atO.OO6 f o r x=l and *0.017 f o r x=2) and f u r t h e r ra p id i n ­
c re a s e of o ' through th e microwave re g io n («un w ith n*0.65 fo r x=l and
n*0.85 fo r x=2) i n d i c a t e t h a t o ' c o n ta in s c o n t r i b u t i o n s from p ro c e ss e s
o th e r than th e i n t e r s i t e hopping of io n s . I t i s l i k e l y t h a t th e lo c a l
motion o f Na+ io n s , r a t h e r than hopping between n eig hb oring Na s i t e s ,
i s r e s p o n s ib le f o r th e la r g e o ' a t low te m p e ra tu re s . In th e NASICON
s t r u c t u r e , th e d is ta n c e s from Na s i t e s to th e c o o rd in a tin g oxygen atoms
a r e la r g e r than th e sum o f io n ic r a d i i le a v in g space fo r lo c a l motion;
t h i s i s r e f l e c t e d by th e la r g e tem p e ratu re f a c t o r s fo r Na atoms ob­
serv ed in our neutron d i f f r a c t i o n study of th e s e compounds (see Chapter
V I). The c o n t r i b u t i o n from th e lo c a l motion i s c o n s i s t e n t w ith a low
a c t i v a t i o n energy a t low te m p e ra tu re s. A t r a n s i t i o n to predom inantly
i n t e r s i t e hopping, a s probed a t microwave f r e q u e n c ie s , occurs fo r both
compounds around 200°C, and a t h ig h e r te m p e ra tu re s a sim ple a c t i v a t i o n
b eh av io r o f th e microwave c o n d u c tiv ity i s observed (se e F ig s. 7.13 and
7 .1 4 ).
The s t r u c t u r a l phase t r a n s i t i o n in th e x=2 compound i s accompanied
by a ra p id in c re a s e in th e c o n d u c tiv ity between 120 and 180°C a s seen
in th e behavior of
and o ' . At te m p e ra tu re s above th e phase t r a n s i ­
t i o n , th e a c t i v a t i o n energy becomes s m a lle r than t h a t found between 70
220
and 120°C. This d istin g u ish es the high io n ic conductivity phase of
Na3Zr2S i2P012 from the x=l compound which, in the high-temperature
region, e x h ib its the highest a c tiv a tio n energy for microwave conductiv­
it y . An increase of a ctiv a tio n energy with temperature has been re­
ported for the microwave co n d u ctiv ities of h ollan d ite-typ e compounds
[6 4 ], ot-RbAg4I5 [56] and Na B-alumina [6 0 ,6 1 ]. This can be in tu itiv e ly
explained by noting th a t, at higher temperatures, ions move over larger
d istan ces during the time in terval probed at a given frequency and have
to cross higher p o ten tia l barriers than i s the case for the low e f fe c ­
tiv e barriers of the lo c a l bounce-back hopping which dominates at low
temperatures.
To summarize, the frequency dependence of the conductivity in the
microwave region observed for the two NASICON compounds can be ration­
a liz ed on b asis of ion-hopping transport. Together with the r e su lts of
structure stu d ie s, an extension of the frequency range beyond that
covered in th is work should provide a base for explaining the mechanism
of io n ic transport in those complicated stru ctu res.
VIII. SUMMARY
Dense, p o l y c r y s t a l l i n e samples of two compounds, Na2zr2SiP20j2 and
Na3Zr2S i 2POj2 , (x=l and x=2 in th e NASICON s o l i d s o l u t i o n range
^Nal+ x Zr2S*xP3 -x °1 2 ' 0>x>3)> bave been pre p a red by s o l i d - s t a t e re a c ­
t i o n s between m echanically-m ixed and compacted re a g e n ts which occurred
d u rin g s i n t e r i n g .
The e l e c t r i c a l p r o p e r t i e s of th o s e compounds have been i n v e s t i ­
g a ted by th e impedance sp ec tro sc o p y u sin g c e l l s w ith io n -b lo c k in g e l e c ­
tr o d e s in th e frequency range 1 to 7xlO^Hz.
The method fo r n o n lin e a r
l e a s t - s q u a r e s a n a l y s i s of th e complex im pedance/adm ittance s p e c t r a has
been g r e a t l y improved here by use of a w eig hting scheme based on r e s o ­
l u t i o n of th e measuring equipment.
A p r a c t i c a l computer program has
been developed f o r i n t e r a c t i v e , n o n lin e a r l e a s t - s q u a r e s a n a l y s i s and
g r a p h ic a l p r e s e n t a t i o n of th e complex impedance (a d m itta n c e ) s p e c t r a .
S e v e ra l models of th e a . c . respo nse of s o l i d e l e c t r o l y t e / e l e c t r o d e s
c e l l s have been in c o rp o ra te d in to th e program in ord er to f a c i l i t a t e
i t s a p p l i c a t i o n to a v a r i e t y of system s.
The a . c . impedance s p e c t r a of both Na2Zr2SiP20 12 an<J Na3Zr2s i 2P012
samples w ith io n -b lo c k in g e l e c t r o d e s were f i t t e d by th e use of equ iva­
l e n t c i r c u i t s which c o n ta in e d c o n s ta n t phase elem ents to model th e
observed s p e c t r a .
In th e case of Na2Zr2SiP20 12 , a g ra in-bo u nd ary
p o l a r i z a t i o n was found to e x i s t over th e te m p e ratu re range 50 to 410°C.
In th e case of Na3Zr2S i 2P0^2> two d i s t i n c t p o l a r i z a t i o n p ro c e ss e s were
observed in th e h ig h - and th e low -tem p erature re g io n s .
221
In both c a s e s ,
222
th e io n -b lo c k in g e le c tr o d e s were a d eq u a te ly modelled by a c o n s ta n t
phase elem ent.
For both compounds, th e o n set of an a d d i t i o n a l d i s p e r ­
s io n observed in th e h ig h -fre q u en c y p a r t of th e impedance s p e c t r a a t
low te m p e ratu res was i n t e r p r e t e d in terms of frequency-dependent
hopping c o n d u c tiv ity .
O bserv ation s of th e frequency dependence of th e c o n d u c ti v ity were
extended in to th e microwave re g io n .
New te c h n iq u e s fo r m easuring th e
complex p e r m i t t i v i t y a t microwave fr e q u e n c ie s have a l s o been developed.
In one in s ta n c e , n o n lin e a r l e a s t - s q u a r e s e s tim a tio n was a p p lie d fo r
d a ta re d u c tio n in v a r i a b l e - t e r m i n a t i o n measurements in a waveguide; in
a second c a s e , a rig o ro u s method fo r d e te rm in a tio n of th e complex
p e r m i t t i v i t y of a c e n te re d E-plan s la b in r e c ta n g u la r waveguide was
developed in o rd e r t o e lim in a te th e l o s s of c o n ta c t problem encountered
in our e a r l i e r measurements.
The l a t t e r method employed a complete
s o l u t i o n of th e boundary v alu e problem to s o lv e th e in v e rs e problem of
fin d in g th e c o n s t i t u t i v e p aram eters of a c e n te re d s la b from th e ob­
serv ed s c a t t e r i n g of th e fundamental waveguide mode.
The microwave
c o n d u c t i v i t i e s o f both compounds a r e h ig h e r than th e r e s p e c tiv e lowfrequency bulk c o n d u c t i v i t e s , and in c r e a s e w ith frequency acc o rd in g to
a power-law dependence. The exponent o f th e power-law dependence de­
c r e a s e s w ith te m p e ra tu re . The frequency dependence of c o n d u c tiv ity
observed in th e microwave reg io n a g re e s w ith th e g e n e ra l p r e d i c t i o n s of
hopping models. At low te m p e ra tu re s , l o c a l motion of io n s , w itho ut
s i g n i f i c a n t c ro s s in g of th e p o t e n t i a l b a r r i e r s , dominates th e microwave
c o n d u c tiv ity . The a c t i v a t i o n energy fo r th e microwave c o n d u c tiv ity
223
d e c re a s e s w ith frequency and i s lower than t h a t fo r th e low -frequency,
bulk io n ic c o n d u c ti v ity . The e f f e c t s of th e phase t r a n s i t i o n o c c u rrin g
in th e x=2 compound a r e a p p a re n t in th e observed microwave c o n d u c tiv ity
and p e r m i t t i v i t y . The c o n d u c tiv ity beh av io r of th e two compounds a g re e s
w ith th e fin d in g s of th e s t r u c t u r e s tu d y .
The c r y s t a l s t r u c t u r e s of Na2z r 2SiP20 12 and Na3Zr2S i2P02 were
determ ined a t room te m p e ratu re and a t 300°C by th e n eu tro n powder
d i f f r a c t i o n u sin g p r o f i l e refin em en t based on th e R ie tv e ld te c h n iq u e .
The high io n ic c o n d u c ti v ity o f th e x=2 compound observed a t tempera­
t u r e s around 300°C was r a t i o n a l i z e d on th e b a s i s of th e wide openings
between th e N a(l) and Na(2) s i t e s and a high r e l a t i v e occupancy f a c to r
of th e Na(2) p o s i t i o n s which o ccurred only fo r th e x=2 compound a t
e le v a te d te m p e ra tu re s .
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(1983).
12. J . B. Goodenough, "S k eleton S t r u c t u r e s " , in Ref. 5 pp. 393-415.
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224
225
S u p erio n ic Conductor, Na3 Sc 2 P3 0 12", in Ref. 1, pp. 431-433.
14. N. Nagai, S. F u j i t s u and T. Kanazawa, " Io n ic C on d u ctiv ity in th e
System NaZr2 (P 0 4 ) 3 - Na3 Y2 (P04 ) 3" , J . Am. Ceramic S o c ., 63. 476-477
15. C. Delmas, J . C. V ia la , R. Olaczuaga, G. LeFlem and P. Hagenmuller,
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S o lid S ta te I o n ic s , M l . 209-214 (1981).
1+x z_x x
* J
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9 & 10, 809-812 (19837?
X Z_X i+X
17. H. Y-P. Hong, J . A. K a fa la s , and M. Bayard, "High Na+-Io n Conducti­
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18. R. D. Shannon, H-Y. Chen and T. B e rz in s, " Io n ic C o n d u ctiv ity in
Na5 GdSi 4 0 12", Mat. Res. B u l l . , 12. 969-973 (1977).
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H. Y-P. Hong, " C ry s ta l S tr u c t u r e and Io n ic C o n d u ctiv ity of
Li, .Zn(GeO-). and Other New Li+ S u p e rio n ic C onductors", Mat.
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Res.
20. H. U. B ey eler, " C a tio n ic Short-Range Order in th e H o lla n d ite
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(1976).
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22, 805-807 (1977).
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APPENDICES
Appendix A. D e ta il s o f The Impedance Matching and C a l ib r a tio n fo r The
C om puter-C ontrolled Impedance Measuring System.
Impedance Matching
The d e te rm in a tio n of th e unknown impedance r e l i e s on comparison of
two a . c . v o lta g e s V and V„ [se e P i g . 3.1 and E g .( 3 . 1 ) ] . The b e s t accu0
i
racy i s expected when th e v a lu e o f th e re fe re n c e r e s i s t a n c e , Rr> i s of
th e same ord er as th e a b s o lu te v a lu e of th e unknown impedance, IZ |.
The r a t i o of th e two impedances i s equal to th e r a t i o , SD, of th e
am plitude of th e a . c . v o lta g e over th e unknown |Vs | to th e am plitude of
th e a . c . v o lta g e over th e r e fe re n c e r e s i s t o r IVR| . This r a t i o i s c a lc u ­
l a t e d u sin g th e measured v a lu e s of |VL | , |V5 I and
SD =
'V
/|V r
lVR| ~
V s
-
- cos0
\2
•
+ sin
2
0
1 /2
(Al)
0
The re fe re n c e r e s i s t a n c e i s accepted by th e program (s u b ro u tin e
MEASA) when th e c a l c u l a t e d r a t i o , SD, f a l l s in th e range between
bounds: low er, SDU, and upper, SDO; which a re chosen a s follow s
SDO = minf (2.268 - 2 .2 52 co s0)“ 1 / 2 , 2 .2 1
SDU = maxir (10 .6 - 6.2cos0 ) - 1A// 2 , 0.36 1J
The above bounds, which depend on th e phase d i f f e r e n c e ,
(A2)
0
, were e s t a b ­
l i s h e d e x p e rim e n ta lly by t r i a l and e r r o r in o rder to f a c i l i t a t e smooth
changes of th e re fe re n c e r e s i s t o r s in th e sequence 1, 3, 10, 3 0 . . . fo r
th e unknown impedances e x h ib itin g v a r i e t y of phase a n g le s . Mhen SD>SDO,
238
239
th e next h ig h e r matching r e s i s t a n c e i s chosen; when SD<SDU, th e next
lower matching r e s i s t o r i s sw itched. When th e h ig h e s t a v a i l a b l e r e s i s ­
t o r i s in use (300 kg) and s t i l l SD>SD0, th e program s i g n a l s th e
problem and g iv e s th e u s e r an o p tio n to proceed w ith extended mismatch.
When th e extended mismatch i s allow ed, th e r a t i o SD can be f i v e tim es
h ig h e r. When SD>5xSD0, th e unknown impedance i s c o n sid e re d to be out of
range fo r th e m easuring system.
Gain- Phase C a l ib r a tio n
The g a in and th e phase c h a r a c t e r i s t i c s of th e s i g n a l channel
r e l a t i v e to th e r e f e r e n c e channel a re approximated by f u n c tio n s of th e
freq u en cy , f , of th e form
g ( f ) = a^ + a 2 f + a 3f 2 + a ^ / f + a g/ f 2 + a gl n f
(A3)
Six s e t s o f c o e f f i c i e n t s a ^ - a g a re includ ed in th e c a l i b r a t i o n f i l e
(CALIBR.DA) fo r th e g a in and fo r th e phase a t th r e e d i f f e r e n t s ig n a l
le v e l s e t t i n g s . During impedance measurement, th e measured s ig n a l am­
p l i t u d e i s d iv id e d by th e c a lc u la te d g a in and th e phase c o r r e c t i o n i s
s u b tr a c te d from th e measured phase.
In o rd e r to e s t a b l i s h th e v a lu e s of th e g a in -p h a se c o e f f i c i e n t s ,
c a l i b r a t i o n measurement of a w ell c h a r a c te r iz e d r e s i s t o r , Rc *100 Si, i s
made over th e d e s ir e d frequency range u sin g a c a l i b r a t i o n program
AICAL. The re f e r e n c e r e s i s t o r range i s s e t to #8 (Rr0 =lOO.O8 si) and
kept unchanged ( f ix e d range o p tio n ) . The c a l i b r a t i o n f i l e used fo r t h i s
measurement must have a l l a R c o e f f i c i e n t s equal 0, ex cept fo r a ^ l fo r
th e g a in . The r e s u l t i n g d a ta f i l e i s p ro c e ssed by program FIGAIN (on
PDP11) which uses l e a s t - s q u a r e s f i t t i n g to e s tim a te th e b e s t v a lu e s of
240
c o e f f i c i e n t s in E g .(A3). A ccurate v a lu e s of th e r e f e r e n c e , Rf l and th e
c a l i b r a t i o n r e s i s t o r s , Rc , must be s u p p lie d to th e program, a s w ell as
approxim ate v a lu e s o f th e s p u rio u s c a p a c ita n c e s and in d u c ta n ce s a s s o ­
c i a t e d w ith th e r e f e r e n c e (C r
Lr ) and w ith th e in stru m e n t in p u t (C_,
s
Ls ) (see F i g . 3 . 2 ) . During th e l e a s t - s q u a r e s e s ti m a ti o n , th e c o e f f i c i e n t s whose e stim a te d s ta n d a rd d e v ia t io n s a r e l a r g e r than t h e i r abso­
l u t e v a lu e s should be s e t equal zero and f ix e d . The r e s u l t i n g v a lu e s of
c o e f f i c i e n t s a r e s t o r e d in f i l e CALIBR.DA u sin g e d i t o r ( e . g . TECO on
PDP8).
C a l i b r a t i o n of Spurious Impedances
The param eters o f th e c i r c u i t r e p r e s e n tin g th e a . c . s ig n a l path in
th e impedance m easuring system (se e F i g . 3 .2 ) a r e determ ined in th e
fo llo w in g way. Values of th e re fe re n c e r e s i s t o r s a r e measured u sin g a
p r e c i s i o n d i g i t a l ohmmeter which can be connected between th e two
in p u ts of th e r e f e r e n c e r e s i s t o r s switchbox. The r e f e r e n c e r e s i s t o r s
can be sw itched manually u sin g a d i a l provided by th e c o n t r o l l e r i n t e r ­
fa c e u n i t .
The c a l i b r a t i o n measurements aimed a t d e term in in g th e s p u rio u s
impedances a re made u sin g program AICAL. The c a l i b r a t i o n f i l e used fo r
th e s e measurements must c o n ta in proper v a lu e s of th e g a in -p h a se c o e f f i ­
c i e n t s . The range o f th e r e fe re n c e r e s i s t a n c e should be f r e e l y a d j u s t ­
a b le by th e program. The c o r r e c t v alu e o f th e in p u t impedance should be
s u p p lie d in form of r e s i s t a n c e , CR, in s e r i e s w ith in d u c ta n c e , CL, and
p a r a l l e l to c a p a c ita n c e , CC.
The i n i t i a l c a l i b r a t i o n , which e s t a b l i s h e s th e v a lu e s of Lr , Cf
and Rp , i s c o n v e n ie n tly performed u sing c a p a c i t o r s and r e s i s t o r s plug­
ged d i r e c t l y in to th e in p u t of th e re fe re n c e r e s i s t o r s sw itchbox. The
e f f e c t s of sp u rio u s ind u c ta n ce s a re most sev e re a t low impedance ra n ­
g e s, whereas th e e f f e c t s of sp u rio u s c a p a c ita n c e s a re s i g n i f i c a n t a t
high impedance ran g es. At l e a s t th r e e measurements over a range of
fre q u e n c ie s should be made on high impedance range; e . g . , u sin g an open
in p u t, a 300 kfl r e s i s t o r and a 50 pF c a p a c i t o r . For measurements of an
open in p u t and of a c a p a c i t o r , th e frequency scan should be lim it e d to
th e frequency range whe^e impedance matching i s p o s s ib le . Two measure­
ments a r e needed a t low impedance rang es; e . g . , u sin g a 4 S r e s i s t o r
and a 5 uF c a p a c i t o r . The r e s u l t i n g f i l e s which c o n ta in c a l i b r a t i o n
d a ta f o r a given s ig n a l le v e l o b ta in e d u sing d i f f e r e n t in p u t impedances
can be merged using program MERCAL (on POP 11).
Program FICATO which i s used fo r f i t t i n g th e v a lu e s of sp u rio u s
impedances u ses th e merged c a l i b r a t i o n d a ta f i l e . The e s tim a tio n of
param eters and d ia lo g w ith th e program a re s i m i l a r to th o se employed in
program FIRDAC (se e Appendix C). The u s e r s p e c i f i e s th e re f e r e n c e range
fo r which c a l c u l a t i o n s a re re q u ire d and th e program ta k e s in to account
only measurements made u sin g t h a t range (an o p tio n i s provided fo r
in c lu d in g measurements made w ith extended mismatch on range #1). The
choice of range #1 (Rr ^a<300 kfl) i s recommended fo r d e te rm in a tio n of the
s p u rio u s c a p a c ita n c e s Cr , c , and th e e f f e c t i v e r e s i s t a n c e of th e
v o lta g e probe, Rp , whereas range #11 (Rr j^a«3.18 Q) i s most u s e f u l fo r
d e te rm in a tio n of sp u rio u s in d uctan ces LIT, L5 , and s e r i a l r e s i s t a n c e ,
242
FSt . During f i t t i n g f o r th e range #1, t h e v a l u e s of L_,S
L_t and R_5 must
be kept c o n s t a n t ( f i x e d a t approximate v a l u e s ) . S i m i l a r l y , during
f i t t i n g fo r the range #11, th e v a lu e s of C , C and R
"
*
must be f ix e d.
r
Once t h e v a lu e s of the s p u ri o u s impedances a s s o c i a t e d Nith the
r e f e r e n c e r e s i s t o r s switchbox (Cr , Lf ) and th e v o l t a g e probe (Rp ) a re
e s t a b l i s h e d , th e c a l i b r a t i o n fo r th e in duc ta nce and c a p a c i t a n c e of
sample hold er in v o lv e s only two measurements. Using program AICAL, the
impedance of an open ho ld e r i s measured (with the c o n t a c t s s e p a r a t e d
more than du rin g measurements with a t y p i c a l sample) in ord e r t o d e t e r ­
mine th e t o t a l s p u r i o u s c a p a c i t a n c e , C
5
A well c h a r a c t e r i z e d r e s i s t o r
R *3 S i s placed between th e c o n t a c t s in p lace of th e sample (or
c
connected elsewhere in s e r i e s with s h o r t - c i c u i t e d c o n t a c t s ) f o r d e t e r ­
mining the t o t a l s p u r i o u s in d uc ta nc e , L_.
Program FICATO i s used again
5
and only C_
s i s a d j u s t e d f o r measurements on range #1 whereas L_
s and R_s
a r e a d j u s t e d usi ng range #11.
The v a lu es of t h e r e f e r e n c e r e s i s t a n c e s , th e s p u r io u s induc ta nce s
and c a p a c i t a n c e s a r e i n c o r p o r a te d (u s in g e d i t o r ) i n t o c a l i b r a t i o n f i l e
CALIBR.DA and a r e used by program AIMESR d u rin g measurements of unknown
impedances. The format of f i l e CALIBR.DA c a l l s fo r s e p a r a t e v a l u e s of
Rp. Cr , Cg for t h e t h r e e s i g n a l l e v e l s . Although t h e v a lu e s of Rp , Cf ,
Cs should not depend on the s i g n a l l e v e l , the e s t i m a t e d v a lu es a re
u s u a l l y s l i g h t l y d i f f e r e n t f o r c a l i b r a t i o n a t d i f f e r e n t l e v e l s which
may i n d i c a t e d i s c r e p a n c i e s in th e ga in -p h a se c a l i b r a t i o n .
243
Appendix B. Algorithm fo r Nonlinear R e g r e s s i o n .
Our a lg o r it h m fo r min imization of t h e o b j e c t i v e f u n c t i o n
[ E g . ( 3 . 1 0 ) ] has been developed and optimized g r a d u a l l y , f i r s t in con­
n e c t i o n with t e s t i n g of a r t i f i c i a l l y - g e n e r a t e d d a t a , and then when
improvements were needed fo r s u c c e s s f u l a n a l y s i s of th e exp erimental
d a t a . I t i s a combination of th e Gauss-Newton method with an a d j u s t a b l e
s t e p and a d a p t i v e Marquardt method [116]. Control of th e bounds on
pa ram ete rs and c a l c u l a t i o n of Marquardt terms i s s i m i l a r t o t h a t found
in "Nonlinear Reg ression Routines" developed a t U n i v e r s i t y of
Wisconsin, Madison [117]. Use of s t e p bound, a d j u s t a b l e on th e b a s i s of
l i n e a r i t y of changes of t h e o b j e c t i v e f u n c t i o n , i s p a t t e r n e d a f t e r an
a l g o r i t h m developed in Argonne Na tional Laboratory [118,119].
The b a s i c formula f o r c o r r e c t i o n of t h e para mete rs in each i t e r a ­
t i o n i s given by E q .( 3 . 1 4 ) . P r i o r to m a t r i x i n v e r s i o n , th e system i s
T
s c a l e d by t h e square r o o t s of th e di ag onal elements of m a tr ix B=A GyA.
The s c a l i n g m a tr ix i s updated in each i t e r a t i o n
°»}j =
° m j ■ 0 £or
(B1>
and t h e s c a l e d c o r r e c t i o n i s c a l c u l a t e d as
AXS =- s (Dn AT Gy ADn + Xpl)’ 1 DyATGy [Y-F(X)]
(B2)
where xp i s t h e Marquardt term, s i s s t e p - r e d u c t i o n f a c t o r , i n i t i a l l y
s = l , XQ=0. In v e rs io n of th e symmetric, p o s i t i v e d e f i n i t e m a tri x in
Eq.(B2) i s accomplished n um e ric al ly v i a th e Cholesky decomposition
[161]. I f t h i s m a tr ix i s found a l g o r i t h m i c a l l y not p o s i t i v e d e f i n i t e ,
then th e l a r g e r Marquardt term i s used Xp+1 = xp + 0 .1 .
The squared norm of scaled correction
244
cL = I (Ax.s ) 2
X i=l
1
(B3)
i s a measure of th e s t e p s i z e . I f d A i s g r e a t e r than the a d j u s t a b l e
s t e p bound,
a
,
c o r r e c t i o n i s reduced by f a c t o r /A/d X . F ur th er r e d u c ti o n
of AX may be n e ce s sa ry in o rd er to keep t h e parameters w i t h i n p re de ­
termined bounds. Whenever t o t a l re d u c ti o n f a c t o r , s , f a l l s below c e r ­
t a i n l i m i t ( s e t t o 0 . 0 5 ) , th e next Marquardt term i s c a l c u l a t e d a c ­
cording to th e formula
U X 5)T 4XS
P+1
Xp
+ (4XS)T (B + Xpl)"1 4XS
(
'
and Eq.(B2) i s e v a l u a t e d a gai n; v a lu e s of Xp a r e s t o r e d . I f , d ur in g th e
c u r r e n t i t e r a t i o n , t h e squared norm of th e c o r r e c t i o n p r i o r to reduc­
t i o n f a l l s below
a/
2 then Xp i s r e pl a ce d by X ^ a t the beginning of
th e next i t e r a t i o n ; o t h e r w i s e , th e c u r r e n t x
is retained.
P
A new value of t h e o b j e c t i v e fu n c t i o n S(X+aX) i s compared with
S(X) and th e change of S i s compared with a l i n e a r term
S = 2AXTATGy [Y-F(X)]
(B5)
I f S(X+AX)<S(X) and [S(X+aX )-S (X )]/ aS > 0 .2 5 , t h e c o r r e c t i o n X i s
acce pte d and a new i t e r a t i o n s t a r t s . I f S(X+aX) > 10 S(X), th e next
Marquardt term i s used [see Eq.(B4)]. Otherwise, th e c o r r e c t i o n AX i s
reduced by a f a c t o r s f which minimizes
aq u a d r a t i c
S(X+saX) a t p o i n t s s=0, s=l with d e r i v a t i v e
at
fu n c t i o n equal
s=0 e q u a l l i n g
to
as
aS/2
s
=
( 86)
AS+S(X)-S(X+AX)
The factor s r is further restricted to the interval 0.3<sf<0.6 .
245
The s t e p bound
i s reduced by
sr
a
i s updated in fo ll ow in g way. I f th e
of Eq.(B6)
i f [S(X+aX)-S(X)]/aS > 0 .5
A =
correction
s r s r A;
then
A
= max(4d
X
(B7)
.A)
The t e s t of convergence i s based on th e change of S(X) and on th e
squared norm of t h e c o r r e c t i o n d
X
When X =0 and t h e r e i s no s t e p
O
r e d u c t i o n caused by bounds of p a ra m et e rs , convergence i s a s s e r t e d and
r e g r e s s i o n i s completed when t h e fo ll o w i n g c o n d i t i o n s a r e f u l f i l l e d :
|S(X+AX)-S(X)| < t xS(X)
dx < t 2S(X)
(B8)
where t ^ , t 2 a r e t o l e r a n c e c o n s t a n t s ( t y p i c a l l y t ^ = 0.0001, t 2 =
0 . 00001 ) .
In c a s e s where some pa rame te rs a r e not w e l l - d e f i n e d , t h e c o n d i t i o n
on dx may be not f u l f i l l e d and no s i g n i f i c a n t change in S(X) t a k e s
p l a c e . I f such o s c i l l a t i o n s of pa ram ete rs p e r s i s t f o r s e v e r a l consecu­
t i v e i t e r a t i o n s , then min im izat ion i s completed with a warning.
The only s i t u a t i o n , which l e a d s t o f a i l u r e of m in im i z a ti o n , i s
accumulation of more Marquardt terms than the d e c l a r e d l i m i t (20
te rm s ). This means t h a t a d e c r e a s e of o b j e c t i v e f u n c t i o n cannot be
accomplished by t h e a l g o r i t h m . This occu rs when e i t h e r t h e i n i t i a l
guess es of th e v a l u e s of t h e pa ram ete rs a r e v e ry poor or t h e model
f u n c t i o n does not resemble th e d a t a .
The a lg o r it h m m a i n t a i n s v a l u e s of t h e pa ra me te rs w i t h i n bounds
s p e c i f i e d in d e f i n i n g t h e model f u n c t i o n . A f te r c o r r e c t i o n of
t h e para mete rs aX has been
c a l c u l a t e d acc o rd in g to Eq.(B2),
th e a l g o ­
rith m t e s t s whether t h e new val ue X+aX i s w i t h i n bounds. When, p r i o r to
246
c o r r e c t i o n , t h e parameter
i s equal t o i t s bound, then t h e s i g n o£
th e c o r r e c t i o n i s checked. I f th e s i g n i s such t h a t ax^ d i s p l a c e s x^
o u t s i d e of th e bounds, then th e parameter x^ i s f i x e d by removing the
column and t h e row j from Eq.(B2) and th e c o r r e c t i o n i s r e c a l c u l a t e d .
The l a r g e s t s t e p r e d u c t i o n f a c t o r , s b , (0<sb < l ) , such t h a t X+sbAX
s a t i s f i e s t h e bounds, i s found and th e c o r r e c t i o n i s reduced acc ord ­
ingly.
I t should be poin te d out t h a t s t r i c t obedience of th e bounds may
pre v e n t f i n d i n g a minimum w it h in th e bounds when th e min im iz ation pa th ,
p r e f e r r e d by th e a l g o r i t h m , goes through p o i n t s o u t s i d e t h e bounds. I f
such a problem o c c u r s , th e
bounds can be re la x e d by th e u s e r . On the
o t h e r hand, imposing narrower bounds may speed up convergence in a case
when approximate v a l u e s of c e r t a i n para mete rs a r e known; however, for
t h e remaining pa ram ete rs c o r r e c t i n i t i a l v a l u e s w i l l not be a v a i l a b l e .
247
Appendix C. Program FIRDAC fo r Nonlinear Le a st -S a ua re s A na lys is of the
Complex Impedance/Admittance S p e c t r a .
Embedding New Model Functions i n t o the Program FIRDAC
In orde r t o add a new model f u n c t i o n t o ch oi ce s a v a i l a b l e in pr o ­
gram FIRDAC, one has to update the s u b r o u ti n e CHOFIT by w r i t i n g two
s u b r o u t i n e s : Dmodel which i s used du rin g f i t t i n g and Smodel used for
p l o t t i n g . C a l l s t o them a r e e f f e c t e d through t h e s u b r o u t i n e s DERIVF and
SYZYIR, r e s p e c t i v e l y . Changes in e x i s t i n g s u b r o u t i n e s and the s t r u c t u r e
of new s u b r o u t i n e s a r e d e s c r i b e d he re . Upper case de no tes re q u ir e d
p a r t s of code, lower case i n d i c a t e s and e x p l a i n s model dependent code.
The model f u n c t i o n can be removed from program FIRDAC by making changes
o p p o s it e t o th o s e given below.
The fol lo wi ng l i m i t a t i o n s on the number of parameters a r e imposed
by d e c l a r a t i o n s of a r r a y s ; fo r example,
( i ) th e maximum t o t a l number of parameters NRT i s 26, which i n c lu d e s
parameters of a s s o c i a t e d with measurement c o r r e c t i o n s i f t h e i r
e s t i m a t i o n i s r e q u i r e d (3 fo r frequency range up t o 0.5MHz plu s 5
fo r t h e r a d i o frequency ra ng e );
( i i ) t h e maximum number of s im u lta ne ous ly e s ti m a te d pa rameters NRA i s
15.
The l i m i t on t h e t o t a l number of pa rameters can be i n c re a s e d by
changing d e c l a r a t i o n s of X, XNAM, DX in the main program FIRDAC, IPFX,
LIM, XFX in COMMON/FIXP/, XLIM in s u b r o u ti n e NLMFLX and XMIN i s subrou­
t i n e DERIVF. The number of e st i m a te d para mete rs i s l i m i t e d by d e c l a r a ­
t i o n s of CX in FIRDAC and XI, XT, DN, ETA, A in NLMFLX.
248
The s u b r o u t i n e CHOFIT allows th e us er t o choose a model f u n c ti o n
from th e implemented s e t . The number IDER a s s ig n e d t o th e new model
f u n c t i o n (s u c c e s s i v e to t h a t p r e v i o u s l y used) and an acronymic name
must be inc luded in t h e 216 FORMAT. A new l a b e l , corresp on din g to IDER,
i s added to th e computed GOTO st a t e m e n t and a new s e c t i o n of th e code,
s t a r t i n g with t h i s l a b e l , must d e f i n e th e number of model parameters
NR, 4 c h a r a c t e r acronyms of t h e parameters XNAM(l)
XNAN(NR) and
acronym of th e model XNAM(NR+1). I t i s recommended t h a t parameters a r e
ordere d in a way which s i m p l i f i e s c a l c u l a t i o n of th e p a r t i a l d e r i v a ­
t i v e s in th e s u b r o u t i n e Dmodel. Following IF(IDR.LE.l) and GOTO 50, a r e
WRITE s t a t e m e n t s producing, t o g e t h e r with th e co rresponding FORMAT
statements, a
p r i n t o u t of th e d e s c r i p t i o n of th e model.
The model fu n c t i o n f( «, X) i s d e fi n e d by th e s u b r o u t i n e Dmodel
which c a l c u l a t e s r e s i d u a l s C (I) =Y (I )- f(0 M (I1 ),X ) and Jacobian A ( I , J )
[ E q . ( 3 . 1 5 ) ] . When c a l l e d with NST=-1, th e Dmodel d e f i n e s d e f a u l t v a lu es
of th e bounds on th e parameter v a r i a t i o n : lower XLIM(1,I) and upper
XLIM(2,I), as well a s "zero" v a lu e s fo r pa ra m ete rs . Those v a lu e s a r e
a s s ig n e d to f i x e d pa rameters whose i n i t i a l guesses were e n te r e d as 0
f i x e d (skipped e n t r y in i n i t i a l guess qu e ry ). The s u b r o u ti n e Dmodel
should have fol lo wi ng form (see l i s t i n g of D12C0R as an example):
SUBROUTINE Dmodel( IDER,X,Y,A,C,NR,NRA,MF,NST,MAXST
1 ,XLIM,XMIN)
DIMENSION X(NR),XLIM(2,NR),XMIN(NR)
COMPLEX Y(MF),A(MF,1),C(MF), ZT.ZM
INCLUDE ’ FREQ.COM'
INCLUDE ’TYPR.COM’
INCLUDE ’FIXP.COM’
IF(NST.GE.O) GOTO 8
C INITIALIZATION
249
460
C
8
C
46
50
WRITE(LPR,460)
F0RMAT('+ mo de l' )
MAXST= i n i t i a l maximum number of i t e r a t i o n s
XLIM(1, j) = lower bound on parameter j
XLIM(2,j)= upper bound on parameter j
j= l,2,...N R
XMIN(j)= " z e r o ” va lu e of parameter j
RETURN
COMPUTATIONS
CONTINUE
c a l c u l a t e p a r t s of e x p r e s s i o n s which do not depend
on frequency
LOOP OVER FREQUENCIES
DO 50 1=1,MF
I1=IST+I-1
C a l c u l a t e va lu e ZT of complex model f u n c t i o n and
complex p a r t i a l d e r i v a t i v e s A ( I , j ) a t a n g u la r frequency
0M(I1). Only d e r i v a t i v e s with r e s p e c t t o a d j u s t e d
parameters (IDFX(j)=.FALSE.) should be inc lud ed in A.
Computation of d e r i v a t i v e s with r e s p e c t t o f ix e d
parameters can be skipped (a s in D12C0R) or columns
corre spo ndin g to f i x e d pa rameters can be removed from A
l a t e r in a n o th er loop (as in D13C0M).
I t i s recommended
t h a t both impedance (IDER>0) and admittanc e
r e p r e s e n t a t i o n s (IDER<0) a r e implemented. I f
c a l c u l a t i o n s a r e f i r s t made in impedance r e p r e s e n t a t i o n
then fo ll ow in g code tr a ns fo rm s r e s u l t s to admittance.IF(IDER.GT.O) GOTO 50
ZT=1./ZT
ZM=-ZT*ZT
DO 46 K=1, no of parameters a c t i v e a t t h i s p o i n t
A(I,K)=A(I,K)*ZM
C(I)=Y(I)-ZT
NRA= no of a d j u s t e d parameters
RETURN
END
C a l c u l a t i o n of th e model f u n c t i o n and t h e p a r t i a l d e r i v a t i v e s i s
u s u a l l y th e most time-consuming p a r t of t h e n o n l i n e a r r e g r e s s i o n and i t
i s s t r o n g l y a dvis ed t o optimize th e e xec ut io n speed of th e s u b r o u ti n e
Dmodel. The s u b r o u t i n e Dmodel, in th e form given above, uses complex
a r i t h m e t i c which s i m p l i f i e s coding. All computations can a l s o be made
in r e a l a r i t h m e t i c - see s u b r o u ti n e DI7INC.
A c a l l to th e Dmodel s u b r o u ti n e must be inc luded in s u b r o u ti n e
250
DERIVF with a new l a b e l , corresponding to model number IDER, added to
computed GOTO s t a t e m e n t s l a b e l l e d 44 and 58 (assuming t h a t both impe­
dance and adm ittanc e r e p r e s e n t a t i o n s a r e implemented).
P a r t of Dmodel code which c a l c u l a t e s v a l u e s of model f u n c t i o n must
be d u p l i c a t e d in s u b r o u t i n e Smodel used fo r p l o t t i n g . This s u b r o u t i n e
should have th e form:
8
SUBROUTINE Smodel( ZT,0M1,X,NI)
COMPLEX ZT
DIMENSION X(l)
LOGICAL NI
IF(NI) GOTO 8
c a l c u l a t e p a r t s of e x p r e s s io n s which do no t depend on
frequency 0M1 - Smodel i s f i r s t c a l l e d with NI=.FALSE.
CONTINUE
c a l c u l a t e complex value of model f u n c t i o n in impedance
r e p r e s e n t a t i o n ZT
RETURN
END
As in t h e case of Dmodel, r e a l a r i t h m e t i c can be used by d e c l a r i n g
DIMENSION ZT(2) (see s u b ro u ti n e SY7INC). A c a l l t o Smodel must be i n ­
cluded in th e s u b r o u t i n e SYZYIR with a l a b e l , co rresponding t o model
number IDER, included in the computed GOTO s t a t e m e n t .
A f te r s u c c e s s f u l com pil ation of new and modified s u b r o u t i n e s , th e
new v e r s i o n of t a s k has to be b u i l t . The names Dmodel and Smodel a re
included in th e o v e rl a y d e s c r i p t i o n f i l e FIRDAC.ODL, foll owing the
names of the s u b r o u t i n e s correspo nd ing t o o t h e r models. The new task
FIRDAC.TSK i s b u i l t by th e command TKB @FIRDAC where command f i l e
FIRDAC.CMD c o n t a i n s a l s o assignment of l o g i c a l u n i t 7 t o Tekt ron ix 4010
Graphics Terminal.
A new model should be thoroughly t e s t e d . T e s ts of l e a s t - s q u a r e s
e s t i m a t i o n s on s y n t h e t i c d a t a ge ner at e d us in g a new model f u n c t i o n a re
251
recommended f i r s t . S y n t h e t i c d a t a can be g e n e r a te d by th e program
GENTES, a new v e r s i o n of which must be b u i l t (TKB 0GENTES) with th e
modified s u b r o u t i n e s CHOFIT, SYZYIR and in c l u d i n g Smodel i n t o ov e rl a y
d e s c r i p t i o n f i l e GENTES.ODL. The program GENTES can a l s o be used fo r
p l o t t i n g th e s y n t h e t i c s p e c t r a .
V a ri a b le p r i n t o u t o p t i o n s , governed by
pa rame ters IDR, ITY, ITDR in program FIRDAC, f a c i l i t a t e debugging of
new s u b r o u t i n e s .
U s e r ' s Guide t o Program FIRDAC
FIRDAC i s an i n t e r a c t i v e program which combines t h e n o n l i n e a r ,
l e a s t - s q u a r e s e s t i m a t i o n of para mete rs in the models of a . c . response
o f s o l i d e l e c t r o l y t e / e l e c t r o d e s system with g r a p h i c a l p r e s e n t a t i o n of
th e d a t a and r e s u l t s of f i t t i n g . I t i s implemented on a computer e quip ­
ped with a g r a p h i c s t e r m i n a l ( T e k tr o n ic s 4010-1 or emulat or) an d /o r a
p l o t t e r (Hewlett Packard 7470A or a no th er a c c e p t i n g HPGL i n s t r u c t i o n s ) .
When FIRDAC i s used w it hout on l i n e g r a p h i c s , th e r e s u l t s of f i t t i n g
can be s t o r e d fo r l a t e r p l o t t i n g by a n o th e r program (PLOTFI).
FIRDAC a c c e p t s d a t a from a d a t a f i l e or typed from t h e t e r m i n a l .
The s u b r o u t i n e DAREAD used fo r d a t a e n t r y accommodates a simple l i n e
e d i t o r which f a c i l i t a t e s c o r r e c t i o n of ty p in g e r r o r s . There a r e two
s t a n d a r d kinds of th e d a t a f i l e s : one fo r th e impedance d a t a e n te r e d
manually; th e o t h e r f o r d a t a c o l l e c t e d usi ng computer c o n t r o l l e d impe­
dance a n a l y z e r . The f i r s t l i n e of t h e d a t a f i l e i s for matted
(2 1 2,2(6A1),2E11.4) and contains.MF - number of d i s c r e t e f r e q u e n c i e s in t h e spectrum,
252
ISR - number of th e s i g n a l l e v e l a p p l i e d by t h e impedance a n a ly z e r or 0
f o r d a t a c o l l e c t e d manually,
IFNAM - f i l e name (6 c h a r a c t e r s ) ,
ISAMP - specimen name (6 c h a r a c t e r s ) ,
GEOM - L/A g e o m e t ri c a l f a c t o r - t h e r a t i o of th e d i s t a n c e between e l e c ­
t r o d e s to t h e a r e a of e l e c t r o d e s (1/cm),
TEMP - te mp era tu re in de gre es C e l s i u s .
The fol lo wi ng MF l i n e s fo rm att ed (3 E 1 1 . 4 .I2 ) a r e i n t e r p r e t e d a c ­
co rdi ng t o va lu e of ISR. Uhen ISR>0, each l i n e c o n t a i n s th e frequency
(Hz), th e r e a l and t h e imaginary p a r t s of t h e ad mi tta nce (fl- *) and IRZ
( t h e number of the r e f e r e n c e r e s i s t o r used by t h e impedance a n a l y z e r ) .
Markers which r e p r e s e n t d a t a in graphs correspond to th e v a l u e s of IRZ.
When ISR=0, t h e l i n e c o n te n t in v o lv e s th e frequency (Hz), th e a b s o l u t e
valu e ( a ) and th e phase (d e gre e) of th e impedance, IRZ=0 fo r d a t a
measured with a low frequency impedance meter and IRZ=-1 fo r ra dio
frequency d a t a t a k e n with th e HP 4815A v e c t o r impedance meter. Proper
marking of t h e d a t a with IRZ i s r e q u i r e d when c o r r e c t i o n s fo r t h e s t r a y
in s t r u m e n t a l impedances a r e a p p l i e d by th e program - IRZ<0 ra d io f r e ­
quency c o r r e c t i o n , IRZ=0 e q u i v a l e n t lumped c i r c u i t c o r r e c t i o n for th e
s t r a y impedances, IRZ>0 no i n s t r u m e n t a l c o r r e c t i o n .
The c o n te n t of p r i n t o u t produced by FIRDAC a r e c o n t r o l l e d by t h r e e
para mete rs IDR, ITY, ITDR which a r e s t o r e d in COMMON/TYPR/. ITY a l s o
det ermi ne s th e i n t e r a c t i o n of th e u s e r with th e r o u t i n e performing the
n o n l i n e a r r e g r e s s i o n . The normal s e t t i n g i s IDR=0, ITY=0, ITDR=0. With
ITY
0, when th e l i m i t on th e number of i t e r a t i o n s i s reached or
253
o s c i l l a t i o n s of para mete rs p e r s i s t w it ho ut a s i g n i f i c a n t change of the
o b j e c t i v e f u n c t i o n , t h e us er can d e cide whether to t e r m i n a t e r e g r e s ­
s i o n , a cc e p t th e r e s u l t s or co nt in ue i t e r a t i o n s . Info rmation included
in a normal p r i n t o u t i s d e s c r ib e d below. The name of t h e s u b r o u ti n e
g e n e r a t i n g a given item i s included in b r a c e s .
(i)
t i t l e - time, d a t e and in fo rm at io n from th e f i r s t l i n e of the
d a t a f i l e ( f i l e name, sample name, L/A f a c t o r , te m p e ra t u re ,
number of f r e q u e n c i e s , l e v e l number) {DAREAD};
(ii)
(iii)
(iv)
(v)
(vi)
l i s t i n g of d a t a i f r e qu e st e d by th e u s e r {DAREAD};
parameters of in s t r u m e n t a l c o r r e c t i o n s i f a p p l i e d {CHCORR};
acronym of chosen model f u n c t i o n {CHOFIT};
type of weighting scheme with c or re spo ndi ng para mete rs {CONVEG};
i n i t i a l guess of parameters - acronym, number, v a l u e , f i x i n g
f l a g (-1 f i x e d pa rameter, 0 s u b j e c t t o e s t i m a t i o n ) {GESPAR};
(vii)
frequency range of d a t a used f o r f i t t i n g {FIRDAC},
( v i i i ) number of r e a l e qu a ti o n s and t o t a l number of pa rameters
{NLMFLX};
(ix)
acronym of t h e model f u n c t i o n and th e r e p r e s e n t a t i o n of the d a t a
used f o r f i t t i n g (impedance or a d m it ta n c e ) {DERIVF & Dmodel a c t u a l l y c a l l e d s u b r o u ti n e } ;
(x)
re c ord of ad jus tm ent of p a ra m e te rs ' bounds by th e u s e r , changes
of parameter v a l u e s made to conform with t h e bounds, number of
pa rameters f i x e d , a d j u s t e d ( e s t i m a t e d ) and equal t o th e bounds
{DERIVF};
(xi)
f a t a l message when s i z e of problem (number of e s t i m a t e d p a ra -
me ters times number of measurements) i s l a r g e r than d e c l a r e d
dimension of m a tr ix A in s u b r o u t i n e NLMFLX {DERIVF}, r e g r e s s i o n
i s a b o r te d ;
( x i i ) warnings about problems with convergence ( o s c i l l a t i o n s , l i m i t on
number of i t e r a t i o n s reached, a l l para mete rs on bounds, e t c . )
with v a lu e s of parameters {NLMFLX};
( x i i i ) no convergence message or f i n a l in for ma ti on about c o n d i t i o n s of
convergence: number of i t e r a t i o n s , weighted root-mean-square
r e s i d u a l Rmsr, l a s t v a lu e s of convergence t e s t parameters r e l a t i v e change of o b j e c t i v e f u n c t i o n TES and norm of t o t a l
s c a l e d c o r r e c t i o n of pa rameters DX [ Eq. ( B3)] {NLMFLX};
( x iv )
e s t i m a t e d v a l u e s of pa rameters x^, u n c e r t a i n t i e s ax^ (c o n fi d e n ­
ce) of e s t i m a t e s acc ord ing t o E q . ( 3 . 1 6 ) , r e l a t i v e u n c e r t a i n t i e s
aXj/Xj i f Xj=0). Estimated parameters a r e he re numbered consecu­
t i v e l y with fi x e d parameters removed from t h e l i s t .
{NLMFLX &
DRUM}
(xv)
p a i r of pa rameters which e x h i b i t s th e h i g h e s t a b s o l u t e va lu e of
c o r r e l a t i o n or a l l p a i r s of parameters with a b s o l u t e v a lu e s of
c o r r e l a t i o n s hi g h e r than 0 .9 {NLMFLX};
( x v i)
f i n a l l i s t of a l l parameters l a b e l e d by numbers and acronyms,
t h e i r e s ti m a te d v a lu e s followed by u n c e r t a i n t i e s , fi xe d v a lu es
followed by zero {RENUMP}.
S e t t i n g IDR=1 produces a d d i t i o n a l p r i n t o u t of a co nc is e d e s c r i p ­
t i o n of model in ( i v ) {CHOFIT} and f u l l c o r r e l a t i o n m a tr ix in (xv)
{NLMFLX & DRUMSP}. IDR=3 adds printout of measurements in represents-
255
t i o n used fo r f i t t i n g and t h e co rresponding weight m a tr ix G Y (l, i) =gJ^,
GY(2,i)=92 2’ GY( 3 , i ) = 9 j 2 ^see E(IS - ( 3 -8) “ ( 3 1 ° ) ] a f t e r ( v i i i ) , and f i n a l
l i s t of r e s i d u a l s C ( i ) = Y ( i ) - f ( w . , X ) a f t e r (xv) {NLMFLX & DRUM}. IDR=4
i s u s e f u l only when one t e s t s and debugs a new model f u n c t i o n or a
m o d i f i c a t i o n of r e g r e s s i o n a lg o ri th m - i t produces l a r g e p r i n t o u t s of
Jacobian m a tr ix A [ E q . ( 3 . 1 5 ) 3 , v e c t o r XT=DNAr Gy (Y-F(X)), s c a l e d system
T
m a tr ix GX^DyA GyADN+xpI p r i o r and a f t e r i n v e r s i o n and s c a l e d v e c t o r of
c o r r e c t i o n s XI=aXS [Eq.(B2)] {NLMFLX, DRUMSP & DRUM).
S e t t i n g ITY>0 r e s u l t s in automatic t e r m i n a t i o n with warning and
with p r i n t o u t of r e s u l t s in c as e s of o s c i l l a t i o n s or exceeded l i m i t on
number of i t e r a t i o n s . ITY=-1 produces p r i n t o u t of th e p ro g r e s s of mini­
mi za tio n of the o b j e c t i v e f u n c ti o n which c o n t a i n s {NLMFLX}:
- v a lu e s of pa rameters a f t e r each c o r r e c t i o n ,
- val ue of o b j e c t i v e f u n c ti o n SM [ E q . ( 3 . 1 0 ) ] a f t e r each c a l c u l a t i o n of
th e model f u n c t i o n ,
- norm of s c a l e d c o r r e c t i o n DX [Eq .(B 3)] ,
- d e c r e a s e of SM p r e d i c t e d by l i n e a r i z a t i o n ,
- r a t i o of a c t u a l to p r e d i c t e d de cr ea se of SM (0 i f SM i n c r e a s e d ) ,
- a d j u s t a b l e s t e p bound a,
- v a lu e s of s t e p r e d u c ti o n (dumping) parameter s and
Marquardt term xp
[Eq.(B2)3 a f t e r each change.
Such p r i n t o u t t r a c e s performance of the min imization a lg o ri th m and i s
u s e f u l whenever doubts re g a rd in g i t s proper f u n c t i o n i n g a r i s e . S e t t i n g
ITDR=-1 adds p r i n t o u t of s c a l i n g v e c t o r [diagona l of m a tr ix D^,
E q .( B l )] and a p p l i e d c o r r e c t i o n s of parameters {NLMFLX}, ITDR=-2 giv es
256
v a lu e s of r e s i d u a l s Y-F(X) a f t e r each c a l c u l a t i o n of model f u n c t i o n
{DERIVF}.
An i n t e r a c t i v e s e s i o n i s s t a r t e d by t h e RUN command. Name of the
d i r e c t o r y c o n t a i n i n g FIROAC should be inc luded ( e . g . R $FIRDAC fo r the
system d i r e c t o r y ) . Following i s th e d e s c r i p t i o n of i n t e r a c t i v e prompts
is su e d by th e program in t y p i c a l l y encountered o r d e r . Names of subrou­
t i n e s producing prompts a r e include d in b r a c e s . Each prompt, quoted as
i t appea rs on the s cr ee n of t e r m i n a l , i s s e p a r a t e d by / / from names of
th e v a r i a b l e s whose v a lu e s a r e a s s ig n e d by th e us er an d/o r from the
format s p e c i f i c a t i o n in p a r e n t h e s i s . Keyboard inp ut a f t e r each prompt
should be te rm in a te d by <RETURN>. I f more than one va lu e i s e n t e r e d ,
numbers should be s e p a r a t e d by commas. Real numbers can be e n t e r e d in
th e f i x e d format Fp.g whenever Ep.q format i s s p e c i f i e d and v i c e - v e r s a ,
bu t t h e decimal p o i n t must be p r e s e n t .
(a ) {DAREAD}
P r i n t o u t s i z e IDR (<0 o n l i n e [ , 1 n os po ol ]) 0 . + - 1 , + - 2 , + - 3 ? //IDR.ISPO
(213)
IDR>=0 d i r e c t s a l l p r i n t o u t to f i l e FIRDAC.OUT ( l o g i c a l u n i t 2)
which i s c lo se d and e n t e r e d i n t o p r i n t queue (s pooled) a f t e r normal
e x i t from program or when new d a t a s e t i s r e q ue st e d a f t e r f i t t i n g . Mhen
o p t i o n a l parameter ISP0=1 i s e n te r e d fol lo wi ng comma, f i l e FIRDAC.OUT
i s c r e a t e d but not p r i n t e d . In e i t h e r case c onse c u ti v e v e r s i o n s of
FIRDAC.OUT remain in the u s e r ' s d i r e c t o r y and should be l a t e r d e l e t e d .
IDR<0 d i r e c t s ou tp ut to p r i n t e r on l i n e (system p r i n t e r - l o g i c a l u n i t
6) and should not be used when o th e r u s e r s a r e a c c e s s i n g t h i s p r i n t e r
257
in ord e r to avoid d e la y and mixing of p r i n t o u t s . The va lu e of IOR
c o n t r o l l i n g d e t a i l s of p r i n t o u t i s s e t to m i n ( | I D R | - l , 0 ) th u s e n t r i e s 1 , 0 , 1 r e s u l t in the same minimal p r i n t o u t .
(b) {DAREAD}
-1 type in d a t a , or read from PILE: //IFNAM (24A1)
Name of e x i s t i n g d a t a f i l e should be e n t e r e d with d i r e c t o r y iden­
t i f i c a t i o n , i f d i f f e r e n t than the u s e r ' s d i r e c t o r y . Extension by de­
f a u l t i s .DAT. The ne xt prompt a f t e r rea d in g t h e f i l e i s ( c ) . Typing -1
s wi tc h e s t o d a t a e n t r y from te r m i n a l ( b l ) - ( b 6 ) .
( b l ) Specimen //ISAMP (6A1)
Acronym of specimen (6 c h a r a c t e r s ) ;
(b2) L/A ( l . / c m ) = //GEOM (E10.3)
R a t io of d i s t a n c e between e l e c t r o d e s to t h e i r a r e a , f o r l a b e l s
on ly, not used in c a l c u l a t i o n s ;
(b3) TEMP (deg C) //TEMP (E10.3)
Temperature a t which d a t a were c o l l e c t e d ; (b4) St o r e on f i l e
//INAM (24A1)
Name of f i l e to be c r e a t e d fo r s t o r a g e of d a t a . The d e f a u l t name i s
F0R003.DAT;
(b5) No
of f r e q s //MP (13);
(b6) NO
*
1
2
FREQ
* ABS(IMP)
* / / (E10.3)* / / ( E 1 0 . 3 )
*
* PHASE
* //(E10.3)
.................................................
Prompt with i n c r e a s i n g d a t a number i s r e p e a te d u n t i l d a t a f o r a l l
MF f r e q u e n c i e s a r e e n t e r e d . E n t r i e s of frequency (Hz), a b s o l u t e value
258
of impedance (0) and phase ang le of impedance (deg) a r e t e rm in a te d by
<RETURN>. Cursor r e t u r n s t o th e same l i n e a f t e r e n t r i e s of FREQ and
ABS(IMP).
(c) {DAREAD}
-2 Add RF d a t a , -1 P r i n t , 0 OK, Edi t d a t a No //NFO (13)
NF0=-2 s t a r t s
e n t r y of a d d t i t i o n a l d a t a in t h e sequence ( b 5 ) - ( b 6 ) ,
which w i l l be marked by IRZ=-1 f o r o p t i o n a l RF in s t r u m e n t a l c o r r e c ­
t i o n s . NF0=-1 produces p r i n t o u t of d a t a (No, freq ue ncy , | Z | ,
0
, Re(Y),
Im(Y) ) and oth e rw is e i s e q u i v a l e n t to NF0=0 - d a t a a r e acce pted for
a n a l y s i s . NF0>0 i s used f o r e d i t i n g d a t a number NFO. I t i s followed by
d i s p l a y of d a t a a t frequency number NFO, in th e same r e p r e s e n t a t i o n
as
s t o r e d in d a t a f i l e , and prompt
( c l ) -1 Del, 0 Acc, 1 Chg, 2 Ins //IED (13)
IED=-1 t h e d a t a p o i n t i s d e l e t e d and a l l d a t a numbers g r e a t e r than IED
a r e reduced by 1;
IED=0
no change, a c c e pt ed as i s ;
IED=1
change d a t a , followed by prompt f o r new v a l u e s [same as (b6)]
and by
IRZ=//IRZ(NF0) (13)
-1 r f d a t a , 0 low fr equency, >0 range of impedance a n a l y z e r ; IED=2
i n s e r t d a t a , followed by the same prompt f o r new d a t a p o in t as IED=1,
but th e new d a t a a r e i n s e r t e d in f r o n t of d a t a
number IED and a l l d a t a
numbers >IED a r e in c r e a s e d by 1.
A f te r e d i t i n g of d a t a number NFO prompt (c ) i s re p e a te d u n t i l
t e rm in a te d by - 2 , -1 or 0. I t should be noted t h a t d a t a typed in and
c o r r e c t i o n s of d a t a a r e not saved on d i s k u n t i l e d i t i n g i s te rm in a te d ,
(d) {FIRDAC}
Display: 0 none, 1-4 Complex Plane: 1 Adm, 2 Imp, 3 Cap, 4 Mod; 9 Bode;
5-8 l o g - l o g ( f ) : 5 Adm, 6 Imp, 7 Cap, 8 Mod [ , - 1 P l o t ] //ITYPN,IDEVN
(213)
Optional on l i n e d i s p l a y of d a t a on a g r a p h i c s te r m i n a l when
IDEVN=0 or ou tp ut f o r a d i g i t a l p l o t t e r ( l o g i c a l u n i t LHP=1) when
IDEVN=-1. The ou tp ut de vic e should be tur ne d on and ready when used on
l i n e . Output f o r th e p l o t t e r can be s t o r e d in th e f i l e FIRDAC.HPP and
l a t e r s e n t to p l o t t e r . IDEVN=-2 produces a d d i t i o n a l prompt
( d l ) {HINITT}
-1 Enter HPGL command, 0 p l o t , 1 change margins / / I I (13) Il= -1 g iv e s
o p ti o n fo r i s s u i n g HPGL commands fo r p l o t t e r ( d 2 ) {HINITT}
Command: //(70A1)
which may, fo r example, be used t o reduce t h e speed of pen when drawing
l i n e s ( HPGL code VSIO;); 11=1 i s followed by prompts fo r l o c a t i o n of
lower l e f t edge of p l o t t i n g a r e a and f o r s i z e of p l o t
(<33) {HINITT}
Lower l e f t i s
53, 27 mm, t o be //MARX,MARY (214)
Size i s 193, 147 nun, to be //LSIX,LSIY (214)
numbers a r e e n te r e d in p a i r s , s e p a r a t e d by a comma. I t should be ob­
ser ved t h a t p l o t t i n g a r e a i s the image of PL0T10 g r a p h i c s s c r e e n , whose
s i z e in r a s t e r u n i t s i s 1023 h o r i z o n t a l by 780 v e r t i c a l . Graphs a r e
s c a l e d acc ord ing to t h e s re en a s p e c t r a t i o , and equal s c a l e s of r e a l
and imaginary axes on complex plane diagrams a r e r e t a i n e d only when
260
LSIY/LSIX i s ap proxim ately equal 0.762=780/1023. The d e f a u l t p l o t t i n g
a r e a has th e proper a s p e c t r a t i o and le a v e s re a so n a b le margins on 8.5
by 11 inch paper. When graphs of d e f a u l t s i z e a r e reduced approximately
0.75 t im e s , they f i t h o r i z o n t a l l y on 8.5 by 11 inch paper.
Meaning of th e a b b r e v i a t i o n s d e s c r i b i n g d i f f e r e n t forms of graphs
which correspond t o d i f f e r e n t v a l u e s of ITYPN i s :
f - frequency (Hz);
Adm - complex
ad m itt a nce : Re(Y), Im(Y) (G- *);
Imp - complex
impedance: Re(Z), -Im(Z) (Q);
Cap - complex c a p a c i t a n c e (C=Y/i»): Re(C), -Im(C) (nF);
Mod - complex modulus (M=i«Z): Re(M), Im(M) (nF” '*');
Bode - s p e c t r o s c o p i c p l o t s of a b s o l u t e va lu e |Z| and phase -0 of impe­
dance .
The p l o t s a r e
s c a l e d a u t o m a t i c a l l y and a r e designed t o f i l l
of th e g r a p h i c s te r m i n a l (780 v e r t i c a l
th e scr ee n
by 1023h o r i z o n t a l r a s t e r
u n i t s ) . S c a le s of r e a l and imaginary axes of complex plane diagrams a re
always i d e n t i c a l in ord er t o avoid d i s t o r t i o n of c h a r a c t e r i s t i c shapes
( e . g . s e m i c i r c l e s ) of complex l o c c i . P l o t s of d a t a e x h i b i t i n g much
l a r g e r span in imaginary than in r e a l p a r t a r e s p l i t i n t o two frames
with d i f f e r e n t s c a l e s in ord er to re v e a l d e t a i l s of complex l o c c i near
o r i g i n of complex p la n e. Sp e c tr o s c o p ic p l o t s v e rs u s decimal lo ga rit hm
of frequency g e n er at e d when ITYPN=5,6,7,8 d i s p l a y r e a l and imaginary
p a r t s of complex d a ta in two s e p a r a t e frames with th e same s c a l e expan­
si o n used in both. Decimal lo g a ri th m s of r e a l and imaginary p a r t a re
d i s p l a y e d i f a l l v a l u e s a r e p o s i t i v e , l i n e a r s c a l e s a r e used oth e rw is e.
261
Graphs produced on HP p l o t t e r a r e enhanced with r e s p e c t t o th e scr ee n
d i s p l a y s thanks to t h e b e t t e r r e s o l u t i o n and programmability of the
p l o t t e r , but the arrangement of frames and s c a l i n g a r e the same.
(e) {FIRDAC}
-3 Next d a t a , -2 I n s t r c o r r , -1 E d i t , 0 F i t , >0 Display [ , - 1 P l o t ]
//ITYPN,IDEVN (213)
This prompt appears only i f ITYPN>0 response was given t o (d) and
graph was c r e a t e d . ITYPN=-3 r e t u r n s c o n t r o l to (a ) and thu s allows to
use FIRDAC f o r p l o t t i n g d a t a w ith out f u r t h e r a n a l y s i s . ITYPN=-1 r e t u r n s
to d a t a e d i t i n g (c) and i s u s e f u l when e r r o r i s n o t i c e d on th e d i s p l a y .
ITYPN=-2 i n i t i a t e s prompt ( f ) f o r in s t r u m e n t a l c o r r e c t i o n s . This prompt
i s skipped when ITYPN=0 and ex ecu tio n advances d i r e c t l y to ch oice of
model ( i ) . ITYPN>0 produces a n o th er d i s p l a y or p l o t acc ord ing to
ch oi ce s in (d ) .
( f ) {FIRDAC}
C o r r e c t i o n s : 0 none, NEG f ix e d t o d a t a , POS t o model (1 f i x e d , 2
a d j u s t ) //ICOR (13)
In st ru m e n ta l c o r r e c t i o n s a r e a p p l i e d only w i t h i n program, without
change t o d a t a f i l e .
IC0R<0 s p e c i f i e s t h a t s t r a y i n s tr u m e n ta l imped­
ances w i l l be s u b s t r a c t e d from d a t a in an at te m p t t o o b t a i n th e t r u e
impedance of the specimen. When IC0R=1, s t r a y impedances a re included
in th e impedance ge ne ra te d by model f u n c t i o n , in ord er t o reproduce
the= experimental d a t a as measured. This second method i s recommended
f o r l e a s t squ ar es a n a l y s i s , e s p e c i a l l y when in s t r u m e n t a l c o r r e c t i o n i s
s i g n i f i c a n t , because i t le av e s d a t a u n a l t e r e d fo r c a l c u l a t i o n of
262
weights based on ex perimental r e s o l u t i o n [ E q s . ( 3 . 8 ) - ( 3 . 1 1 ) ] . ICOR=2
op tio n f o r e s t i m a t i o n of c o r r e c t i o n pa rameters can be used when c o r r e c ­
t i o n i s s u b s t a n t i a l and i t s parameters a r e not known e x a c t l y . P a r t i ­
c u l a r l y , when c a l i b r a t i o n d a ta a r e measured with known c i r c u i t in place
of specimen, para mete rs o f s t r a y impedances can be e st i m a te d and used
l a t e r f o r c o r r e c t i n g d a t a . The c i r c u i t elements modeling s t r a y imped­
ances of a p p a r a tu s may a l s o be t r e a t e d as an e x te n s io n of model f o r th e
e l e c t r o l y t e / e l e c t r o d e s system.
(g) {CHCORR}
Parameters from f i l e CALIBR.DAT, RETURN OK,
-1 type i n , +1 pr e v io u s v a l u e s , or from f i l e //INAM (24A1)
This prompt follows IC0R=0 response t o ( f ) . Parameters of i n s t r u ­
mental c o r r e c t i o n s a r e by d e f a u l t s t o r e d in f i l e CALIBR.DAT in th e
u s e r ' s d i r e c t o r y . I f they a r e s t o r e d in a n o th er f i l e the f i l e name must
be give n. Format of such f i l e i s (8E10.3). Parameters a r e org an iz ed as
foll ows :
3 pa rameters fo r c o r r e c t i o n of low frequency d a t a (up t o 1 MHz) r e s i s t a n c e in s e r i e s with specimen RHO ( 8 ) , c a p a c i t a n c e CHO (F) p a r a l ­
l e l t o specimen and ind uctance HIN (H) in s e r i e s with specimen;
5 pa rameters fo r c o r r e c t i o n of r a d io frequency d a t a measured with
Hewlett-Packard 4815A v e c t o r impedance meter - le n g t h of t r a n s m i s s i o n
l i n e from meter probe to specimen T1 (m), c h a r a c t e r i s t i c impedance of
t h i s t r a n s m i s s i o n l i n e ZO ( a ) , c a p a c it a n c e (F) p a r a l l e l with t h e me­
t e r ' s probe, induc ta nce in s e r i e s with the probe PRIN (H), r e s i d u a l
r e s i s t a n c e of th e probe RPR (Q). I t i s assumed t h a t t r a n s m i s s i o n l i n e
263
i s e l e c t r i c a l l y te r m in a te d by impedance of th e specimen.
Five parameters fo r r a d i o frequency c o r r e c t i o n may be l e f t equal
t o 0 . 0 i f d a t a measured with HP4815A a r e not analyzed ( i . e . none of the
d a t a i s marked with IRZ=-1).
Response -1 i s followed by prompts f o r parameters in th e same
ord er as s t o r e d in f i l e CALIBR.DAT. +1 may be used when c o r r e c t i o n
para mete rs have a l r e a d y been s p e c i f i e d d ur in g c u r r e n t s e s s i o n and same
values are desired.
When IC0R=-1 was e n te r e d in ( f ) prompt (e) r e t u r n s a f t e r d a ta a r e
c o r r e c t e d . I f d a t a were p l o t t e d p r i o r to c o r r e c t i o n s , then c o r r e c t e d
d a t a can be d i s p l a y e d in th e same frame i f th e same ITYPN i s e n te r e d
a ga in .
( i ) {CHOFIT}
Choice of model, IDER>0 impedance f i t , IDER<0 ad mi tta nce f i t : 7INC +1,
6TSA - 2 , 6CFC +-3, 13C0M +-4, 17T0T +-5, 12C0R +-6, 18MAC +-7 ___
IDER?=// IDER (13)
See Se c tio n 3.4 fo r d e s c r i p t i o n of model f u n c t i o n s l a b e l l e d by
acronyms. Note t h a t 7INC can only be used in impedance r e p r e s e n t a t i o n
and 6TSA only in ad m itt a nc e . Computations of th e o t h e r f u n c t i o n s a re
s l i g h t l y f a s t e r in t h e impedance r e p r e s e n t a t i o n .
( j ) {CONVEG}
R e s o lu ti o n weights: dABS(Z)=l.00%, dPH= 0. 6deg , 1 t o change //IW (13)
The d e f a u l t we ightin g scheme i s acc ord in g to E q s . ( 3 . 5 ) —(3-12) with
a|Z |/|Z |
and A<t> given in t h i s prompt. <RETURN> a c c e p t s the d e f a u l t .
When 1 i s e n t e r e d c hoi ce of weighting scheme i s given
264
( j l ) {CONVEG}
Weights: -1 EQUAL, 0 %Z & dPH, 1 P r o p o r t i o n a l Re & Iro, 2 1/1 Z|
/ / IW (13)
IW--1 equal weights for real and imaginary parts of a l l data.
Program calcu la tes numerical value of weight which i s expected to
resu lt in root-mean-square residual of the order of 1. IW=0 gives the
same weighting scheme as default but with choice of instrumental reso­
lu tion . It i s followed by prompts
dABS(Z)% / / (F6.2)
dPH(deg)// (F6.2)
IW=1 produces weights based on th e assumption t h a t e r r o r s of r e a l and
imaginary p a r t s of complex d a t a a r e independent and p r o p o r t i o n a l to
t h e i r r e s p e c t i v e v a l u e s (weighting scheme used in r e f . [ 10 7] ). I t i s
followed by prompt
Prop c o n st / / (F6.2)
IW=2 r e s u l t s in weights p r o p o r t i o n a l t o th e in v e r s e of a b s o l u t e val ue
of complex d a t a - same f o r r e a l and imaginary p a r t s .
(k) (GESPAR)
Guess of model para mete rs : V a l u e [ , - 1 ] ( t o f i x i t ) : X (l ) =R bu =( F1 0.4 ,I3 ),
X(2)=Rgb=(F10.4 , 1 3 ) , X(3)=CPgb=(F10.4,13) X(4)=ANgb=
Prompts f o r c o n s e c u ti v e para mete rs l a b e l e d by numbers and acronyms
(here taken from model 12C0R) appear t h r e e per l i n e . See d e s c r i p t i o n of
model f u n c t i o n s in Se c ti o n 4.4 fo r d e f i n i t i o n of para mete rs . When
nonzero val ue i s e n t e r e d i t i s t r e a t e d as i n i t i a l val ue for l e a s t
sq ua re s e s t i m a t i o n . Value followed by ,-1 s p e c i f i e s fi x e d pa rameter,
not to be a d j u s t e d . Zero valu e or simply <RETURN> d e s i g n a t e s f ix e d zero
265
va lu e of parameter. E f f e c t i v e l y , i f zero would cause numerical prob­
lems, a nonzero number, s e v e r a l o r d e r s of magnitude s m a l le r than any
p h y s i c a l l y s i g n i f i c a n t v a l u e , i s l a t e r a s s ig n e d to th e fi x e d parameter
in s u b r o u t i n e DERIVE. Such i n t e r p r e t a t i o n of a skipped parameter i s
conv enie nt fo r speeding up s e t t i n g of t h e model. When va lu e of a r e ­
s i s t a n c e i s skipped i t i s re p la c e d by a s h o r t c i r c u i t ; an open c i r c u i t
i s in tr oduce d by s k ip p in g va lu e of c a p a c i t o r s or CPE elements in the
e q u i v a l e n t c i r c u i t . Zero i n i t i a l va lu e f o r e s t i m a t i o n of a parameter
must be followed by ,1.
( k l ) {GESPAR}
I n i t i a l model pa ra me te rs : VALUE,IPFX, 0 OK, 1 Change, -1 Fix X(l)= Rbu=
1.0 0E - 6.- 1/ /I C A ......................
Prompts with v a l u e s of para mete rs r e p l a c e (k) when t h e same model
i s used a gai n a f t e r e s t i m a t i o n of pa ra m et e rs . P r e v io u s ly e st i m a te d or
f i x e d v a l u e s a r e d i s p l a y e d followed by 0 or -1 r e s p e c t i v e l y . ICA=-1
changes e s t i m a t e d parameter i n t o fi x e d pa rameter. ICA=0 (simply
<RETURN>) a c c e p t s t h e d i s p l a y e d va lu e and s t a t u s of parameter. ICA=1 i s
followed by e r a s u r e of old va lu e from s c r e e n and new va lu e with o p t i o n ­
a l ,- 1 f o r fi x e d parameter can be e n t e r e d as in (k ).
(1) {GESPAR}
I n i t i a l v a l u e s of CORRECTION params, 0 OK, -1 Fix , 1 Change X(13)= CHO=
1 . 31E-11,//ICA (13)
...............
When IC0R=2 was s e t in ( f ) , i n s t r u m e n t a l c o r r e c t i o n parameters
from (g) a r e inc lud ed a t the end of parameter v e c t o r . They a r e t r e a t e d
as i n i t i a l v a lu e s f o r e s t i m a t i o n u n l e s s -1 i s e n te r e d in or d er to f i x
266
parameter. When 1 i s e n t e r e d , new va lu e can be typed in and t r e a t e d as
i n i t i a l guess or as f i x e d parameter i f followed by ,-1 .
(m) {FIRDAC}
40 f r e q s , r e s t r i c t f i t from FREQ / / I S T ( 13) to FREQ //MF1(13)
When ISTCO ( simply <RETURN>) i s e n te r e d a f t e r the f i r s t prompt
a l l d a t a (40 f r e q u e n c i e s in t h i s example) a r e used for th e e s t i m a t i o n
of par am ete rs . When IST>0 i s e n t e r e d , a second prompt a pp ear s and only
f r e q u e n c i e s numbered from 1ST t o MF1 a r e taken i n t o account. This
simple f e a t u r e t o g e t h e r with p r o v i s i o n f o r f i x i n g parameters provid es a
powerful t o o l fo r a n a l y s i s of complicated s p e c t r a . P a r t of th e spectrum
can be f i t t e d with a simple model with a few parameters and the new
pa rameters ( e . g . elements of an e q u i v a l e n t c i r c u i t ) can be added to the
model as frequency range i s g r a d u a l l y extended.
(n) {FIRDAC}
P r i n t o u t c o n t r o l : ITY,ITDR,LPT ? ( j u s t RETURN i s OK) / / ( 3 I 3 )
At t h i s p o i n t c o n t r o l parameters ITY.ITDR a r e s e t - see explana­
t i o n of t h e i r f u n c t i o n s a t th e beginning of t h i s u s e r ' s guide. Recom­
mended response i s <RETURN>, which e n a b le s i n t e r v e n t i o n in case of
problems with convergence but does not g e n e r a t e e x t e n s i v e p r i n t o u t . LPT
i s th e l o g i c a l u n i t number f o r p r i n t o u t g e n e r a te d when ITY=-1. I f
skipped or e n t e r e d 0 i t i s s e t equal t o LPR - th e l o g i c a l u n i t number
f o r th e r e s t of p r i n t o u t (6 on l i n e , 2 s p o o le d ) . S e t t i n g LPT=5 d i r e c t s
p r i n t o u t of p ro g r e s s of f i t t o the te r m i n a l f o r immediate examination,
(o) {DERIVF}
Range of PARAM ? / / I L (13) i s from l.OOE-18 t o 1.00E-02 to be from
267
//XLIM(1,I ) (E10.3)
to //(XLIM(2,I) (E10.3)
The second p a r t of querry appea rs only i f 0<IL<NR+1 i s e n te r e d
a f t e r th e f i r s t prompt. Lower XLIM(l.IL) and upper XLIM(2,IL) bounds on
v a r i a t i o n of parameter IL can be changed from th e d e f a u l t v a lu e s l i s t e d
in th e prompt (here l. E - 1 8 and l . E - 2 r e s p e c t i v e l y ) . In p a r t i c u l a r , when
XLIM(1,I)=XLIM(2,I) i s e n t e r e d , parameter IL becomes fi x e d a t t h i s
v a lu e. R e s u lt i s the same as when val ue i s followed by ,-1 in (k) or
( k l ) . The bounds s e t here o v e r r i d e inp ut from (k ) . I f n e c e s s a r y , the
i n i t i a l v a lu e s a r e changed t o be w it h in bounds. S t a t u s of a parameter
f ix e d in (k) may be changed t o e st i m a te d by s e t t i n g
XLIM(1,I)<XLIM(2,I). Prompt (o) i s r e pe at e d u n t i l IL^O i s e n t e r e d . At
t h i s moment i t e r a t i v e min imization of th e o b j e c t i v e f u n c t i o n b e gi ns . I t
should be remembered t h a t th e bounds of parameters r e t u r n t o t h e i r
d e f a u l t v a l u e s a t t h e beginning of next r e g r e s s i o n , and nonstandard
bounds must be e n t e r e d ag ain i f r e q u i r e d .
( o l ) {DERIVF}
Cannot a d j u s t 21 parameters with 200 measurements, m a tr ix A(2700) too
small !!! 1 to f i x more pa ra m et e rs , e l s e give up //NR2 (13)
This warning appears when the dimension LA (he re 2700) of m a tr ix A for
s t o r a g e of p a r t i a l d e r i v a t i v e s , d e c l a r e d in s u b r o u t i n e NLMFLX, i s not
s u f f i c i e n t to handle th e l e a s t sq uar es e s t i m a t i o n , i . e . LA i s s m al le r
than number of th e e s ti m a te d parameters times twice th e number of
fr e q u e n c i e s taken i n t o account. The user has a choice between f i x i n g
more parameters - r e t u r n to prompt (o) or t e r m i n a t i n g th e r e g r e s s i o n next prompt ( s ) .
268
(p) {NLMFLX}
ITER
1 SM= 1.3452E+01
*****
ITER
2 SM= 1.0187E+01
*****
Each c a l c u l a t e d val ue of th e o b j e c t i v e f u n c t i o n i s d i s p l a y e d on
the t e rm in a l (no i n p u t i s e x p e c te d ). The v a l u e s of SM may not be
d e c r e a s i n g and s e v e r a l v a lu es may be c a l c u l a t e d in one i t e r a t i o n when
s t e p s i z e an d/o r Marqurdt term i s changed (see d e s c r i p t i o n of the
a lg o r it h m - Appendix A). The l a s t valu e in i t e r a t i o n i s th e acce pted
one and should be lower than SM from pr e vi ou s i t e r a t i o n (small i n c r e a s e
^ l . E - 4 of th e val ue i s t o l e r a t e d ) .
( p i ) COVARIANCE MATRIX IS NOT POSITIVE DEFINITE ! ! ! !
This i s a warning t h a t th e system m a tr ix of Eq.(B3) i s found not
p o s i t i v e d e f i n i t e by th e a lg o r it h m of Cholesky decomposition {CHOLPI}.
I t s i g n a l s poor c o n d i t i o n i n g of th e problem, i . e . caused but too many
h i g h l y c o r r e l a t e d para mete rs or at te m pt t o e s t i m a t e para mete rs which
have i n s i g n i f i c a n t i n f l u e n c e on th e model f u n c t i o n . The a lg o ri th m t r i e s
t o make th e system m a t r i x p o s i t i v e d e f i n i t e by adding p o s i t i v e
Marquardt term t o t h e diagon al and min imiza tion i s co nti nu ed . Conver­
gence i s , however, slow and, i f poor c o n d i t i o n i n g p e r s i s t s , e s t i m a t i o n
of confide nce l i m i t s and c o r r e l a t i o n s i s not p o s s i b l e . When warninig
( o l ) appea rs only a t th e beginning of r e g r e s s i o n i t i s u s u a l l y a s s o ­
c i a t e d with poor i n i t i a l guess of para mete rs which was overcome by the
a lg o ri th m .
(p2) O s c i l l a t e s a f t e r 20 ITER ??
PARAM: 1.234E+03 . . . . .
269
or
(p3) Limit of 20 ITER !!!
PARAM: 1.234E+03 .............
and
(p4) {NLMFLX}
-1 Terminate, 0 Accept, >0 No of s t e p s [ITY,ITDR,IDR.LPT]
//MAXST, ITY, LPT, ITOR, IDR (513)
Response -1 t e r m i n a t e s n o n l i n e a r r e g r e s s i o n witho ut p r i n t o u t of
r e s u l t s and witho ut op ti o n fo r p l o t t i n g - same as (p5). MAXST=0 (simply
<RETURN>) a c c e p t s c u r r e n t v a lu e s as r e s u l t s of l e a s t s q u ar es e s t i m a t i o n
and program continu ou s in t h e same way as a f t e r s u c c e s s f u l convergence.
MAXST>0 i s th e new l i m i t on t o t a l number of i t e r a t i o n s and should be
g r e a t e r than number given in (p2) or (p3). MAXST can be followed by
o p t i o n a l p r i n t o u t c o n t r o l para mete rs s e p a r a t e d by commas as in (n ) .
Here IDR i s a l s o in c lu d e d , which pe rm it s c r e a t i o n of an e x te n s iv e
d i a g n o s t i c p r i n t o u t . Decision on whether t o t e r m i n a t e , acc e pt or con­
t i n u e r e g r e s s i o n can be based on v a l u e s of th e o b j e c t i v e f u n c ti o n SM in
s e v e r a l pre ceding i t e r a t i o n s , which a r e d i s p l a y e d by th e t e r m i n a l , and
on v a l u e s of par am ete rs .
(p5) NO CONVERGENCE AFTER 220 STEPS ! ! ! ! ! ! ! !
PARAM: 8 . 453E+05 ...............
This f a t a l message s i g n a l s accumulation of more than 20 Marquardt
terms. In r a r e c a s e s convergence can be o b ta in e d a f t e r r e g r e s s i o n i s
r e s t a r t e d v i a (k) with v a lu e s of para mete rs given in (p5). G e ne ra lly ,
however, (p5) s i g n a l s t h a t e i t h e r i n i t i a l v a l u e s of parameters were too
270
f a r o f f or model f u n c t i o n i s not a deq ua te . Next prompt i s ( s ) .
(p6) All parameters on bounds a f t e r 23 s t e p s !!!
PARAM: 1.000E-2..........................
-1 Terminate, 0 Accept, 1 Try t o r e l a x bounds //NST (13)
Warning when a l l para mete rs a r e equal t o t h e i r bounds and c a l c u ­
l a t e d c o r r e c t i o n would move them o u t s i d e th e bounds. A minimum of the
o b j e c t i v e f u n c t i o n found on t h e border of t h e domain fo r v a r i a t i o n of
pa ra m et e rs . When ITY>0 only th e f i r s t l i n e a pp ear s and th e r e s u l t i s
a c c e pt ed . NST=0 and NST=-1 have same e f f e c t a s in (p 4) . NST=1 r e s t a r t s
r e g r e s s i o n with c u r r e n t v a lu e s of para mete rs and c o n t r o l v a r i a b l e s of
m in im iz at io n a l g o r i t h m ( s t e p bounds, dumping, Marquardt te r m s ) , prompt
(o) r e t u r n s and t h e u ser has a chance t o r e l a x th e bounds of chosen
pa ra m et e rs . Search f o r th e minimum i s con tin ue d over th e extended
range.
(p7) {NLMFLX}
CONVERGED IN 27 STEPS, WTD RMS RES 1.2345E+00, TES=4. 125E-06, DX=
1 . 343E-07
PARAM: 1 . 203E+01 ...........
RCONF: 1.234E-02 ...........
Display of r e s u l t s of t h e l e a s t s q u a r e s estimation.- WTD RMS RES i s
✓SM/(2m-r) , where m i s th e number of f r e q u e n c i e s , r number of th e
e s t i m a t e d pa ra m ete rs ; TES and DX a r e t h e l a s t v a l u e s of convergence
t e s t parameters - r e l a t i v e change of t h e o b j e c t i v e f u n c t i o n and norm of
th e t o t a l s c a l e d c o r r e c t i o n of pa rameters [Eq.(B3)] r e s p e c t i v e l y . PARAM
l i s t s e st i m a te d para mete rs and RCONF t h e i r r e l a t i v e u n c e r t a i n t i e s (con­
271
fid e n c e ) -
ax
^ / x ^,
where
ax
i s given by E q .(3 .1 4 ) (RCONF i s equal to -
AXj fo r param eters equal z e r o ) . Only th e e stim a te d param eters a re
in c lu d e d .
( r ) {FIRDAC} - c o n tr o l o f g ra p h ic s
( r l ) -1 to draw new f i t OVER th e l a s t p l o t //IDEVN (13)
This prompt i s is su e d i f , du rin g p re v io u s f i t on th e same d a ta ,
th e l a s t graph was produced on p l o t t e r . Response -1 p e rm its drawing
more than one f i t t e d curve in th e same frame. I f th e l a s t grap h, a f t e r
p rev io u s f i t on th e same d a t a , was d is p la y e d on g ra p h ic s te rm in a l then
new f i t t e d curve i s drawn over w ith o u t a q u e rry .
( r 2 ) -1 C reate p l o t f i l e , 0 no more p l o t , >0 d is p la y or p l o t :
1-4
Complex Plane: 1 Adm, 2 Imp, 3 Cap, 4 Mod; 9 Bode; 10 R e s id u a l; 5-8
l o g - l o g ( f ) : 5 Adm, 6 Imp, 7 Cap, 8 Mod [ , - 1 p l o t ] //ITYPN,IDEVN (213)
Cursor r e t u r n s t o t h i s prompt a f t e r graph or f i l e i s c re a te d u n t i l
te rm in a te d by ITYPN=0 ( j u s t <RETURN>). The ch oice of graphs i s s i m i l a r
to t h a t of ( d ) . ITYPN=1 to ITYPN=9 produces c o n tin u e s curve p l o t s of
th e f i t t e d model fu n c tio n in th e same frame a s th e d i s c r e t e d a ta
p o in ts . The e n t i r e frequency range of th e e xp erim ental d a ta i s covered,
and th e c a l c u l a t e d spectrum i s re p re s e n te d by dashed l i n e o u ts id e th e
range tak en i n to account d urin g f i t t i n g . Comparison w ith th e experimen­
t a l d a ta of th e c urv es e x tr a p o la t e d o u ts id e th e f i t t e d range i s h e lp f u l
f o r d e c id in g how th e model response should be extended over a wider
frequency range d u rin g a step w ise c o n s tr u c ti o n of a model fu n c tio n
( e . g . adding elem ents to an e q u iv a le n t c i r c u i t ) . In th e complex plane
diagrams (ITYPN=1,2,3 or 4) th e c a l c u l a t e d cu rves have t i c marks a t th e
272
p o in ts corresp on din g to th e exp erim en tal fr e q u e n c ie s ( i f s e p a r a tio n of
th o se p o in ts a s s u r e s s u f f i c i e n t r e s o l u t i o n ) . Data p o in ts a t th e f r e ­
q uen cies n e a r e s t to f u l l decades ( f * l.E n ) a re lin k e d by dashed l i n e s to
th e co rresp on ding p o in ts on f i t t e d c u rv e s.
ITYPN=10 produces d i s c r e t e p l o t of r e s i d u a l d e v ia t io n s of d a ta
from f i t t e d model fu n c tio n . R e la tiv e d e v ia t io n of th e a b s o lu te v a lu e of
impedance ( p e r c e n t) and d e v ia t io n o f th e phase of impedance (d eg ree)
a re p l o t t e d v e rs u s lo g a rith m of frequency. P o in ts o u ts id e th e f i t t e d
range a r e includ ed .
As b e fo re , o p tio n a l ,-1 (IDEVN=-1) can be used to c r e a t e th e en­
hanced graph on th e p l o t t e r . When IDEVN<-1 i s s p e c i f i e d th e a u x i l i a r y
prompt ( d l ) can be used to change th e s i z e of th e p l o t t i n g a re a and/or
to is s u e an HPGL command.
ITYPN=-1 produces a f i l e which c o n ta in s th e in fo rm atio n about th e
d a ta (from th e f i r s t l i n e of d a ta f i l e ) and th e model (number and
acronym o f model; names, v a lu e s and u n c e r t a i n t i e s of param eters) f o l ­
lowed by th e l i s t of th e e xp erim ental d a ta and th e corresponding f i t t e d
v a lu e s in th e same r e p r e s e n t a t i o n as used d urin g f i t t i n g {PLOUTR}. Name
o f t h i s f i l e i s a combination of th e number o f e stim a te d p aram eters,
acronym of th e model and name of th e d a ta f i l e w ith e x te n sio n .PLO
( e . g . 8 param eters e stim a te d in model 12C0R fo r d a ta f i l e N1P018 8C0NP018. PLO). Name of th e c re a te d f i l e i s in clu ded in th e p r i n t o u t .
Inform ation s to r e d in f i l e *.PLO can l a t e r be used fo r p l o t t i n g with
th e a id of program PLOTFI which has th e same g ra p h ic c a p a b i l i t i e s as
FIRDAC. This o p tio n can be u s e f u l when g ra p h ic s hardware i s not a v a i l ­
273
able during session Nith program FIRDAC.
( s ) {FIRDAC}
-1 Stop, 0 nex t d a ta , 1 new g u e ss , 2 o th e r UTS a n d /o r MODEL / / I Q (13)
Response -1 te rm in a te s th e i n t e r a c t i v e s e s s io n w ith program
FIRDAC, 0 r e t u r n s to th e beginning (a ) fo r new d a ta . In both c a se s the
p r i n t o u t f i l e FIRDAC.OUT i s c lo se d ( i f p r i n t o u t i s not on l i n e ) and
e n te r e d in to p r i n t queue u n le s s ISP0=1 was given in response to prompt
( a ) . IG=1 b rin g s back qu erry fo r i n i t i a l v a lu e s of param eters ( k l ) and
IG=2 allow s to choose a n o th e r model an d /o r w eighting scheme ( i ) and ( j )
fo r an a d d i t i o n a l a n a l y s i s of th e same d a ta .
The above d e s c r i p t i o n of u s e r ' s i n t e r a c t i o n w ith program FIRDAC
pro v id es th e b a s ic in fo rm atio n re q u ire d fo r perform ing th e n o n lin e a r
l e a s t sq u ares a n a l y s i s of complex impedamnce/admittance d a ta . More de­
t a i l e d in fo rm atio n about o p e ra tio n of th e program can be found in th e
sou rce l i s t i n g . Although program FIRDAC p ro v id e s q u ite powerful and
r a t h e r f r i e n d l y to o l fo r d a ta a n a l y s i s , c e r t a i n s t r a t e g y fo r i t s e f f i ­
c i e n t use must be developed by th e u s e r , e s p e c i a l l y when e x act form of
th e model fu n c tio n i s not known a p r i o r i . Such s t r a t e g y cannot be
e a s i l y form ulated a s a c lo se d a lg o rith m because i t depends both on th e
ex p erim en tal spectrum and on th e phenomenological model. Wise u t i l i z a ­
t i o n of f e a t u r e s such l i k e f ix i n g th e v a lu e s of p a ra m ete rs, lim i t i n g
th e frequency range o f d a ta to be f i t t e d , d i s p la y o f d a ta and f i t t e d
spectrum in d i f f e r e n t r e p r e s e n t a t i o n s i s e s s e n t i a l fo r s u c c e s s f u l
modeling of th e experim ental a . c . response of e le c tro c h e m ic a l systems.
274
APPENDIX D. The L east Squares E stim atio n of Param eters in N onlinear
I m p l ic it Model.
Consider a system in which n m easureable v a r i a b l e s y^ and r un­
known param eters x^ a r e r e l a t e d by m fu n c tio n s f k w ith th e p ro p e rty
f k(x x
xr ,y 1>. . . ,y n ) = 0
k = l,2 ,...,m
(Dl)
o r , in v e c to r n o t a t i o n ,
F(X,Y) = 0
(D2)
where n>m>r.C e rta in assum ptions about c o n t i n u i t y of f u n c tio n s f^ and
behavior o f t h e i r d e r i v a t i v e s a r e made [ 6 ] . The measurements of th e
v a ria b les
c o n ta in random ex p erim en tal e r r o r s :
Ym = Y+E
(D3)
In o rd e r to e s tim a te th e p aram eters X, i t i s n e c e ssa ry to e s tim a te
th e tr u e v a lu e s o f th e measured v a r i a b l e s Y. The e r r o r v e c to r E i s
u s u a ll y assumed to be a n o r m a l l y - d i s t r i b u t e d , random v e c to r having a
T
zero mean and a known p o s i t i v e d e f i n i t e c o v arian c e m a trix Cy=<EE >. The
maximum lik e lih o o d e s tim a te of (X,Y) i s o b ta in e d when th e o b je c t iv e
fu n c tio n
Q(Y) = ( Y m - Y ^ C / ^ - Y )
(D4)
i s minimized w ith Eq.(D2) a s a c o n s t r a i n t . The c o n s t r a i n t may be ad­
j o i n t to th e o b j e c t i v e fu n c tio n by a m-vector X of Lagrange m u l t i p l i e r s
Qe (X,Y,X) = (Ym-Y )TCy_1(Ym-Y) + XTF(X,Y)
(D5)
An i t e r a t i v e a lg o rith m fo r m inim izatio n of fu n c tio n QE was given
by B r i t t and Luecke [110]. D e ta il s can be found in t h e i r paper. In s te p
1 of i t e r a t i o n , th e c o n s t r a i n t s [Eq.(D 2)] a re l i n e a r i z e d around th e
e s tim a te s of th e param eters X^_^ and th e o b s e rv a b le s Y^_^ from s t e p 1-
275
1. C o rre c tio n s fo r param eters X and o b s e rv a b le s Y a re c a l c u l a t e d by
s o lv in g a system of l i n e a r e q u a tio n s . Measured v a lu e s a re taken as th e
i n i t i a l v a lu e s fo r th e o b s e rv a b le s
Good i n i t i a l g uesses XQ of
th e param eters a r e re q u ire d to o b ta in convergence in s tr o n g ly n o n lin e a r
c a s e s . The i t e r a t i o n s a r e continued u n t i l c o r r e c t i o n s of th e param eters
and th e o b se rv a b le s become s u f f i c i e n t l y sm a ll. A s i m i l a r a lg o rith m
given by Brandt [113] i s no t e x a c t in th e n o n lin e a r case because the
o b je c t iv e f u n c tio n , which i s minimized t h e r e , c o n ta in s c o r r e c t i o n s to
th e v a r i a b l e s in tro d u ce d in given i t e r a t i o n
in p la ce of the
t o t a l d i f f e r e n c e between measured and e s tim a te d v a lu e s of o b serv ab le s
( W
The e x te n t of th e com putations and computer memory re q u ire d by th e
a lg o rith m depends on th e n a tu re of th e problem. M atrice s of p a r t i a l
d e r i v a t i v e s of th e c o n s t r a i n t fu n c tio n s [E q .(D l)] w ith r e s p e c t to p a ra ­
m eters
i .
a
!i»
lt-1,2
Hi
1*1,2
r
(D6)
ki
and w ith re s p e c t to o b s e rv a b le s
3f
k = l , 2 , . . . ,m
j = l , 2 , . . . ,n
(D7)
»2j X1 - 1 ’Y1-1
a r e c a lc u la te d in each s t e p of i t e r a t i o n . I t i s n e c e ssa ry to i n v e r t an
B
1 =
— -
1,3
(m*m) m a trix
CB=BCyBT
(D8)
cA=ATcB" l A
(D9)
and an ( r * r ) m a trix
M atrix Cfi i s u s u a ll y n e a r ly d ia g o n al and, t h e r e f o r e , can be e a s i l y i n ­
276
v e r te d . C o rre c tio n s o f p aram eters and o b s e rv a b le s in s te p 1 a re
xr xl _ l = -8 Ca " 1ATCb‘ 1 [P(X1_1>Y1_1 )+B(Ym-Y1_1)]
(DIO)
V ym = -cYBTcB 1[F(xi - i ' Yi - i )+A(xr xi - i )+B(YM"Yi - i )]
(D11)
where s i s th e s t e p - r e d u c t i o n f a c t o r , i n i t i a l l y s e t equal to 1.
Convergence of th e a lg o rith m cannot be a s s u re d . I f m u ltip l e s t a ­
ti o n a r y p o in ts a r e p o s s i b l e , th e r e s u l t o f an e s tim a tio n depends on th e
i n i t i a l g u esses of th e p a ra m ete rs. In our im plem entation of th e a lg o ­
rith m , th e fo llo w in g q u a n t i t i e s a r e used t o monitor th e p ro g re s s of th e
ite ra tio n s:
( i ) change of th e o b j e c t i v e f u n c tio n , Eq.(D4);
( i i ) weighted fu n c tio n r e s i d u a l
Q1 = fT( x1- 1 ’Y1- 1>cb1f (X1- 1 ’Y1-1) j
(D12)
( i i i ) squared norm of s c a le d c o r r e c t i o n o f p aram eters
dX = (x l _xl - l ) \ t xl _xl - l )
where DNi i =( CA
i
(D13)
i s th e d i a 9 ° n a l s c a l i n g m a trix ;
( i v ) squared norm o f c o r r e c t i o n o f v a r i a b l e s in i t e r a t i o n 1
dY = ( y 1_y1 - 1 )TcY( y1“ y 1 -1 ) •
(D14)
f
I f both Q o f Eq.(D12) and Q of Eq.(D4) in c r e a s e more th an 10%
from i t e r a t i o n 1-1 t o 1, then th e s t e p - r e d u c t i o n f a c t o r i s reduced by
h a l f , c o r r e c t i o n s of v a r i a b l e s E q .(D ll) a r e r e c a l c u l a t e d w ith new X^X1_1 and th o se v a lu e s a r e used in next i t e r a t i o n . The f a c t o r s i s
m u l t i p l i e d by 1 .3 in each i t e r a t i o n , as long as i t i s l e s s than 1. This
sim ple ad ju stm en t of s t e p s i z e improves s i g n i f i c a n t l y th e convergence
range of th e a lg o rith m .
277
S u c c essfu l convergence of r e g r e s s io n i s o b ta in e d when a l l o f th e
fo llo w in g c o n d itio n s a r e f u l f i l l e d :
Ql<10"6 ,
(Q1-Q1_1 )/Q 1_1<0.001,
dx<10"4 ,
dY<10"4 .
The num erical v a lu e s given above may be too s t r i c t in c e r t a i n c a s e s :
when e r r o r s a re l a r g e , or th e c o n s t r a i n t Eq.(D2) a r e n o t e x a c tly
obeyed. They worked w e ll, however, in most c a s e s s tu d ie d h e re .
Mhen th e maximum li k e l i h o o d e s tim a te (XE,YE) i s found, th e a lg o ­
rith m p ro v id e s in fo rm atio n on th e d i s t r i b u t i o n of th e e s tim a tio n e r ­
r o r s . In a l i n e a r ap p ro x im atio n , th e c o v arian c e m a trix of th e e stim a ­
t i o n e r r o r s fo r p aram eters i s
CX = <(Xe-X)(X e -X)T> = o2Ca _1 = 02 [AT(BCyBT) “ 1A ] '1
where
a
2
(D15)
i s th e minimized o b j e c t i v e fu n c tio n Q given by Eq.(D4) d i ­
v ided by th e th e number o f d eg rees of freedom ( th e root-m ean-square
r e s i d u a l - Rmsr)
^ =„2=jyvV <v v
m-r
Square r o o ts of th e d ia g o n a l elem ents o f th e c o v arian c e m a trix Cy given
by Eq.(D14) a re used to e s ti m a te , in l i n e a r a pp rox im ation , th e 95%
co nfidence l i m i t s f o r each param eter when a l l o th e r param eters a re
ignored xEi±2(Cx i i ) 1 / 2 .
I t should be n o te d , t h a t only th e r a t i o s o f th e v a r ia n c e s and the
c o v a ria n c e s in m a trix Cy e f f e c t th e e s ti m a te s . As can be seen from
Eqs.(D lO ), ( D l l ) , (D15) and (D16), when Cy i s re p la c e d by cCy (c a
p o s i t i v e s c a l a r ) , th e e s tim a te s o f param eters and t h e i r con fidence
l i m i t s remain unchanged.
278
A p p lic a tio n s of th e g e n e ra l a lg o rith m fo r a n a l y s i s of v a r i a b l e te rm in a tio n measurements of waveguide d i s c o n t i n u i t i e s and, s p e c i f i c a l ­
l y , complex p e r m i t t i v i t y of m a te r ia l samples in waveguides a re p re ­
se n te d in Chapter 4.
APPENDIX E. D e riv a tiv e s and o th e r Formulas fo r th e V a ria b le Termination
Method.
P a r t i a l d e r i v a t i v e s of th e in p u t- o u tp u t r e l a t i o n fo r homogeneous­
l y - f i l l e d s e c tio n of waveguide [ Eqs. ( 4 . 3 4 ) —( 4 .3 5 ) ] a re
3f
_
2 j Y l ( 8 + Z B2 Y2/ l 3 ) + 4 Z n Y S i n 2 j Y l + 2 ( Z n 2 Y2 / 8 - 8 ) s i n j Y l c o s j Y l
= ------------ 5-------------- §------------------J --------------
3y
(E l)
[ ( Zg+1) ycosjY l + (8-ZgY / 8 ) s i n j Y l ]
3f
2 y2
(E2)
3Zg
[(ZB+l)YCOsjYl + (8-ZBY2/ 8 ) s i n j Y l ] 2
af
—
= -1
(E3)
3RA
3f
2 j Y 2 ( 8 + Z 0 2 Y2 / B )
(E4)
31
[ ( Z g + 1 ) y c o s j y 1 + ( B - Z BY2 / B ) s i n j Y l ] 2
2 _____________________
3RS
^
ad
(E5)
[(l+ R g)cosB (d-do ) - j( l - R g ) s i n B ( d - d 0 ) ] 2
____________________
(E7)
[(l+Rg)cos8(d-d0 ) - j ( l - R g ) s in 8 ( d -d 0 ) ] 2
where th e s u b s c r i p t i fo r th e v a rio u s measurements has been om itted fo r
th e sake of s i m p l i c i t y . The same e q u a tio n s apply to th e fu n c tio n f , as
w r i t t e n in E q .( 4 .4 ) .
Derivatives of the reflection coefficient R^ [Eq.(4.6)] with res-
279
pect to th e p o s i t i o n o£ v o lta g e minimum s and VSWR W=201og1QVs a re
3RX
= 2j8Rx ,
3s
3R.
lnlO
—^ = ------------ §— Ra
3W
10
V ^-l M
(E8)
APPENDIX P. Computer P r i n t o u t s fo r Examples of Data Reduction in the
V a ria b le-T e rm in a tio n Method.
The computer p r i n t o u t s fo r th e examples d is c u s s e d in S e c .4 .3 .3 .
a re given below
a) c a l i b r a t i o n
Meaning of th e a b b r e v ia tio n in terms of symbols used in Chapter 4
i s as follow s
AKC=ti/a m"1 ,
BEZ0=8,
CLOS=R„,
O
MIN POS - s ,
FIRO - phase of RA>
MR - number of p a ra m ete rs,
SC POS - d,
FIRS - phase of RnD ,
ABS(RO)=|RAl ,
MY - number o f a d ju s te d o b s e rv a b le s fo r
each measurement, FRS=Qf [ Eq. (D12) ] ,
CONF - e stim a te d s ta n d a rd d e v ia ­
t i o n s of p a ra m ete rs, a ls o l i s t e d a f t e r +- fo r some p aram eters.
In th e p r i n t o u t o f c o r r e c te d measurements, th e a d ju s te d v a lu e s a re
followed by d e v ia t io n s from th e measured v a lu e s . For th e a b s o lu te v alu e
of r e f l e c t i o n c o e f f i c i e n t RO=|RAl th e d e v ia t io n i s given as a change of
VSWR in dB. The f i r s t e s tim a tio n i s fo r th e s c a t t e r i n g param eters Sh
o n ly , th e second in c lu d e s refin em en t of th e re fe re n c e p o s itio n dQ and
th e e f f e c t i v e r e f l e c t i o n c o e f f i c i e n t Rg of a v a r i a b l e s h o r t c i r c u i t .
280
P r in to u t from program CAHOSW:
17-APR-85 CAHOSW, FILES: DATA - WX8WIN .RDS, RESULTS - WX8WIN .CAW
AKC= 1.374E+02, DETECTOR EXP. 2.0000, BEZO= 1.090E+02 1/M,
FREQ= 8.370E+09, LAMBDASL0TTED/2= 29.078 MM, LAMBDAL0AD/2= 28.814
REFERENCE D0= 19.00 MM, S0= 13.63 MM, RS=0.9950 , CL0S=0.9819
RESOLUTION: DRS=0.0100 DDS=0.03MM, DVSWR=0.2DB+3.0*, DSA=0.03MM
* VSWR DB * ABS(RO)
FIRS # MIN POS * FIRO
NO # SC POS *
38.60 # 0.9768
258.71 *
19.95 * -1 0 1 .6 9 *
12.70 *
1 *
-63.00 *
23.08 *
38.50 * 0.9765
9.52 * 298.38 *
2 *
38.20 * 0.9757
338.05 *
26.20 * -2 4 .3 8 *
6.35 *
3 #
14.87 #
37.90 # 0.9748
29.37 *
3.17 *
377.72 *
4 *
*
*
*
#
*
54.67
32.58
38.45 * 0.9764
5
0.00
417.38
6
7
8
9
*
*
*
*
15.88 *
19.05 *
22.23 *
25.40 *
219.04
179.38
139.71
100.04
*
#
*
*
16.80 * -140.82 #
13.59 * -1 8 0 .5 0 *
10.32 * -2 2 0 .9 2 *
98.37 *
36.12 *
38.45 * 0.9764
38.65 # 0.9769
38.25 # 0.9758
37.80 * 0.9746
MR= 6
MY= 4
##########
FIT OF GENERAL SCATTERING MATRIX
AFTER
3 ITER, RMS RESIDUAL 7.9077E-02, FRS= 5.6067E-10
PARAM: 4 . 998E-03-1. 265E-02 9.804E-01-2.255E-02-5.882E-03-1.224E-02
CONF.: 2.570E-04 2.573E-04 2.765E-04 2.393E-04 2.647E-04 2.651E-04
CORRECTED MEASUREMENTS
FIRO
DFIRO
DRS
FIRS
DFIRS
RO DVSWR
NO
RS
0.00
0.0 0 0.9768 0.00 -101.70
1 0.9951 0.0001 -101.28
-0 .0 2
0.0 2 0.9765 -0 .01 -63.03
2 0.9952 0.0002 -6 1.6 0
-0.0 1 0.9757 0.01 -24.36
0.02
3 0.9949 -0.0001 -21.97
14.85
-0 .0 2
17.73
0.02 0.9749 0.03
4 0.9945 -0.0005
54.71
0.03
57.35
-0 .0 3 0.9762 -0 .0 5
5 0.9960 0.0010
-0 .0 3 0.9764 0.02 -140.79
0.03
6 0.9947 -0.0003 -140.98
7 0.9954 0.0004 179.38
-0.01
0.01 0.9769 -0 .0 2 179.50
0.0 0 0.9758 0.00 139.08
0.0 0
8 0.9949 -0.0001 139.71
98.35
-0 .0 2
0.02 0.9747 0.04
9 0.9943 -0.0007 100.06
FIT OF GENERAL SCATTERING MATRIX
ADJUSTMENT OF SHORT CIRCUIT POSITION X(7) AND REFLECTION X(8)
MR= 8
MY= 4
##########
AFTER
2 ITER, RMS RESIDUAL 7.9071E-02, FRS= 3.1116E-08
PARAM: 4 . 998E-03-1. 265E-02 9 . 809E-01-2. 465E-02-5. 911E-03-1. 224E-02
2 . 127E-03 9 . 945E-01
CONF.: 2.570E-04 2.573E-04 4.982E-04 4.475E-04 2.631E-04 2.639E-04
5 . 167E-04 5 . 777E-04
CORRECTED MEASUREMENTS
DVSWR
NO
DRS
FIRS
DFIRS
RO
FIRO
DFIRO
RS
1 1.0000 0.0000
0 .00
0 .0 0 0.9819 0.0 0 180.00
0.0 0
2 1.0001 0.0001 -78.72
0.0 0 0.9768 0.00 -101.70
0.00
-0 .0 2
3 1.0002 0.0002 -118.40
-0 .0 2 0.9765 -0.01 -63.03
4 0.9999 -0.0001 -158.03
0.01 0.9757 0.01 -24.36
0.02
14.85
-0 .0 2
5 0.9995 -0.0005 -197.73
-0 .0 2 0.9749 0.03
281
0.03 0.9762 -0.05
54.71
0.03
6 1.0010 0.0010 - 237.35
0.03 0.9764 0.02 -140.79
0.03
7 0.9997 -0.0003 -39.02
-0 .0 1 0.9769 -0 .0 2 179.50
-0.0 1
0.62
8 1.0004 0.0004
0.0 0 0.9758 0.0 0 139.08
40.29
0.00
9 0.9999 -0.0001
-0 .0 2 0.9747 0.04
79.94
98.35
-0 .0 2
10 0.9993 -0.0007
ADJUSTED D0= 19.010 +- 0. 002 RS==0.9945 +- 0.0006
ATT*BEZO= 0.00E-01 +- 0.00E-01
SCATTTERING PARAMETERS
4 . 9977E-03 -1.2650E-02 9.8091E-01 -2.4652E-02 -5.9110E-03 -1.2237E-02
b) measurement o f p l e x i g l a s s
Explanation of th e a b b r e v ia ti o n s used i s as follows.a) In th e l i s t i n g of d a ta RO, FIRO a r e th e a b s o lu te v a lu e and th e phase
of RAh; ROC, FIRA a r e th e a b s o lu te v a lu e and th e phase o f R^ a t th e
f r o n t s u rfa c e of th e sample, c o r r e c te d a cco rd in g to E g .( 4 .3 8 ) ; SC POS
_3
g iv e s th e p o s itio n o f th e s h o r t c i r c u i t as measured in inches*10 ;
b) +1* d en o tes an im aginary p a r t , DIE=e', CON=o, CORRELATION i s th e
c o r r e l a t i o n between th e e s ti m a te s of th e r e a l and th e imaginary p a r t s
of th e p e r m i t t i v i t y , TANDTA=tan8,
GY=y ;
c) Estim ated s c a t t e r i n g param eters a r e given a s r e a l and imaginary
p a r t s in th e l i s t of param eters PARAM and a s th e a b s o lu te v alu e and th e
phase below th e l i s t of c o r r e c t e d measurements;
d) In th e l i s t of th e c o r r e c te d measurements fo llo w in g th e e s tim a tio n
of th e p ro pagation f a c t o r , th e phase of th e r e f l e c t i o n c o e f f i c i e n t of
th e load i s given as FIRL=B(di - d o )*180/TT (modulo 360), w hile in th e
l i s t fo llow in g th e f i t of th e s c a t t e r i n g m a trix , th e phase i s
FIRS=-2B(di - d o )xl80/n+180 (modulo 180), (This i s th e same as in th e
l i s t of d a t a ) .
282
P r in to u t from program TLDCEW:
06-MAY-85, TLDCEW, CALIBRATION FILE WX8WIN .CAW
ARC= 1 . 374E+02, DEX= 2.0000, BEZO= 1.090E+02 1/M, FREQ= 8.370E+09
LAMBDASL0T/2= 29.078mm , LAMBDAL0AD/2= 28.814mm
RESOLUTION: DRS=0.0100 DDS=0.03mm, DVSWR=0. 2DB+3. 0 %, DSA=0.03mm
REFERENCE: D0= 19.01mm, S0= 13.63mm, RGS=0.9945
CALIBRATION SCATTERING:
4.9977E-03 -1.2650E-02 9.8091E-01 -2.4652E-02 -5.9110E-03 -1.2237E-02
If If II If t f f l t f f l If ff t f If If If I f I f I f I f If If I f t f I f If If II I f If If If I f If If If If If II If If If If If I f I f If I I If If If If I I I I If I I I I I I I I I I I I I I I I l l l l l l I I I I If I I I I I I I I
DATA PLEBRA . DTR, SAMPLE LENGTH 23.75 +- 0.020 mm, TEMP. 20.00 DEG C
* FIRO *
ROC * FIRA
RO
NO * SC POS* FI RGS * VSWR *MIN POS*
*
*
*
*
#
6 1 . 2 3 * 0 . 9 7 1 6 *-232.83
0.9524 *
1
0.0
57.51
32.25
33.12
3 9 . 2 6 # 0 . 9 6 0 7 *-254.53
2 * 160.0 *
6.73 * 30.48 * 31.34 * 0 .9419 *
*
*
#
#
#
*
319.13
28.33
29.13
0.9262
1 1 . 8 4 * 0 . 9 4 4 0 * -281.90
3
310.0
*
*
*
*
*
*
0
.
9
0
1
7
2
5
.
4
2
3
4 . 1 0 * 0 . 9 1 7 1 *-328.51
4
450.0
274.70
25.73
5 * 5 5 0 . 0 * 2 4 2 . 9 7 * 2 5 . 0 0 * 2 0 . 5 9 * 0 . 8 9 3 5 # - 9 3 . 8 3 # 0 . 9 0 7 8 * -2 9 .8 4
6 * 6 5 0 . 0 * 2 1 1 . 2 3 * 2 8 . 1 0 * 1 4 . 6 2 * 0 . 9 2 4 3 # . ■ 1 6 7 . 7 4 * 0 . 9 4 1 5 *-104.58
7 * 7 9 0 . 0 * 1 6 6 . 8 1 * 3 3 . 5 5 * 9 . 2 2 # 0 . 9 5 8 8 *. ■ 2 3 4 . 6 6 * 0 . 9 7 7 7 *-170.28
8 * 9 3 0 . 0 # 1 2 2 . 3 8 * 3 4 . 2 8 * 6 . 4 9 # 0 . 9 6 2 1 * . - 2 6 8 . 3 3 * 0 . 9 8 1 3 *-203.06
9 * 1 0 5 0 . 0 * 8 4 . 3 0 * 3 3 . 3 2 * 4 . 9 3 * 0 . 9 5 7 8 #. - 2 8 7 . 6 5 * 0 . 9 7 7 0 *-221.93
1 0 * 5 0 0 . 0 * 2 5 8 . 8 4 * 2 5 . 1 2 # 2 3 . 3 1 * 0 . 8 9 4 9 * - 6 0 . 2 2 * 0 . 9 0 9 4 *-355.31
FIT OF GENERAL SCATTERING MATRIX
AFTER
4 ITER, RMS RESIDUAL 1.9717E-01, FRS= 9.3111E-10
PARAM:- 2 . 452E-01 3.317E-01 2.467E-01 7.519E-01-2.577E-01 3.186E-01
CONF.: 6.461E-04 6.795E-04 6.603E-04 7.503E-04 7.635E-04 7.560E-04
CORRECTED MEASUREMENTS
DVSWR
RO
FIRO
DFIRO
RGS
DFIRS
NO
DRGS
FIRS
-0 .1 0
1 0.9939 -0.0006
57.55
0.04 0.9721 0.17 127.06
6.75
0.02 0.9611 0.0 7 105.43
-0 .0 4
2 0.9942 -0.0003
78.11
0.01
3 0.9945 0.0001 -40.88
-0 .0 1 0.9440 -0 .0 1
0.06
-0 .0 7 0.9170 -0.0 1
31.55
4 0.9947 0.0002 -8 5.3 7
-0 .0 1
5 0.9946 0.0001 ■-117.01
0.02 0.9078 0.00 -29.85
0.03
-0 .0 5 0.9419 0.05 -104.55
6 0.9936 -0.0009 ■-148.81
-0 .0 2
7 0.9943 -0.0002 166.83
0.0 2 0.9778 0.03 -170.31
-0 .0 5
8 0.9951 0.0006 122.40
0.02 0.9810 -0 .1 5 156.89
0.18
84.23
-0 .0 7 0.9767 -0 .1 2 138.25
9 0.9949 0.0004
-0 .0 5
4.63
0.08 0.9089 -0 .0 4
10 0.9950 0.0006 •-101.08
S l l =0.4125 +-0.0006 / 126.47 +- 0.09deg
S22;= 0.4098 +-0.0008 / 128.96 +- o.: Lldeg
S12S12=0.7913 +-0.0007 / 71.83 +- 0.05deg
IS22-S111=0.0027 + -0.0010, f s 2 2 - f s l l = 1.25 +-0.14deg, s h i f t 0.100mm
FIT OF SYMMETRIC SCATTERING MATRIX
AFTER
3 ITER, RMS RESIDUAL 1.0642E+00, FRS= 1.4555E-07
PARAM:- 2 . 520E-01 3.271E-01 2.472E-01 7.512E-01
CONF.: 2 . 739E-03 2.691E-03 3.494E-03 3.895E-03
CORRECTED MEASUREMENTS
NO
RGS
DRGS
FIRS
DFIRS
RO
DVSWR
FIRO
DFIRO
283
0.48
-0 .1 8 0.9711 -0 .1 3 127.65
57.33
1 0.9950 0.0006
0.52
-0 .2 2 0.9605 -0 .0 5 106.00
6.52
2 0.9949 0.0004
0
.30
0.03
78.39
0.9442
-4
1.0
6
-0
.1
9
3 0.9945 0.0001
31.37
-0 .1 2
0.14 0.9180 0.10
4 0.9932 -0.0013 -85.15
-0 .3 2
0.61 0.9091 0.12 -3 0 .1 6
5 0.9923 -0.0022 -116.42
-0 .3 2
0.57 0.9429 0.22 -104.90
6 0.9916 -0.0029 -148.20
-0 .1 6
0.14 0.9779 0.07 -170.44
7 0.9940 -0.0005 166.94
0.25
-0 .1 2 0.9804 -0 .4 3 157.18
8 0.9963 0.0018 122.25
0.69
84.03
-0 .2 7 0.9758 -0 .4 7 138.76
9 0.9962 0.0017
4.36
-0 .3 2
0.51 0.9101 0.07
10 0.9930 -0.0015 -100.65
S ll =0.4129 +-0.0024 / 127.61 +- 0.34deg
S12S12=0.7909 +-0.0038 / 71.79 +- 0.25deg
PROPAGATION CONST: 9.206E-01 +1* 2.663E+02
DIE=
2.917 +- 0.040 CON= 7.409E-03 +- 2.258E-02 l./(OHM*M)
CORRELATION 0.01 46 , TANDTA= 5.464E-03 +- 1.665E-02
INDIVIDUAL SOLUTIONS
1* GY=7.275E-01 +I*2.438E+02, DIE= 2.546+-0.008 C0N=5. 359E-03+-2. 2E-03
2* GY=7.673E-01 +I*2.433E+02, DIE= 2.537+-0.007 C0N=5. 638E-03+-1. 9E-03
GY BECAME -7.894E+00 +1* 2.186E+02 NEW 7.000E-01 +1* 2.440E+02
3* GY=8.061E-01 +I*2.429E+02, DIE= 2.531+-0.006 C0N=5. 915E-03+-1. 9E-03
4* GY=8. 574E-01 +I*2.427E+02, DIE= 2.528+-0.007 C0N=6.287E-03+-2.2E-03
5* GY=8.949E-01 +I*2.429E+02, DIE= 2.531+-0.007 C0N=6.567E-03+-2.8E-03
6* GY=9.355E-01 +I*2.440E+02, DIE= 2.548+-0.009 C0N=6.895E-03+-4.1E-03
7* GY=7.355E-01 +1*2.458E+02, DIE= 2.578+-0.012 C0N=5.462E-03+-5.1E-03
8* GY=6.228E-01 +I*2.452E+02, DIE= 2.567+-0.010 C0N=4.613E-03+-3.4E-03
9* GY=6.596E-01 +1*2.445E+02, DIE= 2.556+-0.008 C0N=4.871E-03+-2.5E-03
10* GY=8. 651E-01 +I*2.427E+02, DIE= 2.528+-0.007 C0N=6.343E-03+-2.4E-03
DIE=
2.545 +- 0.016 CON= 5.795E-03 +- 7.029E-04 1/(0HM*M)
PROPAGATION CONST: 7.872E-01 +1* 2.438E+02
MR= 2
MY= 4
########«#
FIT OF PROPAGATION CONST X(1)+I*X(2)
AFTER
5 ITER, RMS RESIDUAL 2.9664E+00, FRS= 8.4371E-07
PARAM: 7.965E-01 2.432E+02
CONF.: 2.262E-01 1.695E-01
CORRECTED MEASUREMENTS
DVSWR
FIRO
DRS
RO
DFIRO
NO
RS
FIRL
DFIRL
1.07
1 0.9960 0.0015
61.45
0.21 0.9703 -0 .3 8 128.24
0.07
2 0.9953 0.0008
86.65
0.01 0.9598 -0 .2 1 105.54
77.22
-0 .8 8
3 0.9938 -0.0006 110.15
-0 .2 8 0.9436 -0 .0 7
30.56
4 0.9917 -0.0028 132.10
-0 .5 5 0.9180 0.10
-0 .9 3
5 0.9913 -0.0032 148.23
-0 .2 9 0.9098 0.20 -30 .16
-0 .3 2
0.54
6 0.9914 -0.0031 164.84
0.45 0.9422 0.10 -104.03
7 0.9956 0.0011
1.89
7.44
0.84 0.9768 -0 .3 4 -168.40
8 0.9981 0.0036
29.36
2.09
0.55 0.9795 -0 .8 1 159.03
1.82
9 0.9975 0.0030
48.23
0.38 0.9749 -0 .7 7 139.90
3.95
-0.5 5 0.9106 0.13
-0 .7 4
10 0.9918 -0.0027 140.03
PROPAGATION CONST: 7 . 965E-01 +1* 2.432E+02
DIE:=
2.536 +- 0. 003 CON= 5.853E -03 +- 1.662E -03 l./(0HM*M)
284
CORRELATION -0.0098, TANDTA= 4.965E-03 +-
1.410E-03
MR= 4
MY= 4
########*#
PIT OP PROPAGATION CONST X(1)+I*X(2)
REFERENCE SHIFTS: X(3)/BEZ0 INPUT, X(4)/BEZ0 LOAD
INIT VAL: 7.965E-01 2.432E+02 O.OOOE-OI 0.000E-01
AFTER
6 ITER, RMS RESIDUAL 2.8836E-01, FRS= 7.4647E -08
PARAM: 8 . 198E-01 2.413E+02 3.827E-02-1.676E-02
CONF.: 2.281E-02 5.616E-02 9.53IE-04 1.163E-03
CORRELATIONS:
3
1
2
l.OOOE+OO -1.604E-02 3.178E-02 2.539E-03
1 *
2 * -1.604E-02 l.OOOE+OO -6.078E-01 7 . 313E-01
3.178E-02 -6.078E-01 l.OOOE+OO 2 . 660E-03
3 *
2.539E-03 7 . 313E-01 2.660E-03 1 . OOOE+OO
4 *
CORRECTED MEASUREMENTS
DVSWR
RO
FIRO
DFIRO
NO
RS
DRS
FIRL
DFIRL
-0 .1 0
61.23
-0 .0 2 0.9707 -0 .2 6 127.07
1 0.9953 0.0008
-0 .0 4
2 0.9943 -0.0002
86.62
-0 .0 1 0.9609 0.04 105.43
0.01
0.0 0 0.9453 0.21
78.11
3 0.9933 -0.0012 110.44
0.04 0.9191 0.22
0.06
4 0.9921 -0.0023 132.69
31.55
-0 .01
-0 .0 1 0.9091 0.12 -29.85
5 0.9924 -0.0021 148.51
0.03
0.02 0.9420 0.07 -104.55
6 0.9933 -0.0012 164.41
-0 .0 3
7 0.9969 0.0025
6.59
-0.0 1 0.9769 -0 .3 2 -170.31
-0 .05
-0.01 0.9793 -0 .8 9 156.89
28.80
8 0.9983 0.0038
0.18
47.89
0.03 0.9750 -0 .7 4 138.25
9 0.9971 0.0026
-0 .0 5
-0 .0 4 0.9107 0.14
4.63
10 0.9925 -0.0020 140.54
PROPAGATION CONST: 8 . 198E-01 +1* 2.413E+02
SHIFTS: FRONT 0.351 +-0.009 BACK-0 .154 +-Ci.O ll mm
DIE=
2.506 +- 0.001 CON= 5 . 976E-03 +- 1.663E-04 l./(OHM*M)
CORRELATION 0.0230, TANDTA= 5.130E-03 +- 1.428E-04
MR= 6
MY= 4
**********
FIT OF PROPAGATION CONST X(1)+I*X(2)
REFERENCE SHIFTS: X(3)/BEZ0 INPUT, X(4)/BEZ0 LOAD
ADJUSTMENT OF ATTENUATION X(5) LOAD, X(6) INPUT
INIT VAL: 8.198E-01 2.413E+02 3 . 827E-02-1. 676E-02 l.OOOE+OO l.OOOE+OO
AFTER
4 ITER, RMS RESIDUAL 1.9716E-01, FRS= 6.6757E-08
PARAM: 9 . 731E-01 2 . 413E+02 3.828E-02-1.655E-02 1. OOOE+OO 9 . 930E-01
CONF.: 4 . 986E-02 3.845E-02 6 . 513E-04 7.965E-04 2. 125E-03 1 . 556E-03
CORRELATIONS:
2
3
4
5
6
1
1* l.OOOE+OO 2.049E-02 3 . 845E-02 4.053E-02 6 869E-01 - 6 . 191E-01
2* 2.049E-02 l.OOOE+OO -6.060E-01 7.322E-01 -1 747E-02 -6.022E-02
467E-02 - 2 . 892E-03
3* 3 . 845E-02 -6.060E-01 l.OOOE+OO 3.534E-03
4* 4.053E-02 7.322E-01 3.534E-03 l.OOOE+OO
106E-03 -6.059E-02
5* 6 . 869E-01 - 1 . 747E-02 3.467E-02 1 . 106E-03
OOOE+OO 5 . 259E-02
6* -6.191E-01 -6.022E-02 -2.892E-03 -6.059E-02
259E-02 l.OOOE+OO
285
CORRECTED MEASUREMENTS
RO DVSWR
FIRO
DFIRO
DFIRL
NO
RS
DRS
FIRL
-0 .1 0
-0 .0 2 0.9721 0.17 127.06
61.23
1 0.9939 -0.0006
-0 .0 4
-0.0 1 0.9611 0.07 105.43
2 0.9942 -0.0003
86.62
0.01
0.00 0.9440 -0 .0 1
78.11
3 0.9945 0.0001 110.44
31.55
0.06
0.04 0.9170 -0 .0 1
4 0.9947 0.0002 132.69
-0 .0 1
-0.01 0.9078 0 .00 -29.85
5 0.9946 0.0001 148.51
0.03
0.02 0.9419 0.05 -104.55
6 0.9936 -0.0009 164.41
-0 .0 2
7 0.9943 -0.0002
6.59
-0 .01 0.9778 0.03 -170.31
8 0.9951 0.0006
-0 .0 5
28.80
-0 .0 1 0.9810 -0 .1 5 156.89
0.18
47.89
0.04 0.9767 -0 .1 2 138.25
9 0.9949 0.0004
4.63
-0 .0 5
10 0.9951 0.0006 140.54
-0 .0 4 0.9089 -0 .0 4
PROPAGATION CONST: 9 . 731E-01 +1* 2 . 413E+02
SHIFTS: FRONT 0.351 +-0.006 BACK-0 .152 +-01.007 mm
DIE=
2.506 +- 0.001 CON= 7.094E-03 +- 3.635E-04 l./(OHM*M)
CORRELATION -0 .0 2 2 6 , TANDTA= 6.090E-03 +- 3.120E-04
c) measurement o f NASICON
A b re v ia tio n s a r e th e same a s in b ) . Two a d d i t i o n a l c a l i b r a t i o n
p aram eters a re SH=ss , th e d is ta n c e (modulo h a lf-w a v e le n g th ) from sample
to th e end of s t a i n l e s s s t e e l guide on th e s id e o f s l o t t e d s e c t i o n , and
CLS=|Ss l 2 I1/r2- INDIVIDUAL SOLUTIONS re p e a te d a t th e end of th e c a l c u l a ­
t i o n s use th e c a l i b r a t i o n p aram eters a d ju s te d in th e p re v io u s s t e p .
P r in to u t from program TLDCEU:
ll-MAY-85, TLDCEU, CALIBRATION FILE WX8NAS.CAW
AKC= 1.374E+02, DEX= 2.0000, BEZ0= 1.090E+02 1/M, FREQ= 8.370E+09
LAMBDASL0T/2= 29.078mm , LAMBDALOAD/2- 28.814mm
RESOLUTION: DRS=0.0100 DDS=0.03mm, DVSWR=0.2DB+3.0%, DSA=0.03mm
REFERENCE: D0= 17.49mm, S0= 13.63mm, RS=0.9690, SH= 4.81mm, CLS= 0.9740
CALIBRATION SCATTERING
4.9977E-03 -1.2650E-02 9.8091E-01 -2.4652E-02 -5.9110E-03 -1.2237E-02
II t l II If I I I I I I I I I I I I I I If I I I I I I I I I I I I I I I I I I I I I I I t I I I I I I I I I I If I I I I I I I I I I I I If II If I I I I I I I I I I I I I I I I I I M I I I I If I I I I I I I I I I H I I I I I I I t I I I I I I I I I I I I I I I I I I I I I I
DATA NPXH12.DTR, SAMPLE LENGTH 0.88 +- 0.010 mm, TEMP. 123.60 DEG C
NO * SC POS * FIRS * VSUR *MIN POS*
RO * FIRO
*
ROC * FIRA
1 *
0.0 0 * 38.53 * 18.72 * 10.94 * 0.7923 *-213.24 * 0.8328 * 196.40
2 * 200.00 *
335.06 *19.94 *10.84 * 0.8170 *-214.60 * 0.8583 * 195.08
3 * 400.00 *
271.59 *21.48 *10.69 * 0.8444 *-216.40 * 0.8865 * 193.33
4 * 600.00 *
208.12 *25.72 *10.27 * 0.9016 *-221.66 * 0.9451 * 188.20
5 * 725.00 *
168.45 *12.28 * 8.12 * 0.6087 *-248.22 * 0.6456 * 162.33
6 * 750.00 *
160.52 *8.88 * 9.61 * 0.4709 *-229.77 * 0.5029 * 180.15
7 * 775.00 *
152.58 *10.32 *10.80 * 0.5328 *-215.04 * 0.5657 * 194.50
8 * 825.00 *
136.72 *13.60 *11.16 * 0.6544 *-210.58 * 0.6907 * 198.94
286
9 * 925.00 * 104.98 * 16.50 * 11.07 * 0.7397 *-211.63 * 0.7787 * 197.95
10 *1050.00 * 65.31 * 18.33 * 10.98 * 0.7838 *-212.87 * 0.8241 * 196.76
FIT OP GENERAL SCATTERING MATRIX
AFTER
4 ITER, RMS RESIDUAL 2.0047E-01, FRS= 2.7414E-07
PARAM:- 8 . 058E-01-1. 802E-01-1. 594E-02-8. 805E-02-8. 062E-01-1. 500E-01
CONF.: 1.095E-03 5.881E-04 5.130E-04 6.500E-04 1.888E-03 2.152E-03
CORRECTED MEASUREMENTS
DFIRO
DRS
FIRS
DFIRS
RO DVSWR FIRO
NO
RS
-0 .1 6
38.53
0.0 0 0.8328 0.0 0 -163.76
1 0.9691 0.0001
-0 .0 4
0.00 0.8587 0 .0 3 -164.96
2 0.9690 0.0000 -24.94
0.13
0.0 0 0.8862 -0 .0 2 -166.54
3 0.9689 -0.0001 -88.42
-0 .0 7
0.01 0.9461 0.16 -171.87
4 0.9691 0.0001 -151.87
-0 .0 7 0.6519 0.19 162.32
-0.0 1
5 0.9690 0.0000 168.38
0.09
0.05 0.5038 0.0 2 -179.77
6 0.9693 0.0003 160.57
0.02
0.01 0.5661 0.01 -165.48
7 0.9688 -0.0002 152.59
0.01
0.0 0 0.6896 -0 .0 4 -161.05
8 0.9689 -0.0001 136.71
0.05
0.00 0.7785 -0 .0 1 -161.99
9 0.9689 -0.0001 104.98
0.06
65.31
0.00 0.8178 -0 .3 4 -163.18
10 0.9690 0.0000
s ir =0.8257 +-0.0011 /-1 6 7 .4 0 +- 0.05deg
S22=0.8200 +-0.0019 /-1 6 9 .4 6 +- 0.15deg
S12S12=0.0895 +-0.0006 /-1 0 0 .2 6 +- 0.32deg
|S 22 -S 1 1 |= 0 .0057+-0.0022, f s 2 2 - f s l l = -1 .0 3 + -0 .1 6 d eg , s h i f t -0.083mm
FIT OF SYMMETRIC SCATTERING MATRIX
AFTER
5 ITER, RMS RESIDUAL 7.5466E-01, FRS= 8.9620E-08
PARAM:- 8 . 062E-01-1. 796E-01-1. 962E-02-8. 240E-02
CONF.: 3.623E-03 2.137E-03 1.390E-03 1.802E-03
CORRECTED MEASUREMENTS
DFIRO
DRS
FIRO
NO
RS
FIRS
DFIRS
RO DVSWR
-0 .4 3
38.53
0.01 0.8361 0.19 -164.03
1 0.9693 0.0003
-0 .3 7
2 0.9692 0.0002 -24.94
0.01 0.8592 0.06 -165.29
-0.2 1
3 0.9692 0.0002 -88.41
0.00 0.8829 -0 .2 8 -166.88
-0 .0 6
4 0.9711 0.0021 ■-151.89
-0.0 1 0.9323 -1 .8 8 -171.85
5 0.9719 0.0029 168.10
0.05
-0 .3 5 0.6778 1.00 162.38
6 0.9673 -0.0017 160.20
-0 .3 2 0.5013 -0 .0 4 179.67 359.52
7 0.9642 -0.0048 152.66
0.31
0.08 0.5612 -0 .1 2 -165.19
0.48
8 0.9651 -0.0039 136.72
0.01 0.6955 0.16 -160.58
0.04
9 0.9688 -0.0002 104.98
0.00 0.7854 0 .30 -162.01
-0 .1 5
10 0.9692 0.0002
65.31
0 .00 0.8223 -0 .0 9 -163.39
S l l =0.8259 +-0.0037 /-1 6 7 .4 4 +- 0.17deg
S12S12=0.0847 +-0.0018 /-1 0 3 .3 9 +- 0.99deg
PROPAGATION CONST: 1.496E+02 +1* 1.179E+03
DIE= 45.049 +- 3.992 CON= 5.329E+00 +- 1.939E+00 l./(OHM*M)
CORRELATION -0 .0 8 0 1 , TANDTA= 2.545E-01 +- 9.353E-02
INDIVIDUAL SOLUTIONS
1* GY=1. 831E+02 +1*7.966E+02, DIE=20.147+-1.368 C0N=4. 406E+00+-5. 3E-01
2* GY=1. 809E+02 +I*7.862E+02, DIE=19.635+-1.431 C0N=4.296E+00+-5.9E-01
3* GY=1. 767E+02 +1*7.715E+02, DIE=18.939+-1.550 C0N=4.117E+00+-6.9E-01
287
4* GY=1. 49GE+02 +I*6.316E+02, DIE=12.849+-2.457 C0N=2. 855E+00+-1. 3E+00
GY BECAME 1.933E+02 +1* 6.655E+02 NEW 1.810E+02 +1* 7.870E+02
5* GY=2.687E+02 +I*8.337E+02, DIE=20.853+-2.430 C0N=6.767E+00+-l.lE+00
6* GY=2.304E+02 +1*8.370E+02, DIE=21.653+-1.025 C0N=5. 825E+00+-7. IE-01
7* GY=2.146E+02 +I*8.359E+02, DIE=21. 825+-1.273 C0N=5. 419E+00+-5.0E-01
8* GY=2.010E+02 +1*8.281E+02, DIE=21. 5B3+-1.253 CON=5.O27E+OO+-3.1E-01
9* GY=1. 923E+02 +1*8.146E+02, DIE=20.972+-1.331 C0N=4.733E+00+-4.4E-01
10* GY=1. 826E+02 +I*8.096E+02, DIE=20.827+-1.367 C0N=4. 465E+00+-5. 2E-01
DIE= 19.928 +- 2.516 C0N= 4.791E+00 +- 1.007E+00 l./(0HM*M)
PROPAGATION CONST: 1.980E+02 +1* 7.945E+02
MR= 2
MY= 4
FIT OF PROPAGATION CONSTANT X(1)+I*X(2)
AFTER
8 ITER, RMS RESIDUAL 2.3698E+00, FRS= 4.9962E-07
PARAM: 1.975E+02 8.044E+02
CONF.: 7.718E+00 8.881E+00
CORRECTED MEASUREMENTS
DFIRO
DRS
FIRL
DFIRL
RO DVSWR
FIRO
NO
RS
-0 .8 8
70.73
-0 .0 1 0.8282 -0 .2 6 -164.48
1 0.9699 0.0009
-1 .1 7
-0 .0 1 0.8566 -0 .1 1 -166.09
2 0.9699 0.0009 102.46
-1 .4 5
3 0.9706 0.0016 134.18
-0 .0 2 0.8860 -0 .0 4 -168.12
-2 .6 2
4 0.9765 0.0075 165.68
-0 .2 6 0.9473 0 .3 7 -174.42
-0 .7 9
5.80
5 0.9354 -0.0336
0.03 0.6332 -0 .3 7 161.54
0.37 0.4551 -1 .0 8 178.90 358.76
6 0.9552 -0.0138
10.11
0.31
7 0.9586 -0.0104
13.75
0.05 0.5152 -1 .2 4 -165.19
21.67
0.93
8 0.9609 -0.0081
0.03 0.6620 -0 .9 2 -160.13
0.14
9 0.9687 -0.0003
37.52
0.01 0.7669 -0 .5 1 -161.90
10 0.9697 0.0007
57.34
-0 .4 2
0 .0 0 0.8114 - 0 .6 6 -163.66
PROPAGATION CONST: 1 . 975E+02 +1* 8.044E+02
DIE= 20.369 +- 0.450 CON= 4.800E+00 +- 2.070E-01 l./(OHM*M)
CORRELATION 0.2791, TANDTA= 5.069E-01 +- 2.721E-02
MR= 4
MY= 4
*#####*###
FIT OF PROPAGATION CONSTANT X(1)+I*X(2)
REFERENCE SHIFTS X(3)/BEZ0 INPUT, X(4)/BEZ0 LOAD
AFTER
5 ITER, RMS RESIDUAL 2.6221E-01, FRS=
PARAM: 2.292E+02 8.405E+02-2.727E-02 1.113E-02
CONF.: 1 . 442E+00 2.240E+00 7.368E-04 1.422E-03
CORRECTED MEASUREMENTS
RO DVSWR
NO
RS
DRS
FIRL
DFIRL
1 0.9692 0.0002
70.74
0.00 0.8304 -0 .1 4
2 0.9691 0.0001 102.47
0 .0 0 0.8556 -0 .1 7
3 0.9690 0.0000 134.21
0.0 0 0.8824 -0 .3 2
4 0.9697 0.0007 165.95
0.0 0 0.9408 -0 .6 7
5 0.9686 -0.0004
5.82
0.04 0.6528 0.22
6 0.9686 -0.0004
9.71
-0 .0 3 0.5015 -0 .0 3
7 0.9672 -0.0018
13.69
-0 .0 2 0.5653 -0 .0 1
8 0.9685 -0.0005
21.64
0.00 0.6892 -0 .0 5
9 0.9689 -0.0001
37.51
0.00 0.7772 -0 .0 6
10 0.9690 0.0000
57.35
0.0 0 0.8157 -0 .4 4
2.4026E-07
FIRO
-163.81
-165.00
-166.56
-171.80
162.31
-179.75
-165.37
-161.00
-162.02
-163.22
DFIRO
-0 .2 1
-0 .0 8
0.11
0.0 0
-0 .0 2
0.10
0.13
0.06
0.03
0.02
288
PROPAGATION CONST: 2.292E+02 +1* 8.405E+02
SHIFTS: FRONT-O.250 +-0.007 BACK 0.102 +-0.013 nun
DIE= 21.861 +- 0.124 CON= 5.820E+00 +- 3.994E-02
CORRELATION 0.2353, TANDTA= 5.727E-01 +- 5.657E-03
l./(OHM*M)
MR= 6
MY= 4
##########
FIT OF PROPAGATION CONSTANT X(1)+I*X(2)
REFERENCE SHIFTS X(3)/BEZ0 INPUT, X(4)/BEZ0 LOAD
ADJUSTMENT OF ATTENUATION X(5) LOAD, X(6) INPUT
INIT VAL: 2.292E+02 8.405E+02-2.727E-02 1.113E-02 l.OOOE+OO l.OOOE+OO
AFTER
4 ITER, RMS RESIDUAL 2.0047E-01, FRS= 2.3404E-08
PARAM: 2 . 300E+02 8.360E+02-2.672E-02 8.704E-03 9.992E-01 9.939E-01
CONF.: 1 . 711E+00 2.095E+00 6.391E-04 1.310E-03 2.355E-03 1.745E-03
CORRELATIONS:
1
2
3
4
5
6
1# l.OOOE+OO - 1 . 609E-01 -6.336E-01 -3.320E-01 7.451E-01 -3.677E-01
2* - 1 . 609E-01
l.OOOE+OO -4.832E-01 8.587E-01 -4.341E-02 5.870E-01
3* -6.336E-01
- 4 . 832E-01 l.OOOE+OO -2.872E-01 -4.461E-01 -8.162E-02
4* - 3 . 320E-01
8 . 587E-01 -2.872E-01 l.OOOE+OO -1.732E-01 5.515E-01
5* 7.451E-01 -4.341E-02 -4.461E-01 -1.732E-01 l.OOOE+OO -2.269E-01
6* - 3 . 677E-01
5 . 870E-01 -8.162E-02 5.515E-01 -2.269E-01 l.OOOE+OO
CORRECTED MEASUREMENTS
DVSWR
FIRO
RS
FIRL
DFIRL
RO
NO
DRS
DFIRO
70.74
0.00 0.8328 0.0 0 -163.76
-0 .1 6
1 0.9691 0.0001
0 .00 0.8587 0.03 -164.96
-0 .0 4
2 0.9690 0.0000 102.47
0.00 0.8862 -0 .0 2 -166.54
0.13
3 0.9689 -0.0001 134.21
-0 .0 1 0.9461 0.1 6 -171.87
-0 .0 7
4 0.9691 0.0001 165.93
5.81
0.03 0.6519 0 .1 9 162.32
-0 .0 1
5 0.9690 0.0000
-0 .0 3 0.5038 0 .0 2 -179.77
0.09
9.72
6 0.9693 0.0003
7 0.9688 -0.0002
13.71
0.0 0 0.5661 0.01 -165.48
0.02
21.64
0.00 0.6896 -0 .0 4 -161.05
0.01
8 0.9689 -0.0001
0.05
37.51
0.00
0.7785 -0 .0 1 -161.99
9 0.9689 -0.0001
57.35
0.06
0 .00 0.8178 - 0 .3 4 -163.18
10 0.9690 0.0000
PROPAGATION CONST: 2 . 300E+02 +1* 8 . 360E+02
DIE= 21.603 +- 0.113 CON= 5.809E+00 +- 4.777E-02 l./(OHM*M)
CORRELATION 0.2386, TANDTA= 5.785E-01 +- 6.210E-03
INDIVIDUAL SOLUTIONS
1* GY=2.276E+02 +I*8.337E+02 DIE=21. 517+-1. 708 CON=
2* GY=2. 300E+02 +I*8.348E+02 DIE=21. 540+-1. 863 CON=
3* GY=2. 322E+02 +I*8.395E+02 DIE=21. 763+-2. 131 CON=
4* GY=2.320E+02 +I*8.293E+02 DIE=21.214+-3. 929 CON=
5* GY=2.256E+02 +I*8.432E+02 DIE=22.062+-2. 479 CON=
6* GY=2.296E+02 +I*8.344E+02 DIE=21. 522+-0. 933 CON=
7* GY=2.302E+02 +I*8.356E+02 DIE=21. 583+-1. 289 CON=
8* GY=2.299E+02 +I*8.369E+02 DIE=21. 657+-1. 373 CON=
9* GY=2.307E+02 +I*8.365E+02 DIE=21. 621+-1. 546 CON=
10* GY=2.257E+02 +I*8.432E+02 DIE=22.063+-1. 684 CON=
DIE= 21.654 +- 0.243 CON= 5 . 797E+00 +- 4. 406E-02
PROPAGATION CONST: 2.293E+02 +1* 8 . 367E+02
5 . 731E+00+- 6.0E-01
5 . 800E+00+- 7.1E-01
5 . 889E+00+- 8.8E-01
5 . 811E+00+- 1.9E+00
5 . 747E+00+- 9.9E-01
5 . 788E+00+- 7.1E-01
5 . 811E+00+- 5.7E-01
5 . 813E+00+- 3.3E-01
5 . 829E+00+- 4.5E-01
5 . 749E+00+- 5.8E-01
l./(0HM*M)
289
APPENDIX G. D e ta il s o f C a lc u la tio n fo r th e C e n t r a l ly Located E-plane
Slab in a R ectan gu lar Waveguide.
N orm alization
The n o rm a liz a tio n f a c t o r s WR fo r th e normal modes of th e s l a b f i l l e d waveguide [s e e E g .( 4 .4 3 ) ] acc o rd in g to E g .(4 .4 6 ) a re
sin h [Im (k )(a -t)]
W,-2 _
2Im(k)
sin [R e (k )(a -t)]
2Re(k)
*
sin [k (a -t)/2 ] 2
s i n h [ I m ( k ') t ]
\
s i n [ R e ( k ') t ]
(Gl)
2 Im ( k ')
c o s ( k 't / 2 )
2 R e ( k ')
*
/
where th e s u b s c r i p t n i s o m itted fo r W, k and k ' .
R av le ig h -R itz approxim ation
D e r iv ia tio n of th e R a y le ig h -R itz method fo r an inhomogenouslyf i l l e d waveguide i s given by C o llin [165]. The v a r i a t i o n a l i n t e g r a l fo r
2
th e sgu are of p ro p a g a tio n f a c t o r r i s
J g 2 (x)dx =
where
j
~ ) - e (x )k 02g2 (x) dx
1
f o r 0 ^ x ^ ( a - t ) / 2 , (a + t)/2 ^ x ^ a
e
fo r (a -t)/2 < x < (a + t)/2
(G2)
e (x ) =
The approxim ate e ig e n f u n c tio n s g^ a r e e x p ressed a s s e r i e s of LM sym­
m e tric e ig e n f u n c tio n s (normal modes) fo r th e empty waveguide, f ^ [see
E g .( 4 . 4 3 ) ] :
LM
V*>
= S, a sk l s (,!)
S=1
The follo w in g s e t of homogeneous, l i n e a r e g u a tio n s i s o b ta in e d
(G3)
where th e m a trix elem ents a re
(G6)
and can be e x p l i c i t l y w r i t t e n
t
sin [(2 s-l)n t/a ]
(G7)
'( - l ) s - r sin [ ( s - r )n t/a ]
( - l ) s + rs i n [ ( s + r - l ) n t / a ]
(G8)
+
(s-r)ti
(s+ r-l)n
The s e t of e q u a tio n s (G5) c o n s t i t u t e s an eig en v a lu e problem fo r
th e complex, symmetric m a trix T__.
51 I t can be e f f i c i e n t l y solv ed by a
num erical p rocedure. Sub rou tine CGEEV from th e SLATEC m athem atical sub­
r o u ti n e s l i b r a r y [182] has been employed in our program. I t fin d s th e
e ig e n v a lu e s v ia r e d u c tio n of th e m a trix to th e upper Hessenberg form
followed by th e QR a lg o rith m [161]. The
r e s u l t i n g e ig e n v a lu e s
-
_
2
a re
2
then ordered a c c o rd in g to in c r e a s in g Refy^ ).
Assvmptotic expansion fo r h ig h e r o rd e r modes
D e r iv ia tio n of th e a sym p to tic expansion fo r th e pro p ag atio n f a c ­
t o r s o f hig h er o rd er modes i s s i m i l a r to t h a t given by Lewin [164]. The
s e p a r a tio n c o n s ta n ts k, k ' [see E q .( 4 .4 3 )] a re ex pressed by two new
param eters: v and w
k' = n(v+w)/a,
k = r»(v-w)/a
and E q .(4 .4 4 ) i s r e w r i t t e n as
(G9)
291
W COS[tl(~ - tjv)] = V C O S [tl(~ + jjjw)]
(G10)
Upon in tr o d u c tio n of th e expansion param eter p
4 * V = kQ2 ( e - l ) a 2 = a 2 ( k ,2 - k 2 ) = 4ti2 vw
(G il)
th e c h a r a c t e r i s t i c Eq.(GlO) becomes
( p / v ) 2cos Jv[q - ( p / v ) 2 ]
= cos j v [ l - ( p / v ) 2q]
(G12)
where q = l - 2 t / a .
In th e l i m i t o f la r g e v (v>>p) Eq.(G12) approaches co s(n v/2 )= 0 and
th e s o lu tio n i s vae2n-l. This e x p re ss e s th e f a c t t h a t th e h ig h e r order
normal modes f o r th e s l a b - f i l l e d waveguide approach th e c o rresp on ding
2
modes fo r empty waveguide (p =0). An asy m pto tic expansion fo r vR i s
w r i t t e n in th e form
v
n
= 2 n -l + A d2 + B d4 + ............
n
n
(G13)
The c o e f f i c i e n t s An , Bn a r e o b ta in e d by s u b s t i t u t i n g Eq.(G13) in to
Eq.(G12) and r e t a i n i n g terms up to p
4
q
2 ( - l ) nc o s [ ( 2 n - l ) u q / 2 ]
An = ------ + ---------------------5----------n
2 n -l
n (2 n -ir
<G14>
( - l ) n [ ( 2 n - l ) - 1 -qA ] s in [ ( 2 n - l ) t i q / 2 ] - 2 (2 n -l)A 2 + A q
B =---------------------------- 2---------------------2--------2 .
n
( 2 n - l)
(G15)
F i n a lly the a sy m p totic approxim ation fo r k and k ' i s
k^ = 2 n -l + [An+ ( 2 n - l ) _1]p2 +
[Bn+An ( 2 n - l ) ' 2 ]p4
(G16)
kn = 2 n -l + [An- ( 2 n - l ) _1]p2 +
[Bn-An ( 2 n - l ) ' 2 ]p4
(G17)
The r e s u l t i n g s e r i e s of s e p a r a tio n f a c t o r s kR, k^ i s a ccepted a s i n i ­
t i a l approxim ation fo r th e Newton-Raphson i t e r a t i o n of E q .(4 .4 7 ) i f th e
292
corresponding p ro p a g a tio n f a c t o r s Yfl [se e E q .( 4 .4 4 )] s a t i s f y folowing
2
2
c o n d itio n s : Re(Y„)>0
and Re(Y„n - 7)<Re(Y„
n
i
n ). O therw ise, th e number of
inodes approximated by th e R ay le ig h -R itz method i s in c re a s e d to in c lu d e
th e s e modes fo r which th e a sy m ptotic expansion was u n r e l i a b l e .
Newton-Raphson i t e r a t i o n s
For th e system o f two n o n lin e a r e q u a tio n s E q .( 4 .4 7 ) , th e i t e r a t i o n
s te p i s c a lc u la te d a s [157]:
(G18)
where th e r i g h t hand s id e i s c a l c u l a t e d fo r k=kx , k ^ k ' 1 , and J i s th e
Jacobian fo r th e system of two e q u a tio n s
ah^
ah2
3h^
3h2
3k
ak'
ak' sk
(G19)
In ord er to improve convergence o f th e a lg o rith m to th e ro o t which
i s th e n e a r e s t one to th e i n i t i a l a p p rox im ation , a re d u c tio n of s te p
s i z e i s implemented. When th e v a lu e o f Ih jj + lJ^I i s found to in c re a s e
as a r e s u l t of c o r r e c t i o n given by Eq.(G18), th e s te p s i z e i s h alv ed.
Ib t h i s manner chances of s w itch in g to a n o th er ro o t a r e reduced. The
Newton-Raphson a lg o rith m fo r th e system of two e q u a tio n s , given by
E q .( 4 .4 7 ), proved to be more r e l i a b l e fo r c a l c u l a t i o n of th e normal
modes than th e Newton a lg o rith m a p p lie d to a s i n g l e n o n lin e a r eq u atio n
fo r y which can be o b ta in e d by e lim in a tio n of k and k* from E q s .( 4 .4 4 ) ( 4 .4 5 ) .
293
C r o s s - i n t e g r a l o f normal modes
The i n t e g r a l of th e product o f modal f u n c tio n s fo r th e empty and
th e s l a b - f i l l e d waveguides, d e fin e d by E g .( 4 .5 9 ) , i s
a
Pm„
j f
-VT/ V*> •* * * * &
1 f s ln [ P P
- kn ) ¥ ]
dx
.
s ln [ P P
* kn ) ¥ ]
M- '
' ‘ " (“n H *)
cos(‘ k . )
. t k. )
(2m -l)ti
- k'
n
C05( ( 2 - 1
(2m -l)n
. t k. )
(G20)
k'
+ Kn
294
VITA
Name:
J o z e f R. Oygas
Born:
August 10, 1954
Wielun, Poland
Education:
P o li te c h n ik a Warszawska
Warsaw, Poland
1973 - 1978
M. S. in T echn ical Physics
N orthw estern U n iv e rs ity
Evanston, I l l i n o i s
P o s itio n :
A ssista n t
P o li te c h n ik a Warszawska
Awards:
1980 - 1985
1978 - 1980
C ab ell F ellow ship
N orthw estern U n iv e rs ity 1984
P u b li c a tio n s :
"Frequency-Dependent C o n d u c tiv ity of NASICON Ceramics
in th e Microwave Region" (w ith M. E. Brodwin)
"Neutron Powder D i f f r a c t i o n Study and Io n ic C o n d u ctiv ity
o f Na2Zr2SiP20 12 and Na3Zr2S i 2P012" (w ith W. H. Baur,
D. H. Whitmore, and J . Faber)
Both p re s e n te d a t 5th I n t e r n a t i o n a l Conference on S o lid
S t a t e I o n ic s , Lake Tahoe, 1985, acc e p ted fo r p u b lic a tio n
in S o lid S ta te I o n ic s .
35556017043480
3 5556 017 043 480
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