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Some experiments with phonons at microwave frequencies

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Some Experiments With Phonons At
Microwave Frequencies
A th e s is subm itted to th e U n iv e rs ity o f London
fo r the degree o f Ph.D. by:
Ernest P a tte rs o n , M.Sc.
CHELSEA COLLEGE OF SCIENCE AND TECHNOLOGY
MANRESA RD.
LONDON
S.W .3.
ProQuest Number: 10778202
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A b s tra ct
The tem perature dependence o f th e a tte n u a tio n o f two slow tra n s ­
verse, two fa s t tra n s ve rs e and th re e lo n g itu d in a l u ltra s o n ic modes In
'p e r fe c t* natu ral qu artz specimens have been measured a t a frequency
o f 9 GHz and In th e tem perature range 4 to 35® K.
The a tte n u a tio n o f
th e slow tra n s ve rs e waves agrees w ith th e p re d ic tio n s of the theory o f
Landau and Rumer.
account th e f i n i t e
The th e o rie s o f Marls and S h lren , which take In to
life tim e o f thermal phonons w ith which th e micro­
wave phonons In te r a c t, p re d ic t c o r re c tly the a tte n u a tio n o f lo n g itu d i­
nal and fa s t tran sverse waves, but do not account com pletely fo r the
magnitude o f the a tte n u a tio n .
The steps observed In th e tem perature
dependent a tte n u a tio n o f Im perfect q u artz are Id e n tifie d as two peaks
t
^
a t 16 - 3® K and 24 - 3® K, superimposed on a phonon-phonon back­
ground,
P ossible mechanisms fo r these a tte n u a tio n peaks are d is ­
cussed.
S im ila r measurements have been made o f th e tem perature dependence
o f 9 GHz phonons In various modes In magnesium alum lnate s p in e l.
The
a tte n u a tio n of c e rta in modes Is lower than In any o th e r m a te ria l meas­
ured so f a r and perm its th e observation o f 9 GHz echo p a tte rn s from
liq u id helium tem peratures to room tem perature.
tio n s o f th is are discussed.
P ossible a p p lic a ­
The a tte n u a tio n a t room tem perature Is
In te rp re te d In terms of an e f f e c t iv e v is c o s ity damping mechanism and
approximate values o f th e components o f th e v is c o s ity m a trix are given.
The th ir d m a te ria l
In v e s tig a te d In d e ta il fo r I t s a tte n u a tio n of
9 GHz phonons was pure s in g le -c r y s ta l y ttriu m Iron garnet (Y IG ).
Previous measurements on th is m a te ria l have been com plicated by spur­
ious e ffe c ts which have been avoided here, and the measurements made
are compared w ith c u rre n t th e o rie s of a tte n u a tio n by phonon-phonon
in te ra c tio n s using the re c e n tly measured t h ir d o rd e r e la s t ic constants
o f YIG.
The agreement found Is about the same as In many o th e r ma­
t e r i a l s which suggests th a t the I n t r in s ic a tte n u a tio n In YIG Is under­
stood and Is caused by In te ra c tio n w ith thermal phonons.
To e s ta b lis h the frequency dependence, measurements have been
made on the s in g le c ry s ta l YIG specimens a t I GHz.
These revealed a
sharp peak In the a tte n u a tio n o f lo n g itu d in a l waves measured as a
fu n c tio n o f tem peratu re.
Possible mechanisms fo r th is are discussed.
Some microwave u ltra s o n ic studies were a lso made on g a lliu m
arsenide a t low tem peratures.
In p a r tic u la r the In te n tio n was to
a m p lify 9 GHz waves by a p p lic a tio n o f a d r i f t f i e l d , to an Illu m in a te d
sample. In ord er to a c c e le ra te the m obile c a r r ie r s to a v e lo c ity
g re a te r than th a t o f th e u ltra s o n ic wave.
In these circumstances
th e e le c tro -a c o u s tic In te ra c tio n In p ie z o e le c tr ic semiconductors gives
r is e to possible g a in s .
Such a m p lific a tio n was not achieved and
possible reasons fo r th is are discussed.
AcknowIedqements
The author wishes to express his s in cere thanks to Dr, E.L.James
fo r h is encouragement and guidance, to Dr. M .F.Lew is, w ith whom he
worked a t the H ir s t Research Centre (General E le c tr ic C o ., L td .,
Wembley), f o r many h e lp fu l discussions and c o n trib u tio n s and to both
the General E le c t r ic Co. and Chelsea C ollege fo r the p ro v isio n o f
lab o rato ry f a c i l i t i e s .
He Is Indebted to the te c h n ic a l s t a f f s o f the
H ir s t Research Centre and Chelsea C ollege fo r help In th e constructio n
o f apparatus.
F in a lly he would lik e to thank Chelsea C ollege fo r the
p ro v isio n of a g ra n t.
CONTENTS
Chapter
Page
I
In tro d u c tio n
II
1
Experimental d e ta ils and techniques
'
I.
Microwave transducers, t h e i r p ro p e rtie s and
p rep aratio n
(I)
(II)
(III)
2.
General d e s c rip tio n
The c a v itie s
P rep a ra tio n of samples
The d e te c tin g system
Method of measurement
Measurements using q u a rtz
Measurements using CdS film s
Measurements using NI film s
15
16
17
17
17
18
19
19
Experimental observations on q u a rtz , spinel and YIG
1.
In tro d u c tio n
2.
Observations
(I)
(II)
(III)
(Iv )
(v )
IV
12
13
14
The apparatus and measurement techniques
(I)
(II)
(III)
(Iv )
(v )
(v l)
(v ll)
(v lll)
III
Quartz
CdS film s
NI film s
22
N atural q u a rtz
Im perfect q u a rtz
Spinel
YIG
O ther measurements on YIG
23
25
26
27
29
T h e o re tic a l account o f a tte n u a tio n mechanisms
re le v a n t fo r q u a rtz , spinel and YIG
1.
In tro d u c tio n
2.
In te ra c tio n s w ith In d iv id u a l modes
(I)
(II)
3.
4.
Landau Rumer theory
Longitudinal waves
In te ra c tio n s w ith e n tir e
(1 )
31
35
39
la ttic e
Akhleser theory and Improvements
Comparison of th e o rie s f o r q u a rtz
43
48
Chapter
V
Page
Comparison of observations w ith th eo ry and
discussion
1.
Quartz
(I)
(II)
(III)
(Iv )
2.
Im perfect q u a rtz
3.
Spinel
(I)
(II)
4.
5.
V II
50
53
54
55
56
Temperature dependence
Dependence of the roomtem perature
a tte n u a tio n on the u ltra s o n ic mode
59
^2
Y ttriu m Iron garnet
(I)
(II)
VI
Temperature dependence
Frequency dependence
Magnitude o f th e a tte n u a tio n
R elaxatio n tim e
Temperature dependence
Frequency dependence
The I GHz a tte n u a tio n peak
In YIG
66
68
69
Amp 11fI c a tIo n of 9 GHz waves In GaAs
1.
In tro d u c tio n and theory
2.
Experimental d e ta ils and re s u lts
3.
Conclusions
73
Conclusions
L is t of references
83
Index to symbols
90
L is t of ta b le s
93
Tables
94
L is t of fig u re s
98
Figures
*02
CHAPTER I
INTRODUCTION
In the past ten years a g re a t deal o f In te r e s t has focused on the
propagation o f high frequency u ltra s o n ic waves In s o lid s .
This In t e r ­
e s t a ris e s from the knowledge provided about the m a te ria ls through study
o f th e various mechanisms by which u ltra s o n ic waves In te r a c t In s o lid s .
In p a r tic u la r understanding of the a tte n u a tio n processes Involved In the
In te ra c tio n o f u ltra s o n ic phonons and thermal
l a t t i c e phonons could lead
to new m a te ria ls w ith lower phonon a tte n u a tio n s .
These could fin d many
useful a p p lic a tio n s , fo r example as microwave delay lin e s s in c e , u n til
re c e n tly , microwave delays were achieved by passing th e sig nal through
a length (300 m fo r one ys a p p ro xim ately)o f waveguide.
The t r a n s it
tim e o f sound waves ranges from I to 5 ys/cm In various m a te ria ls , so
t h a t an acoustic delay lin e Is fiv e orders o f magnitude s h o rte r than
the e q u iv a le n t length o f waveguide.
Many o f the e ffe c ts associated w ith high frequency waves In s o lid s
are very d i f f i c u l t to In v e s tig a te because of the poor transm ission of
GHz (10® Hz) microwaves In most m a te ria ls .
This has led to the wide­
spread In v e s tig a tio n of means o f a m p lifyin g u ltra s o n ic waves In s o lid s .
One method o f achieving t h i s , which has been In v e s tig a te d here, o r ig in ­
ated from study o f th e a tte n u a tio n of u ltra s o n ic waves In p ie z o e le c tr ic
semiconductors.
The a tte n u a tio n mechanism due to mobile charge c a r r ie r s
Is reversed by means o f the e le c tro a c o u s tic In te r a c tio n and th is gives
ris e to a m p lific a tio n .
The generation of GHz waves by surface e x c ita tio n o f q u a rtz was
discovered about 1957.* ^
The In a c c e s s ib ility o f these very high f r e ­
quencies u n til th a t tim e had been due to d i f f i c u l t i e s
In fa b ric a tin g
4 5
resonant p la te s a t microwave fre q u e n cie s , ' since to achieve frequen­
c ie s o f th e o rd e r o f 100 MHz the thicknesses Involved are about 0.01
mm, which was the lim it of the technology a t th a t tim e .
The new te c h ­
niques enabled GHz frequency generation to be made In q u a rtz In the
form o f rods which fu n c tio n sim ultaneously as sample and tra n s d u ce r.^
Thus th e tem perature-dependent a tte n u a tio n o f GHz ultrasound In q u a rtz
could be measured.
Measurements on o th e r m a te ria ls . I f these were not
p ie z o e le c tr ic o r had a low coupling co n stan t, were made by bonding to
q u a rtz rods.
A lt e r n a t iv e ly , techniques developed s h o rtly a f t e r 1957
using resonant transducers In the form of th in film s (thickness about
a h a lf wavelength o f the ultrasound) were used.
r e s t r ic t io n to q u a rtz as w ell as bonding problems.
These e lim in a te the
Ferromagnetic
film s may be used such as n ic k e l, where th e conversion takes place
because of th e m a g n e to s tric tiv e e f f e c t , ^
dm!
and th in film s of cadmium
su lphide which are p ie z o e le c tr ic have been found very e f f i c i e n t .
11,12
The e a r ly experiments using surface e x c ita tio n and resonant film s
were c a rrie d o u t In the I extended as high as 114 GHz.
10 GHz region but these have since been
Some p rin c ip a l references describing th e
In v e s tig a tio n o f microwave phenomena, p a r t ic u la r ly a tte n u a tio n . In v a r­
ious d ie le c t r ic and semiconducting m a te ria ls are lis te d h e re ,*^
w h ile others may be found In these papers.
A ll the transducer te c h n i­
ques mentioned above have been employed In the work to be described.
using mostly q u a rtz , magnesium alum lnate (s p in e l) and y ttriu m Iron gar­
n et (YIG ) as specimens and a frequency of 9155 MHz.
G allium arsenide
was used to study th e e le c tro a c o u s tic gain In te ra c tio n a t th is frequency.
The q u a rtz re s u lts were published In reference ( 3 5 ), those o f spinel
(3 6 ) and the YIG In (3 8 ) and (3 9 ).
In
These papers are Included a t the
end o f th is th e s is .
U ltra s o n ic waves e x is t n a tu r a lly In s o lid s In th e form o f thermal
la t t i c e v ib ra tio n s of the s o lid .
wave
In te ra c ts
An a r t i f i c i a l l y generated u ltra s o n ic
w ith the thermal
l a t t i c e waves because o f th e enhar­
monic nature o f th e In te ra c tio n forces In a s o lid and th e main p a r t o f
t h is In te ra c tio n Is w ith l a t t i c e waves o f thermal frequency about kT/h
where k Is Boltzmann's constant, h Is P lan c k 's constant and T the tem­
p e ra tu re In degrees a b so lu te.
A t room tem perature th is
Is a frequency
o f 10,000 GHz and a t liq u id helium tem peratures ICO GHz, so, w ith the
generation of waves In th e range 10 to 100 GHz I t Is now possible to
study th e In te ra c tio n s o f thermal
d if f e r e n t a frequency.
l a t t i c e waves w ith others o f not too
The la t t i c e waves are s u b je c t to In te ra c tio n s
between themselves and have a mean fre e path A.
I f the wavelength X
of the ultrasound Is less than the mean fre e path o f the thermal modes
(wT > 1 ;
0)
Is th e u ltra s o n ic frequency and T Is the thermal phonon re ­
la x a tio n tim e) one can assume the wave to In te r a c t w ith In d iv id u a l
thermal modes.
Such th e o rie s have been worked out by Landau and
R u m e r ,S h lr e n ^ * and o t h e r s . T h i s
Is the case fo r q u a rtz be­
low about 40®K (th e region In v e s tig a te d ) fo r a frequency o f 9 GHz.
10
When X >> & (o)T «
I ) one can t r e a t the In te ra c tio n as ta k in g place
not w ith In d iv id u a l thermal modes but w ith the e n t ir e assembly o f the
l a t t i c e waves.
This mechanism was f i r s t described by Akheiser^^ and
extended by o th e r w o r k e r s . T h e
n e arly a l l taken In th is region I . e .
measurements on spinel were
In te rm ed ia te to high tem peratures,
th e a tte n u a tio n being to o low fo r measurements when wr > I .
The meas­
urements on YIG cover both regio ns.
A tten u atio n measurements a t 9 GHz have been made o f two slow tra n s ­
v ers e , two fa s t transverse and th re e lo n g itu d in a l modes In p e rfe c t
n a tu ra l q u a rtz .
th e o r ie s .
These w i ll be compared w ith phonon-phonon In te ra c tio n
Measurements were made as w ell on a number o f Im perfect
quartzes o f various o r ig in s .
The presence o f Im perfections was found
to cause 's te p s ' In the a tte n u a tio n as a fu n ctio n of tem perature.
P ossible mechanisms fo r t h is w i ll be discussed.
S im ila r measurements were made on s t r i c t l y s to ic h io m e tric c ry s ta ls
o f spinel o rie n ta te d In the < I0 0 > , < IIO > and < l l l > d ire c tio n s .
The
a tte n u a tio n o f c e rta in modes was found to be lower than any o th e r m ater­
ia l measured, and perm its the observation o f 9 GHz echo p a tte rn s a t
room tem peratu re.
The a tte n u a tio n a t room tem perature w i l l be In te r ­
preted In terms o f an e f f e c t iv e v is c o s ity damping mechanism, since
enough data Is not a v a ila b le on th is m a te ria l fo r o th e r than an order
o f magnitude c a lc u la tio n o f I t s tem perature-dependent a tte n u a tio n of
microwaves.
Some more microwave u ltra s o n ic a tte n u a tio n measurements a t 9 GHz
have been made on pure s in g le c ry s ta l YIG.
These, as they are not com-
Il
p lic a te d by th e spurious e ffe c ts o f previous w o r k e r s a r e
to determ ine the I n t r in s ic a tte n u a tio n In t h is m a te r ia l.
b e lie v ed
The re s u lts
are compared w ith c u rre n t th e o rie s o f a tte n u a tio n by phonon-phonon
In te ra c tio n s using the re c e n tly measured th ir d o rd e r e la s t ic constants
o f YIG^® and thermal c o n d u c tiv ity measurements.^^
To check the v a r ia ­
tio n of a tte n u a tio n w ith frequency In th is m a te ria l measurements were
also made a t I GHz and a sharp peak was found In the a tte n u a tio n of
lo n g itu d in a l waves as a fu n c tio n o f tem perature.
This w i ll be d is ­
cussed and attem pts made to e x p la in I t In terms o f various mechanisms.
In a d d itio n th e e le c tro a c o u s tic In te ra c tio n a t 9 GHz has been In v e s tI gated In GaAs fo llo w in g the work o f Hutson, McFee and W hite.
50
They propose th a t the e le c t r i c f i e l d accompanying p a r tic u la r waves
couples to the mobile charge c a r r ie r s In GaAs because of the piezo­
e le c t r i c e f f e c t and w ith th e c a r r ie r s a c c e le ra te d to a v e lo c ity
g re a te r than th a t o f the u ltra s o n ic wave energy Is tra n s fe rre d to the
wave.
The experim ental s e t up and procedure fo r GHz microwave meas­
urements Is described In Chapter I I w ith the re s u lts obtained fo r tem­
perature-dependent a tte n u a tio n reported In Chapter I I I .
Chapters IV
and V discuss re le v a n t th e o rie s o f a tte n u a tio n o f microwaves In s o lid s
and comparison w ith the re s u lts obtained.
Chapter VI describes th e
experiments using GaAs to a m p lify 9 GHz waves and some discussion o f
the observations and th e o rie s Involved.
th e s is w ith f in a l discussion and comments.
Chapter V II concludes the
12
CHAPTER 11
EXPERIMENTAL DETAILS AND TECHNIQUES
I.
Microwave transducers, t h e i r p ro p e rtie s and p re p ara tio n
(I)
Quartz
The experim ental arrangements fo r using q u a rtz as a transducer are
Illu s t r a t e d In fig s ( I ) a ,b ,c ,d .* ^
The end face o f a q u a rtz rod Is
placed In the pulsed o s c illa t in g e le c t r i c f i e l d o f a microwave c a v ity .
The manner In which a sound wave Is In i t i a t e d under a p p lic a tio n o f an
e le c t r i c f i e l d has been tre a te d by Jacobsen.*^
He demonstrates t h a t
the source occurs a t the surface o f the c ry s ta l and Bommel and Dransfe ld
2
showed th e d ire c tio n o f propagation to be dependent on the angle
between the surface normal and the c ry s ta l Iographlc a x is .
Thus a
pulsed o s c illa t in g e le c t r ic f i e l d normal to a quai^ rod which has end
faces f l a t and p erp en d icu lar to a s u ita b le a x is
51
w ill
I n i t i a t e a sound
wave o f th e same frequency and d uration which w i ll propagate along the
rod.
The energy may be reconverted, a t th e I n i t i a t i n g o r the f a r end,
by the Inverse process and detected In a superheterodyne re c e iv e r.
The echo p a tte rn s observed In the case o f q u a rtz fo r a frequency o f
9 GHz are Illu s t r a t e d
In f i g .
(2 ) a . and b.
The e v id e n t modulation o r beating e x h ib ite d In f i g .
(2 ) has been
found to be common to a l l such pulse echo experiments a t 9 GHz and the
2 52 53
reasons fo r th is have been In v e s tig a te d by a number of authors. '
'
The p rin c ip a l ones a re :
(I)
non p a r a lle lis m o f th e end fa c e s , and (2 )
o f f pure mode a x is propagation due to m is o rle n ta tlo n .
The former
13
causes acoustic phase averaging I . e .
I f wave fro n ts In te r s e c t the end
face th e p ie z o e le c tr ic p o la r iz a tio n may change sign across th e face
causing p a r t ia l c a n c e lla tio n o f th e s ig n a l.
The r e s u lt of th e second
e f f e c t Is th a t p a rt o f th e wave fr o n t Is s c a tte re d from the w a lls In
t r a n s i t because on ly along pure mode axes do the energy propagation
v e c to r and wave normal c o in c id e .
phase producing no net s ig n a l.
The s c a tte re d p o rtio n has random
Such o f f a x is propagation w i l l aggra­
vate th e e f f e c t o f b e ats, so the c o rre c t face o r ie n ta tio n fo r a given
v ib ra tio n d ire c tio n Is e s s e n tia l
I f th is Is to be avoided.
These and o th e r considerations
lead to the conclusions th a t the
end faces must be p a r a lle l to w ith in a few seconds o f arc and the c ry s ­
t a l o rie n ta te d as w ell as Is p r a c t ic a lly p o s s ib le .
I t Is a lso Impor­
ta n t fo r th e end faces to be f l a t to achieve maximum s ig n a ls .
A sur­
face th a t Is f l a t to w ith in h a lf a sonic wavelength Is d e s ira b le I . e .
a t 9 GHz, o p t ic a lly f l a t .
Of course th e above requirem ents e q u a lly
w ell apply to any o th e r samples In which th e waves are generated,
whether th e samples are bonded to q u a rtz o r resonant film s are used.
(II)
CdS film s
I t has been demonstrated th a t p ie z o e le c tr ic film s may be produced
by vacuum evaporation techniques.
II
'
12 25
'
The experim ental arrange­
ment Is th e same as th a t fo r q u a rtz the f ilm being placed In th e E f i e l d
of a microwave c a v ity .
CdS film s have been used In th is way In the
work to be described and they were produced by two methods.
The f i r s t
was fila m e n t heating of a c ru c ib le containing CdS c ry s ta ls and the
14
second was a dual evaporation process whereby cadmium and sulphur were
heated In separate c ru c ib le s .
The s u b s tra te tem perature was 200®C
o
and f ilm thicknesses o f 2000-5000 A were evaporated.
Moderate suc­
cess was obtained w ith both techniques although these film s were never
very e f f i c i e n t transducers (see f i g .
(3 )a ).
They were used In th e
measurement o f a tte n u a tio n o f lo n g itu d in a l modes In the various speci­
mens except qu artz (In some cases shear waves were a ls o obtain ed ) but
In the case o f such waves In YIG I t was found more s a tis fa c to r y to
measure the a tte n u a tio n a t the lowest tem peratures by bonding the sample
to x -c u t qu artz transducers (see sectio n 2 ( v l ) ) .
The two sets of
measurements (using q u a rtz and CdS film s ) Joined smoothly to g e th e r fig .
(1 9 ).
(Ill)
NI film s
When a NI film
In a steady magnetic f i e l d normal to the f ilm
Is
s u b jec t to an a lte r n a tin g magnetic f i e l d perp en d icu lar to the d .c .
f i e l d the m agnetization o f th e f i e l d processes about the steady f i e l d
and by m a g n e to s tric tiv e coupling th is produces a ro ta tin g shear s tr a in
over the surface o f th e f ilm .
modes are observed (see f i g .
prepared as per section 2 ( 1 ) .
Thus echoes corresponding to transverse
(3 )b ) In any attached specimen s u ita b ly
With th e d .c . magnetic f i e l d p a r a lle l
to the film th e lo n g itu d in a l o r one o r both shear modes are generated
according to the o rie n ta tio n of th e c ry s ta l mode a xis w ith respect to
the microwave f i e l d .
The experim ental method used Is again very s im i­
la r to th a t fo r q u a rtz except t h a t In t h is case th e NI f ilm was s itu a te d
15
In a region o f high magnetic f i e l d
( l)c .
In the re e n tra n t c a v ity ^
- fig .
Th is method was used to generate tra n s ve rs e waves In spinel
o
and YIG.
The film s were 2000-5000 A
th e o p t ic a lly f l a t
th ic k and
vacuumdeposited on
sample surfaces w ith these a t 300®C.
Twomethods
were used to evaporate the layers (1 ) fila m e n t heatin g of a c ru c ib le
co n tain in g th e N I,
2.
and ( I I ) e le c tro n bombardment e vap o ratio n .
The apparatus and measurement techniques
(I)
General d e s c rip tio n
The 9 GHz apparatus Is shown In f ig s .
(4 ) and ( 5 ) .
A 2J5IA
pulsed magnetron e x c ite s a tunable re e n tra n t microwave c a v ity co n tain ­
ing th e transducer and sample.
Echoes are observed by a m p lifyin g the
re tu rn in g sig nal using a superheterodyne system.
The waveguide from
the top p la te to the c a v ity was made of low c o n d u c tiv ity , cryogenic
m a te ria l as were the tubes used to c a rry the e le c t r ic a l
and to evacuate th e system.
leads, e tc .
The c a v ity and sample were enclosed In a
vacuum -tight can In to which helium gas was allowed to e n te r when the
can had been evacuated.
The tem perature o f th e sample was reduced to
th a t o f liq u id helium by precooling the can, contents and surrounds to
liq u id n itrogen tem perature, and then siphoning helium liq u id In to the
Inner dewar
w ith nitro g en In the o u te r one.
The tem perature could
fu r th e r be reduced by pumping on th e liq u id helium .
By th is means I t
was possible to reduce the tem perature to 2°K th is being measured by
observing the vapour pressure w ith a mercury manometer.
16
The 1 GHz system Is much the same except th a t the microwave power
Is c a rrie d by c o axial cables and coupled In to the c a v ity by coupling
loops - f i g .
(II)
(I)c .
The c a v itie s
F ig s . ( I ) a ,b ,c ,d I l l u s t r a t e the microwave c a v itie s used.
These
were a l l tu n ab le by means o f a worm and wheel o r s im ila r mechanism
which moves th e back p la te of the c a v ity .
They were made o f brass and
the surfaces coated w ith s i l v e r to Increase c o n d u c tiv ity , th is In tu rn
being covered by a gold la y e r f o r p ro te c tio n a g ain s t p e e lin g .
In th e
case o f 9 GHz c a v itie s coupling to the microwave energy was achieved
by a coupling hole adjusted so th a t the coupling was a maximum a t liq u id
helium tem peratures.
to be I(XX).
The Q o f one o f the 9 GHz c a v itie s was measured
This agrees w ell w ith the c a lc u la te d v a lu e .
on th e conversion o f electrom agnetic to u ltra s o n ic energy
C a lc u la tio n s
2
In d ic a te
th a t the e ffic ie n c y I . e . r a t io o f e le c t r ic a l output power to In p u t, of
a 9 GHz re e n tra n t c a v ity system w ith a I cm by 3 mm diam eter q u a rtz rod
as transducer Is o f th e o rd er o f 10
This was found to be so In prac­
t i c e , a f t e r a llo w in g fo r sound absorption In the specimen.
The dimensions o f th e systems are In d ica te d In th e fig u re s .
(I)d
F ig .
Is th e experim ental arrangement, a transm ission one, used In th e
attem pts to a m p lify waves In GaAs.
17
(III)
P rep aratio n of samples
The samples were In the form o f rods t y p ic a lly
I cm long and 3 mm
In diam eter w ith end faces polished o p t ic a lly f l a t and p a r a lle l to a
few seconds o f a rc .
They were o rie n ta te d by means o f th e Laue back
r e fle c tio n technique to an accuracy o f 1 /2 °.
The sources of In d iv id ­
ual samples w i l l be given In the re s u lts section w ith t h e i r o th e r p a r t i ­
c u la rs .
(Iv )
The d e te c tin g system
The d e ta lIs o f the superheterodyne d e te c tio n system used are shown
In f i g .
(6 ).
This Is a frequency changing device used In o rd e r to pro­
vid e s u f f ic ie n t a m p lific a tio n o f th e re tu rn in g s ig n a l.
of frequency
w
/ 2i t ,
To th e v o lta g e ,
being detected Is added an a lte r n a tin g v o ltag e o f
another frequency i û ^ / I ï ï generated lo c a lly and the two pass In to a b a l­
anced m ixer.
The output then contains v o ltag e components which flu c ­
tu a te a t (W |-w )/2 n and (W|+w)/2w as w ell as the modulation frequency.
The d iffe re n c e frequency (In te rm e d ia te frequency) Is now s ele cted and
a m p lifie d w ith It s o r ig in a l modulation to a level where I t can be d e te c t­
ed by the main a m p lifie r and then fed In to an o s c illo s c o p e .
(v )
Method of measurement
A p re c is io n ro ta ry vane a tte n u a to r was used to compare the In te n s i­
t i e s o f several p a irs of echoes a t various tem peratures e .g . the power
dBs necessary to reduce th e h eig h t of th e f i r s t echo to th a t o f the te n th
18
was measured as a fu n c tio n o f tem perature.
This procedure e lim in a te s
any n o n -lin e a r itie s In the d e te c tin g system.
A piece of lead attached
to th e c a v ity Increased It s thermal cap a c ity and prolonged th is process
to several hours p e rm ittin g many readings to be taken .
A c a lib ra te d
carbon re s is ta n c e thermometer was used to measure the tem perature to
-
1 /2 ° K.
This was s u ita b le up to 50*K above which a thermocouple was
used.
The experim ental method and d e ta ils fo r th e In v e s tig a tio n o f the
e le c tro a c o u s tic gain In te ra c tio n w i ll be described In Chapter V I.
(v l)
Measurements using q u a rtz
These were q u ite s tr a ig h t forward and as described above, th e echo
p a tte rn disappearing com pletely a t about 35°K.
The re s u lts fo r quartz
were normalized to zero a t 4 .2 °K , but th is was not always th e case w ith
o th e r samples.
This w i ll be j u s t i f i e d
In th e re s u lts s ec tio n s .
Nor­
m a lizin g to any p a r tic u la r tem perature su b tra cts from th e to t a l a tte n ­
u atio n any tem perature-dependent a tte n u a tio n e .g . th a t due to s c a tte r ­
ing processes.
This Is necessary, as w ell as advantageous, since the
r a t io o f in te n s itie s of a p a ir o f echoes w i l l obviously In general be
d if f e r e n t from any o th e r p a ir a t a p a r tic u la r tem perature.
Although the
envelope curve was never exponential a t liq u id helium tem peratures I t
was not In c o n s is te n tly a ffe c te d by changes of tem perature I . e . the read­
ings fo r d i f f e r e n t p a irs of echoes were u s u a lly found to be In good
agreement.
The accuracy was o f the o rd e r o f 0.1 dB/cm but depended
considerably on the In te n s ity and separation o f the echoes.
V/hen z -
19
cut q u a rtz samples were used they were bonded to x -c u t transducers w ith
a r a ld lt e as were th e YIG specimens on occasions.
The tem perature
dependence o f any a tte n u a tio n In th e bonds and o f the a tte n u a tio n In
th e sample due to transm ission In to th e bonds was found to be n e g li­
g ib le .
The form er Is reasonable since the bonds were only about
o
5000 A th ic k ; the l a t t e r In d ic a te s th a t the acoustic Impedance of
c ry s ta ls and bonding m a te ria l are not tem perature dependent.
conclusions were te s te d by
a r a ld lt e
These
bonding samples o f <I00> c u t
s ilic o n and < l l l > c u t semi In s u la tin g GaAs to x -c u t q u a rtz transducers.
No change In th e a tte n u a tio n o f e it h e r specimen could be detected In
warming up from 4 .2 °K to about 25°K because both m a te ria ls have a
n e g lig ib le a tte n u a tio n up to th is te m p e ra tu re ;^ *'^ *
th is v e r if ie s
t h a t th e re Is no change In the a tte n u a tio n due to th e bond.
In a d d i­
t io n , the re s u lts were reproducible from bond to bond.
(v ll)
Measurements using CdS f ilm transducers
This technique was as s tr a ig h t forward as In th e case o f quartz
although some phasing e ffe c ts were observed below 50°K where the a tte n ­
uation Is small In spinel and YIG,
Therefore several runs were u s u ally
taken fo r each specimen and the fin d in g s averaged.
D e ta ils o f accuracy
w i ll be given w ith th e re s u lts fo r p a r tic u la r specimens.
( v l l l ) Measurements using NI film transducers
These measurements presented d i f f i c u l t i e s because, as w ell as the
uniform precession mode, th in ferrom agnetic film s d is p la y a spectrum of
20
spin modes which a lso couple m a g n e to s tric tiv e I y to the l a t t i c e .
8 o
Overlap and In te rfe re n c e between these modes then leads to an echo
p a tte rn which changes d r a s t ic a lly w ith magnetic f i e l d .
Now the de­
m agnetising and s tra in -in d u c e d anisotropy f ie ld s In the NI film s change
w ith tem perature due to th e v a r ia tio n o f s a tu ra tio n m agnetization and
d if f e r e n t ia l c o n tra c tio n between the film and s u b s tra te .
These e ffe c ts
are f a i r l y small from 4 .2 °K to about 60°K and Pomerantz has used mag­
n e tic film s to measure th e a tte n u a tio n o f microwave phonons In many
m a te ria ls a t tem peratures up to and o c ca s io n a lly above liq u id n itrogen
tem peratures.
Hov/ever, between about 6 0 °K and room tem perature thin g s
change c o n sid e ra b ly .
In the experiments reported here th is d i f f i c u l t y
was accentuated In some cases by the smallness o f th e a tte n u a tio n being
measured and, p a r t ic u la r ly w ith s p in e l, by the smallness of the a v a il­
ab le samples.
A ft e r tr y in g several measuring techniques (e .g . using
a fix e d f i e l d ) one was found fo r spinel th a t gave reasonably consis­
te n t re s u lts ;
by c o n s is te n t Is meant t h a t the In d ica te d a tte n u a tio n
was approxim ately the same fo r any p a ir o f echoes.
The method em­
ployed fo r YIG was only s lig h t ly d if f e r e n t and th e procedure was as
fo llo w s .
The specimen o r ie n ta tio n was adjusted w ith respect to the
d .c . and microwave magnetic f ie ld s u n til the echo p a tte rn was as n e arly
as possible e x p o n e n tia lly decaying a t room tem perature (see f i g .
(3 )b )
and where the echo p a tte rn peaks up as a whole a t one p a r t ic u la r f i e l d
a t th e high f i e l d end of the spin wave spectrum I . e . near the uniform
precession mode.
With patience these conditions could u s u a lly be
s a t is fie d sim ultaneously.
With YIG a t 9 GHz room tem perature a d ju s t-
merits w h ile observing the echo p a tte rn s were not possible and a more
a r b it r a r y I n i t i a l placing had to be used, but o f course t h is could be
a lte r e d by re tu rn in g the sample to room tem perature I f the echo p a tte rn
was thought u n s u ita b le fo r measurements o r the re s u lts proved Inconsis­
te n t.
At I GHz both spinel and YIG could be adjusted a t room tempera­
tu re w h ile observing r e fle c tio n s .
As th e tem perature was slow ly
changed th e magnetic f i e l d was adjusted to fo llo w the peak In th e echo
p a tte r n .
T y p ic a lly the applied f i e l d decreased by 500 Oe, about \ %,
as th e tem perature was reduced from room tem perature o r ju s t below to
4 .2 °K .
This In d ic a te s t h a t the change In the s tr a in Induced aniso­
tro p y f i e l d
(see r e f .
specimen;
Is more Im portant than the change In demagnetizing f i e l d
(38) appendix I ) .
Again several runs were taken fo r each
the accuracy w i ll be discussed In the sections where these
are presented.
A fu r th e r d i f f i c u l t y w ith NI film transducers Is mode conversion
o f lin e a r ly p o la riz e d transverse waves on r e fle c tio n from th e NI
film s .
55
This gives r is e to an a d d itio n a l a tte n u a tio n (e .g . fo r
tra n s ve rs e waves In the <IIO > d ir e c tio n ) which has been assumed Inde­
pendent o f tem perature.
Attempts to extend the measurements on
tra n s ve rs e waves to above room tem perature (< IIO > c u t s p in e l) were
la rg e ly fo ile d by a d e c lin e In th e e ffic ie n c y of the transducers, pro­
bably due to spin wave losses.
up to 500°K.
N evertheless echoes have been observed
22
CHAPTER I I I
EXPERIMENTAL OBSERVATIONS ON QUARTZ, SPINEL AND YIG
In tro d u c tio n
This chapter deals w ith th e tem perature-dependent a tte n u a tio n of
phonon modes In various specimens o f q u a rtz , spinel and YIG and o th e r
re s u lts obtained In the course o f these experim ents.
The m a jo rity of
a tte n u a tio n measurements reported are a r b i t r a r i l y norm alized to zero
a t 4 .2 °K , thereby s u b tra c tin g from th e to t a l a tte n u a tio n any tempera­
tu re -in d e p e n d en t a tte n u a tio n .
This procedure Is reasonable fo r the
’ p e r fe c t' q u artz specimens but some of the Im perfect specimens show a
rap id r is e o f a tte n u a tio n ju s t above 4 .2 °K .
th e re Is s t i l l
a tem perature-dependent a tte n u a tio n oy. (4 .2 °K ) a t 4 .2 °K
in a d d itio n to any re sid u al a tte n u a tio n
cesses.
This In d ic a te s th a t
e .g . due to s c a tte rin g pro­
The t o t a l a tte n u a tio n a t tem perature T j Is then given by a ( T j)
= 01^ + cxy (4 .2 °K ) +
(g ra p h ).
A p lo t of the tem perature-dependent
a tte n u a tio n should th e re fo re be co rrected fo r c tj(4 .2 ° K ).
most u n lik e ly
th a t
However, I t
Is
(4 .2 °K ) exceeds o (8 .4 °K ) - a (4 .2 °K ) and I t Is
e a s ily v e r if ie d th a t the a d d itio n of o ij(4 .2 °K ) ^ o (8 .4 °K ) - a (4 .2 * K ) to
a l l poin ts on th e re le v a n t graphs w i ll not s ig n if ic a n t ly a f f e c t them.
The a tte n u a tio n observed In spinel Is so low th a t reducing the tem­
p e ratu re below 4 .2 °K did not show any change.
With the YIG specimens,
although no change In the a tte n u a tio n was observed w ith in the e x p e ri­
mental accuracy fo r the p a r tic u la r specimen measured (tem perature re ­
duced to 2 .2 ° K ), a less pure specimen, which had a tem perature-dependent
23
a tte n u a tio n of about 5 dB/cm a t 2 0 °K (and too high to be measured above
2 0 °K ), showed a reduction In a tte n u a tio n by 0 ,7 dB/cm on cooling from
4 ,2 °K to 2 .2 °K .
This Is presumably due to th e t a l l o f a very large
a tte n u a tio n peak a t about 3 0 °K (see l a t e r ) .
2.
Observations
(I)
N atural q u a rtz
The samples used were n a tu ra l B r a z ilia n q u a rtz and the f i r s t meas­
urements were made on a I cm long 3 mm diam eter a c -c u t q u a rtz c ry s ta l
a t a frequency of 9155 MHz.
The accuracy Is about 0 . I dB/cm.
predominant mode e x c ite d w ith th e geometry o f f i g .
( I ) a o r b Is the
pure tra n s ve rs e mode w ith v e lo c ity 'v 3 .3 x 10® cm/sec.
tio n Is shown In f i g .
The
The attenua­
(7 ) and v a rie s as wT^'3 In f a i r agreement w ith
th e re s u lts o f previous w o rkers.*^
These re s u lts are a check on th e
measuring technique and tem perature s c a le .
The absence of steps In ­
d ic ate s th a t the sample Is r e la t i v e l y fre e from Im p e r fe c tio n s .* *'* ^
Echoes have been observed due to e x tra phonon modes In th is sample.
The v e lo c itie s o f these have been measured by M.F. Lewis (published In
r e f . 35) using NI film transducers since by varying the o r ie n ta tio n o f
th e a p p lied magnetic f i e l d
I t Is possible to enhance and suppress
wanted and unwanted modes re s p e c tiv e ly .
Table ( I )
Is a l i s t of these
measured v e lo c itie s and v e lo c itie s c a lc u la te d from th e e la s t ic constants
o f q u a rtz ,
51
to g e th e r w ith the computed angle 0 ' between the wave v ec to r
and th e d ire c tio n o f energy flo w .
2 .6 5 gm/cm®.
The den sity o f q u a rtz was taken as
The a tte n u a tio n o f several o th e r modes In q u artz have
24
a ls o been measured.
(1 )
They are:
The q u a s i-lo n g itu d in a l mode In a c -c u t q u a rtz .
The re s u lts
are shown In f i g . (7 ) and I t Is c le a r th a t the a tte n u a tio n Is small and
v a rie s fa s te r than T** over a considerable tem perature In te r v a l.
(2 )
10®
The pure tran sverse mode In
cm/sec.
b c -c u t q u a rtz , v e lo c ity
The a tte n u a tio n Is shown In f i g .
magnitude and slope to t h a t o f ( I ) above.
5 .0 x
(8 ) and Is very close In
The low value o f th e a tte n u ­
a tio n In b c -c u t q u a rtz has a lre a d y been noticed by Bommel and Dransf e ld .^
No e x tra modes were observed In th e 2 .5 cm long 3 mm diam eter
specimen.
(3 )
cm/sec.
(9 ).
The lo n g itu d in a l mode In x -c u t q u a rtz , v e lo c ity ^ 5 .7 x 10®
The most p e rfe c t specimen gave the a tte n u a tio n shown In f i g .
This 25 mm long specimen Is considered p e rfe c t because I t gave
1000 echoes a t 4 .2 *K and shov/ed no steps In the a tte n u a tio n versus tem­
p eratu re curve.
Other Im perfect specimens show d if f e r e n t behaviours
as discussed below.
(4 )
10®
The fa s t tran sverse mode In x -c u t q u a rtz , v e lo c ity
cm/sec.
5 .I x
This can be e x c ite d w ith the arrangement of f i g . ( I ) a
o r b by t l l t l n g t h e specimen a t a few degrees to the post o f the
c a v ity .
fig .
The a tte n u a tio n In the most p e rfe c t specimen Is shov/n In
(9 ) and s u rp ris in g ly Is lower than th a t of the lo n g itu d in a l wave.
(5 )
10® cm/sec.
The slow transverse mode In x -c u t a u a r tz , v e lo c ity 3 .3 x
The a tte n u a tio n Is shown In f i g .
T*' over th e range o f measurements.
(9 ) and v a rie s as about
This behaviour Is expected fo r
slow tran sverse phonons on the Landau-Rumer th e o ry .
25
(6 )
cm/sec.
The lo n g itu d in a l mode In z -c u t q u a rtz , v e lo c ity
The samples are bonded to x -c u t transducers.
I n e f f ic ie n t
bonds make th e re s u lts less accurate than f o r o th e r c u ts .
u atio n o f th e most p e rfe c t specimen is shown in f i g .
(1 1 )
6 .4 x 10®
(II),
The a tte n ­
la b e lle d Zg.
Observations on Im perfect qu artz
Measurements o f th e a tte n u a tio n o f lo n g itu d in a l phonons were made
in the fo llo w in g im perfect x -c u t q u a rtz c ry s ta ls :
ian q u a rtz c r y s t a l,
(A) n a tu ra l B r a z il­
(B) a s y n th e tic c ry s ta l grown hydrotherm ally perpen­
d ic u la r to th e basal p la n e, and (C) a s y n th e tic c ry s ta l grown hydro­
thermal ly p erp en d icu lar to the basal plane and d e lib e r a te ly doped w ith
about 0.015% iron so th a t the c ry s ta l has a s lig h t ly blue c o lo u r.
The a tte n u a tio n curves fo r (A ), (B) and (C) are shown in f i g .
to g e th e r w ith th e a tte n u a tio n in a p e rfe c t specimen.
(1 0 )
The steps in the
a tte n u a tio n in the im perfect specimens always occur a t tem peratures o f
16 - 3°K and 24 - 3°K.
In general I t is not possible to determine the
p o s itio n s o f the peaks to b e tte r than - 3®K.
such steps was made by Jacobsen
s u lts are re p lo tte d in f i g .
14
The f i r s t observation o f
using an a c -c u t specimen.
His re­
(7 ) from which I t can be seen th a t the steps
occur a t about 16 and 22®K, i . e .
in the ranges 16 - 3®K and 24 - 3®K.
Attempts were made to produce im perfections in one natu ral a c -c u t rod
by ir r a d ia tin g i t w ith s Co®® source to the e x te n t o f n, 10^ rad.
Although th e c ry s ta l turned black and opaque the e f f e c t on the tempera­
ture-dependent p a rt of th e a tte n u a tio n was m inim al.
26
The a tte n u a tio n of lo n g itu d in a l waves in two im perfect z -c u t quartz
c ry s ta ls has a lso been measured.
One o f these was a s y n th e tic qu artz
rod from the same c ry s ta l as (C) above.
n a tu ra l c r y s ta l.
The o th e r ( Z j ) was cut from a
The a tte n u a tio n curves are shown in f i g .
again th e steps occur a t about 16 and 24*K.
(II)
and
Annealing the n a tu ra l crys­
t a l a t 5(X)®C fo r fiv e days produced no change in the temperature-depen­
dent a tte n u a tio n .
(ill)
Spinel
f
The specimens used were s t r i c t l y s to ic h io m e tric s in g le c ry s ta ls o f
MgAlzOi, grown by the flu x m elt technique a t the General E le c tr ic C o .,
L td ., Wembley, England.
Some contained small concentrations of Cr
(< 0.1%) and were s lig h t ly pink in c o lo u r.
No e x tra phonon atten u a­
tio n was observed when a magnetic f i e l d brought the Cr
sonance a t the phonon frequency.
spins to re­
The samples were o rie n ta te d in the
< I0 0 > , < liO > and < l l l > d ire c tio n s and ty p ic a lly 4 to 5 mm long and 3 mm
In d iam eter.
Thus the pulse length used was reduced to 0 .2 vis In
o rd er to resolve echoes in the s h o rt c ry s ta ls .
For tra n s ve rs e wave generation th in film Ni transducers were used
and fo r lo n g itu d in a l wave generation th in film CdS transducers were used
(see Chapter I I fo r d e t a il s ) .
The a tte n u a tio n measurements a t a f r e ­
quency o f 9155 MHz fo r lo n g itu d in a l and tra n s ve rs e waves In th e above
d ire c tio n s are shown in fig s . (1 4 ), (1 5 ) and (1 6 ).
The curves fo r the
tra n s ve rs e waves in p a r tic u la r are each the averaged r e s u lt of several
runs because o f th e g re a te r d i f f i c u l t y
In measurement, but s t i l l the
27
a tte n u a tio n In d ic a te d a t the lower temperatures cannot be guaranteed to
b e tte r than - 50%,
about -
15% but I t
O verall the accuracy a t th e higher tem peratures Is
Is d i f f i c u l t to allow fo r any remaining system atic
e rro rs which may a ris e from th e transducer p ro p e rtie s .
To in v e s tig a te th e frequency dependence attem pts were made to
measure the a tte n u a tio n a t a frequency of I GHz.
However, f o r th is f r e ­
quency th e apparent a tte n u a tio n o f the lo n g itu d in a l modes in p a r tic u la r
was dominated by phase c a n c e lla tio n e ffe c ts (see f i g .
(3 )b ) and In gen­
e ra l was too small to be measured.
(iv )
Y[G
The s in g le c ry s ta l YIG specimens were the best a v a ila b le i . e .
grown from 99.9999% pure y t t r i a and only those in v e s tig a te d In d e ta il
which gave th e lowest u ltra s o n ic re la x a tio n peaks a t 9155 MHz and low
tem peratures and a ls o th e lowest m agnetoelastic wave losses a t I GHz
and room tem perature.^^
Samples were In th e form of rods t y p ic a lly
7 to 10 mm long and 3 mm in diam eter and o rie n ta te d In the <I00> d ire c ­
t io n , w h ile some measurements were also made on a < l l l > specimen.
Ni and CdS th in film transducers were used fo r tran sverse and lo n g itu ­
d inal wave generation re s p e c tiv e ly , and sometimes bonds to qu artz fo r
the l a t t e r as w e ll, see Chapter I I .
The a tte n u a tio n curves fo r lo n g i­
tu d in a l and tran sverse waves In the <I00> d ire c tio n a t 9155 MHz are
shown in f ig s .
(1 9 ) and (2 1 ).
The accuracy of the lo n g itu d in a l wave
measurements can be estim ated from the s c a tte r of the points taken on
several d if f e r e n t runs, w hile th a t f o r transverse waves is about -
I dB/
28
cm, but betv/een 150 and 250°K the transverse waves echoes were so small
th a t the accuracy could only be - 2 dB/cm.
I t was found th a t the a tte n u a tio n o f the 9155 MHz lo n g itu d in a l wave
was not a ffe c te d by an applied magnetic f i e l d except fo r magnetic f i e l d
values near
K = o i / y \ where
gyromagnetic r a t i o .
57
to is the phonon frequency and
is the
In these circumstances th e re was an a d d itio n a l
a tte n u a tio n when the f i e l d made an angle o th e r than
th e rod a x is , i . e . the u ltra s o n ic wave v e c to r.
0®
and 90® w ith
This behaviour is
'■
58
expected from the work o f Schlomann
and has p revio u sly been observed
a t lower f r e q u e n c i e s . T h e
fig .
d e ta ile d a tte n u a tio n measurements of
(1 9 ) a t 9155 MHz were taken in zero magnetic f i e l d .
In o rd e r to e s ta b lis h the frequency dependence o f lo n g itu d in a l
waves in YIG some measurements were made a t i GHz using CdS th in film
transducers.
The re s u lts are shown in f i g . (2 0 ) where the accuracy
4.
4.
v a rie s from - 0.1 dB/cm a t low tem peratures to - 0 .3 dB/cm a t room
tem peratu re.
The a tte n u a tio n peak which occurs a t about 230®K fo r
I GHz phonons In zero f i e l d can be removed by the a p p lic a tio n o f a large
magnetic f i e l d and appears to be caused by in te ra c tio n s w ith domain
w alls (see Chapter V, section 5 ) .
Previous microwave u ltra s o n ic a tte n u a tio n measurements on tra n s ­
verse waves r e lie d on the m a g n e to s tric tiv e p ro p e rtie s o f YIG I t s e l f . ^ *
However, when using th is technique i t is only possible to s e t an upper
1.1m it on the i n t r in s ic a tte n u a tio n .^ ^
These d i f f i c u l t i e s were over­
come in the re s u lts presented since the method o f transverse wave gen­
e ra tio n was d if f e r e n t i . e . Ni film transducers were used.
The applied
29
magnetic f i e i d , t y p ic a lly 5 kOe, is then s u f f ic ie n t ly high so th a t
throughout th e sample i t exceeds w /y ’ .
(3 8 )^ ^ .)
Between 4.2®K and 50®K the a tte n u a tio n does not change
measurably.
Any peak in th is tem perature range cannot exceed the
accuracy o f measurement ( -
( V)
(See Appendix ( I ) o f r e f .
I dB/cm) o r i t would be detected .
O ther measurements on YIG
The peak observed In the temperature"dependent a tte n u a tio n o f i
GHz lo n g itu d in a l waves In YIG has been in v e s tig a te d fu r th e r .
In two
<I00> specimens th e peak occurs a t 227 - 3®K and in two < l i l > s p e c l-
4.
mens i t occurs a t 250 - 5®K.
o r ig in and p u r ity .
A ll fo u r specimens are o f d if f e r e n t
The a tte n u a tio n peak can be removed by applying
a magnetic f i e l d s u f f ic ie n t ly strong to s a tu ra te th e sample.
Except
fo r the tem perature o f the peak the d iffe re n c e in the a tte n u a tio n s in
th e s a tu ra te d and unsaturated s ta te s has very n e arly th e same shape
and magnitude in a l l the specimens studied;
of th e <I00> specimens is p lo tte d in f i g .
th is d iffe re n c e fo r one
(2 3 ).
I t is not known I f
tra n s ve rs e waves are a ffe c te d by the peak because th e generation of
tra n s ve rs e waves u s u ally re q u ire s a magnetic f i e l d .
On applying a magnetic f i e l d changes in a tte n u a tio n w ith f i e l d
have been observed a t tem peratures in the region o f the peak.
The
excess a tte n u a tio n was found sometimes to decrease monoton Ic a 11 y and
sometimes to peak one o r more tim es before decreasing depending on the
tem perature and magnetic f i e l d o r ie n ta tio n .
in Chapter V.
This w ill be discussed
C e rta in ly these e ffe c ts were not concerned w ith the
30
ferrom agnetic resonance peak since th is was e a s ily resolved a t a l l tem­
p e ra tu re s , w ith f i e l d values and o rie n ta tio n s In agreement w ith
Schlomann
58
.
The re s u lts on one specimen are summarized in f i g .
(2 4 ).
The curve (A) marks the onset of a tte n u a tio n by ferrom agnetic resonance,
as v e r if ie d by I t s
vanishing when G = 0 * o r 90®, and by the f a c t th a t
the magnetic f i e l d
values fo r H p a r a lle l to the rod Is very close to
th e value fo r m agnetostatic pulse propagation when th e sample Is in th e
r f magnetic f i e l d . C u r v e s
tio n s o f the e x tra
peaks.
(B) and (C) represent approximate posi­
(B) is dominant a t tem peratures above 227®K,
w h ile (C) is dominant below th is tem perature, and both seem to be speci­
men dependent.
CHAPTER
tv
THEORETICAL ACCOUNT OF ATTENUATION MECHANISMS RELEVANT FOR
QUARTZ, SPINEL AMD YIG
I.
In tro d u c tio n
A b r ie f account w i ll be given f i r s t of the p ro p e rtie s o f l a t t i c e
waves, and t h is is follow ed by a statem ent o f the problems to be con­
sid ered in fu r th e r sections of th is c h ap ter.
The v ib ra tio n s of a
s o lid can be resolved in to a s e t of e la s t ic o r l a t t i c e waves and I f the
in te ra to m ic forces are p e r fe c tly harmonic and th e s o lid re g u la r these
waves are th e normal modes of v ib ra tio n of the c r y s ta l.
Thus th e d is ­
placement u a t a l a t t i c e s it e X can be w ritte n as a superposition of
waves ;
u(x)
=
— E
Æ q
G, b (q .J ) exp [ K q . x + w t ) ]
■
(I)
where G is th e number o f l a t t i c e s ite s in the c r y s t a l, q Is the wave
v e c to r, w the frequency o f a wave of p o la r iz a tio n e and index j d is ­
tin g u is h e s the various p o la riz a tio n s o f waves q.
The frequency o f a l a t t i c e wave depends on q and fo r each value
o f q th e re are in general th re e values o f w corresponding to the th re e
p o la r iz a tio n branches j .
The re la tio n s h ip between w and Q Is lin e a r
fo r low values o f q but as q increases to a value comparable w ith the
re c ip ro c a l o f the in te rato m ic distance th is
lin e a r re la tio n s h ip f a i l s ;
thus fo r s u f f ic ie n t ly s hort waves the a to m ic ity becomes im portant and
leads to d is p e rs io n .
The th re e p o la r iz a tio n d ire c tio n s e p e rta in in g
32
to any q are m utually perpend icular and in re a l c ry s ta ls i t is oniy fo r
waves along c e rta in symmetry d ire c tio n s th a t the p o la riz a tio n d ire c tio n
bears any sim ple re la tio n s h ip to q .
I t is customary to denote the
p o la r iz a tio n having th e highest w value fo r a given g as the lo n g itu ­
d in a l wave.
There are a s e t o f inverse l a t t ic e vectors b, in te g ra l m u ltip le s
o f th e th re e basic inverse l a t t i c e vectors b j , which are re la te d to the
th re e p e r io d ic ity vectors o f the c ry s ta l by:
.b^j
A tra n s fo rm atio n q
=
2it 6 {J
(2 )
q + b leaves In v a ria n t the displacement u ( x ) o f
eq. ( I ) a t the l a t t i c e s it e and so a wave q + b is p h y s ic a lly the same
as a wave q.
Therefore q space may be divid ed in to zones by planes
which are
normal b is e c to rs o f th e various vectors b and
about th e
o r ig in is s u f f ic ie n t to
the f i r s t
describe a l l l a t t i c e waves
zone
i . e . any
v e c to r o u ts id e the f i r s t zone can be made in to a v ec to r w ith in the
f i r s t zone by an in te g ra l b v ec to r w ith o u t changing the physical nature
o f the corresponding l a t t i c e wave.
The energy o f each l a t t i c e wave Is th a t o f a harmonic o s c illa t o r
g Iven by:
E(q.J) =
Mwf b*(q,J) b(q,J)
where M Is th e mass o f th e atom o r
u n it c e l l .
According to quantum
mechanics th e energy o f such an o s c illa t o r can only take d is c re te
va Iues
(3)
33
where N Is an In te g e r.
The energy of a given mode Is thus composed
o f a zero p o in t energy 1Tü) /2 and N quanta o f energy tTw.
are
c a lle d phonons.
Fors c a tte rin g
tum
mechanicalre la tio n s are
a (q )
These quanta
c a lc u la tio n s th e fo llo w in g
u s e fu l.
quan­
D e fin in g th e dynamicalv a ria b le
=b (q )
(5 )
then a (q ) and i t s complex conjugate a * (q ) must be replaced by m atrices.
The Ham iltonian is given by the symmetrized form:
H
= E
q 2
[a *(q )
-
a (q ) + a (q ) a * ( q ) ]
(6 )
Choosing fo r a (q ) a m a trix whose only non-vanishing elements a re:
a(g)N,N-l
=
w'/?
(7)
and fo r a * (q ) a m a trix whose non-vanishing elements are:
N+l °
(8 )
then H(q) has on ly diagonal elements th e Nth diagonal element being
given by equation ( 4 ) .
Since a (q )
lin k s s ta te s o f N phonons to those
co n ta in in g one phonon less i t Is a d e s tru c tio n
is a c re a tio n o p e ra to r.
o p e rato r w h ile a*(Q )
While the l a t t i c e modes in re al c ry s ta ls are
not tru e normal modes as considered above, in ord er to t r e a t energy
interchanged i t Is assumed t h a t these waves are almost normal modes.
The p ro p e rtie s described a ls o apply to u ltra s o n ic waves Introduced In to
a c ry s ta l as w ell as the thermal
In
c ry s ta l
l a t t i c e waves.
an u ltra s o n ic experiment a number o f modes maybe e x c ite d
to
sucha high degree th a t
In a
the energy content of th is group o f
34
modes, the u ltra s o n ic beam,is much higher than th e to t a i energy, in
thermal e q u ilib riu m , o f a l l the l a t t i c e v ib ra tio n modes o f comparable
and lower freq u en cies.
The energy content of th is beam can be studied
and the a tte n u a tio n deduced.
A t very low tem peratures (M®K) enhar­
monic processes are very small and any a tte n u a tio n a ris e s from s c a tte r ­
ing by s t a t i c im p erfectio n s.
This is a ls o very small s in c e , w ith some
p a r t ic u la r exceptions, c ry s ta ls of very high p u rity are used in such
experim ents.
As the tem perature is raised from about 4®K anharmonic
processes become in c re a s in g ly im portant and the a tte n u a tio n increases
ra p id ly w ith tem perature.
F in a lly the mean fre e path o f the thermal
phonons becomes comparable w ith the wavelength o f the impressed wave,
and a t these and higher tem peratures the theory o f in d iv id u a l mode
in te ra c tio n s breaks down.
d e a lt w ith in two sectio n s:
Thus the th e o r e tic a l considerations w ill be
(I)
wavelength of the Impressed wave less
than th e mean fre e path of the thermal modes i . e . cot > i .
(2) the
extreme opposite case w ith the wavelength much g re a te r than the mean
fre e path o r cot «
i.
In case ( I ) th e wave is assumed to in te r a c t
w ith in d iv id u a l modes,
w h ile fo r ( 2 ) the in te ra c tio n is tre a te d as ta k in g place w ith the e n tir e
assembly of l a t t i c e waves.
The interm ed iate case may be in fe rre d from
( I ) and (2 ) although several attem pts have been made to produce a theory
which covers both regions.
More w i ll be said o f th is
la t e r .
35
2.
In te ra c tio n s w ith In d iv id u a l modes
(i)
Landau-Rumer theory
In most m a te ria ls th e a tte n u a tio n o f transverse p o la rize d phonons
a t low tem peratures Is due to microscopic In te ra c tio n s w ith thermal
phonons by th re e phonon processes and is found e xp e rim en tally to vary
as wT"*.
Landau and R um er^^originally p re d icte d such behaviour a t
tem peratures low enough to s a t is fy the con d itio n
wt
> I.
th is c o n d itio n is s a t is fie d below 3 0 °K fo r 9 GHz phonons.
T y p ic a lly
Klemens^^
has reviewed phonon-phonon In te ra c tio n s processes and deduces the
Landau-Rumer r e s u lt as fo llo w s .
In th e in tro d u c tio n the Ham iltonian fo r an harmonic l a t t i c e v/as
described but i t is the fu r th e r term s, cubic, q u a r tic e tc .
v a ria b le s which are o f in te r e s t now.
in the same
These enharmonic terms g ive r is e
to phonon-phonon in te ra c tio n s and the cubic ones are considered since
these prove to be the most im portant.
I t Is assumed t h a t the depar­
tu re s o f the c ry s ta l from p e rfe c tio n are s u f f ic ie n t ly small th a t per­
tu rb a tio n theory may be used.
A p e rtu rb a tio n in the l a t t i c e Is des­
c rib e d by a p e rtu rb a tio n Ham iltonian H' lin k in g s ta te s i and j .
If
th e system is i n i t i a l l y
in
s ta te j a f t e r tim e t
in s ta te I , the p r o b a b ility of fin d in g i t
is g iven , according to
2 nd
o rd e r p e rtu rb a tio n
th e o ry , as:
W (t)
=
2 (H jj)* ( I -
COS
AE t / f i )
(9 )
where AE = Ej - Ej is the d iffe re n c e in energy between the two s ta te s .
The resonance fa c to r ( I - cos AE t/n )/A E ^ picks out those s ta te s from
36
th e sum over a l l s ta te s J which have the same energy as s ta te I .
The
H am iltonian f o r the l a t t i c e waves is given by:
H
*
E
q
^ Mo)2 [ a * ( q ) a (q ) + . a * ( q ) >
_
A
-
(10)
where a (q ) and a *(q ) have been defined in the in tro d u c tio n .
When the
l a t t i c e undergoes a d ila t io n the e la s t ic constants decrease and so does
th e frequency,
i.e .
w(q)
=
w^(q) [ I
- y (q ) A * ( x ) ]
(II)
where A *(x )
is the d ila t io n and y (q ) the anharmonicc o e f f ic ie n t .
Gruneisen in his o r ig in a l treatm en t took a l l y (q )s to be equal to an
average value y now c a lle d the Gruneisen constant.
Thus the fa c to r
in eqn. ( 10) is changed according to eqn. ( I I ) so th a t H(g) changes by
-2 y A *(x )H (q ) 1*8.
H '(q )
=
-2 y A '(x ) H(q)
(12)
The terms In th is equation are obtained by g e n e ra liz in g eqn. (1 0 ) i . e .
Z
q
0)^
a * (q )a (q )
-
becomes ( l / G )
Z
q q'x
tow’ a * ( q ’ )a (q ) exp [ i ( q - q ’ ) . x ]
_ _
_
and to and to’ are changed according to eqn. ( I I ) ;
s tr a in is a t h ir d
A’ (x )
the d ila t io n a l
l a t t i c e mode and given by:
=
(e .q ) exp ( i q .x ) a (q )
I
(13)
q
( i . e . e q n .( l) using the dynamical v a ria b le o f eqn. (5 ) and the ro ta ­
tio n is given by a s im ila r expression) so the expression fo r the p e rtu r­
b atio n Ham iltonian is of the form:
H’ (q)
=
E
qq'q"
C3 ( q ,q ’ ,q " ) a (q ) a (q ’ ) a * (q " )
(14)
37
Ca contains the fa c to r :
exp [ I ( q + q ’ - q " ) . x ]
which vanishes unless q+q’ -q ” equals zero o r b.
(1 5 )
The former s itu a tio n s
are c a lle d normal processes, w h ile the l a t t e r are known as Umklapp pro­
cesses.
Each term describes an In te ra c tio n where two phonons are
destroyed and a t h ir d Is c rea te d .
In a d d itio n to the r e s t r ic t io n on the wave vecto r im plied by expres­
sion (1 5 ) th e re is a c o n d itio n on th e frequency.
fa c to r in eqn. (9 ) picks out in te ra c tio n s w ith AE =
Since th e resonance
0
only those In t e r ­
actio n s which conserve energy can c o n trib u te i . e . those fo r which
-hoj + tîw ’
=
"hw"
( 16)
The Inverse process Is also possible where a high energy phonon produces
tv/o o th e r phonons.
However, since most o f the thermal phonons have an
energy kT >> h , and each created phonon must have energy less than fiw
one expects th is c o n trib u tio n to be n e g lig ib le .
Using eqns. ( 9 ) ,
(1 4 ) and various quantum mechanical re la tio n s
[eqns. (7 ) and ( 8 ) ] the ra te o f change o f N (q ), th e number o f phonons In
mode q , may be found.
Eqns. (1 5 ) and (1 6 ) r e s t r i c t the possible in t e r ­
actions the form er im plying th a t th e th re e wave vectors must form a
closed t r ia n g le .
For normal processes th is means:
q + q’
q"
(17)
Now th e phase v e lo c ity fo r lo n g itu d in a l waves (w /q )^ is g re a te r than
t h a t f o r tra n s ve rs e waves so eqn. (1 6 ) is only co n sis te n t w ith expression
38
(1 5 ) f o r the fo llo w in g in te ra c tio n s :
tra n s t
long
long
tra n s + tra n s -► long
)
a.
)
b.
except fo r the col I in e ar case w ith the th re e phonons having the same
p o la r iz a tio n s .
In such eqns. as (1 8 ) t h e f i r s t term re fe rs to the
microwave phonons w h ile the others r e fe r to thermal phonons.
Eqn.
(1 6 ) a lso e xp la in s why the lower frequency wave must be transverse to
in te r a c t w ith high frequency waves.
This wi l l be d e a lt w ith in detal
In section 2 ( 1 i ) but meanwhile the discussion is r e s tric te d to tra n s ­
verse waves only in te ra c tin g w ith the l a t t i c e v ib ra tio n s .
To o b ta in a re la x a tio n time from the ra te of change of N(q) a l l
th e modes are considered to be in e q u ilib riu m ex^ejbt the mode q whose
occupation number deviates by amount n.
The re la x a tio n tim e
t
of a
s in g le mode is then defined as:
(I/t)
=
-( l/n )(d N /d t)
Thus an expression may be found fo r t .
(J 9 )
Eqn. ( I 8 )a is th e dominant
process^^ and the re s u ltin g expression f o r the a tte n u a tio n of tra n s ­
verse waves Is :
a =
4 .3 4
-------------------- L_
SO p’
.T ,
(_^)
(2 0 )
where a is the a tte n u a tio n in dB/cm, p is th e d e n s ity , v an average
thermal phonon v e lo c ity and F| is an average of c e rta in second and
t h ir d o rd e r e la s t ic constants.
Eqn. (20) was f i r s t derived by Landau
39
and R u m e r I t
wi l l be shown In section 2 ( i i ) th a t the in te ra c tio n ;
tra n s + tra n s
tra n s
(2 1 )
is a ls o allowed to a s ig n if ic a n t e x te n t in some cases because o f the
fin ite
life tim e o f thermal phonons;*^
so th a t e q n .( 2 0 ) has an addi­
tio n a l term :
a =
4.34TT% w F/ . T 4
4.34tt^wF^^
----------------- L ( ^ ) + ------------------ 1
60p*vi®
60p®v^®
| , t a n ’^ \ Q : 3 2 i ^ k W )
2
-tr^^
( 22)
where L is th e la t t i c e dimension and v is th e v e lo c ity of microwave
phonons under c o n s id e ra tio n .
F^ is again an average of c e r ta in second
and t h ir d o rd e r e la s t ic c o n s t a n t s . T h e procedure fo r computing Fj
and F^ from the e la s t ic constants is s im ila r to th a t fo r the Gruneisen
numbers to which they are r e l a t e d , s e e
section 3 ( 1 ) .
The
second term of eqn. (2 1 ) is a col I inear o r near col I inear in te ra c tio n
and o th e r noncoil in e ar terms may be s ig n if ic a n t.
(ii)
Longitudinal waves:
41
WT > I
A s im ila r c a lc u la tio n to the Landau-Rumer one may be done fo r lon­
g itu d in a l waves.
Now, however, th e re are no s o lu tio n s such th a t w’ o r
w" is much g re a te r than w.
This can be seen from expression (1 5 ) and
eqn. (1 6 ):
u) s
0)”
- Ü)’
=
Ooj’ /B q M .q <: v q
where 7 is the group v e lo c ity of the thermal waves.
(23)
So lower frequency
u ltra s o n ic waves can only in te r a c t w ith higher frequency waves i f the
40
group v e lo c ity o f the higher frequency wave exceeds the phase v e lo c ity
o f the lower frequency one.
Also, since the two high frequency waves
must have n e a rly the same frequency they must belong to the same p o la r­
iz a tio n branch.
In an is o tro p ic e la s t ic continuum (w/q g re a te r fo r
lo n g itu d in a l waves than fo r transverse ones) th is im plies th a t the low
frequency wave must be tra n s v e rs e .^ ^ '^ ^
So a low frequency lo n g itu ­
d in a l wave in te ra c ts mainly w ith waves of comparable frequency, and
th is changes th e Landau-Rumer r e s u lt to one pro p o rtio n al to w*T.
H erring
69
using a s ca lin g o r s im ila r it y argument has proposed th a t the
a tte n u a tio n o f low frequency waves must vary as
anharmonic i t i e s are the dominant in te r a c tio n .
^ provided cubic
Thus the lo n g itu d in a l
wave a tte n u a tio n is expected to be very much weaker than th a t o f tra n s ­
verse waves since the d ir e c t in te ra c tio n w ith higher frequency thermal
waves is fo rb id d e n .
However, experiment has shown th a t the two a tte n ­
uations are of s im ila r magnitude and the lo n g itu d in a l one to vary as
T^ w ith 4 < m < 9.
I t was thought th a t higher order in te ra c tio n s might account fo r
t h i s , namely the q u a rtic anharmonic in te ra c tio n .^ ^ *^ ^ '^ ^
This is a
fo u r phonon process and although i t has been shown to produce an a tte n ­
uation g re a te r than th a t fo r th re e phonon processes, th is is s t i l l much
less than th e transverse wave a tte n u a tio n .
Another mechanism put forward^^ is th a t since a lo n g itu d in a l phonon
can transform in to a lo n g itu d in a l phonon and a transverse one, and
only the l a t t e r of these is removed to the higher frequency thermal
modes, th e population of o th e r low frequency lo n g itu d in a l modes w ill
th e re fo re b u ild up.
The ra te of anharmonic in te ra c tio n s depends on
41
the number o f phonons in modes which receive phonons so th is b u ild up
a c c e le ra te s the a tte n u a tio n of th e primary wave and thus Increases the
p opulation o f the o th e r lo n g itu d in a l and transverse waves.
F in a lly ,
the energy co ntent of a l l o th e r low frequency waves w i ll be comparable
w ith th e prim ary wave and the a tte n u a tio n w i ll then be governed by the
r a te a t which energy is removed from transverse waves to the high f r e ­
quency thermal ones i . e . the a tte n u a tio n of lo n g itu d in a l waves wi l l
comparable w ith th a t of transverse ones.
be
Q u a n tita tiv e considerations^^
support t h is showing th a t the lo n g itu d in a l a tte n u a tio n is governed by
th e tra n s ve rs e wave re la x a tio n tim e ra th e r than the lo n g itu d in a l one
but depends somewhat on in te n s ity and pulse s ep a ra tio n .
Experimental agreement has-been found fo r t h is .^ '^ * '^ ^ but another
e xp la n a tio n has been suggested by K a w a s a k iM a r is ,S im o n s ^ ^ '^ ^ and
Shiren
77
and a p p lie d w ith much success.
fo llo w s .
T h e ir conclusions are as
For lo n g itu d in a l waves th e re are only two processes to be
considered.
long t
long
long + tra n s
long
tra n s
)
)
)
a.
(24)
b.
(th e two high frequency phonons must have the same p o la r iz a tio n )
M aris*^ has shown t h a t , n e g lec tin g v e lo c ity d is p e rs io n , (2 4 )a is only
allow ed f o r col I in ear processes and (24)b cannot occur ot a i l .
However, he
p oints o u t th a t the e f f e c t of dispersion Is such th a t n e ith e r process
wi l l be allow ed;
but when the f i n i t e
life tim e s of thermal phonons are
taken in to account the small amount by which ( I 5 ) a does not conserve
42
energy may not be Im portant.
The thermal phonons have 11W-1me
t
so
t h e i r energy Is uncertain by an amountTT/t and th is allow s processes
which are col I In e ar o r n e arly col lin e a r (th e angle between q and q ’
< I / wt ) .
Maris^^ has carried out the calculations for th is e f fe c t
and th e lo n g itu d in a l wave r e s u lt Is :
4.34n%MwF_*
kT
a = ---------- :----3wlO
60p*v
i^ )
(25)
where the terms have been defined In connection w ith eqn. (2 2 ).
r e s u lt Is only v a lid fo r WT » l
This
as w ith the Landau-Rumer expression.
F o r shear waves a s im ila r s c a tte rin g process is p o s s ib le . Involving
th re e phonons a ll o f the same transverse p o la r iz a tio n .
This term has
been included In eqn. (2 2 ).
Shiren^* has derived the same re s u lts using a d if f e r e n t form ula­
tio n and he a ls o obtains fu r th e r terms which may w ell be im portant de­
pending on the degree o f l a t t i c e anharm onidty o r when the s t r i c t l y
col lin e a r in te ra c tio n s are forbidden.
K a le js , Maris and T ru e lI
have
considered re in s ta tin g (24)b due to a f i n i t e re la x a tio n tim e even though
the energy imbalance (T «
(2 4 )a by a fa c to r (0j^/T )^.
8^
the Debye tem perature) is la rg e r than fo r
However, (2 4 )a gives a maximum rapid tem­
p e ratu re v a r ia tio n o f a tte n u a tio n as T^ w h ile using (2.4)b as w ell gives
T®"®.
A T® dependence fo r lo n g itu d in a l wave a tte n u a tio n has been
observed in a number o f m a te ria ls and I t has been shown
case of
AI2O3
78
th a t In the
the r a t io o f process (24)b to (2 4 )a is c e r ta in ly g re a te r
than u n ity fo r tem peratures between 30®K and 50®K fo r a l l wt.
The In ­
t e n s ity dependent mechanism described on page 40 should be a d d itio n a l to
43
th e above in te ra c tio n s but may w ell be q u ite smali since good e x p e ri­
mental agreement has been found fo r the th e o rie s o u tlin e d above.
78
The a tte n u a tio n o f fa s t transverse waves is expected to be in te r ­
mediate between th e slow transverse and lo n g itu d in a l waves and caused
by the Landau-Rumer and Maris mechanisms, eqn. (2 2 ), w ith perhaps con­
tr ib u tio n s from S h iren ’ s non-coil in e ar process.
For the case of a
tran sverse wave propagating on an even fo ld axis the s t r i c t l y col lin e a r
in te ra c tio n Is forbidden by symmetry so the a tte n u a tio n can only be
caused by the Landau-Rumer term and any non-col I inear in te ra c tio n s
(see Chapter V section
3.
Ui)).
In te ra c tio n s w ith the E n tire L a ttic e ;
(i)
< I.
Akhieser theory and improvements
In th e region wT < I d ire c t
w ith phonons cannot be fo llo w ed .
in te ra c tio n s of the u ltra s o n ic wave
The phonon gas Is then described by
a number o f macroscopic parameters which are p e r io d ic a lly v arie d by
th e Impressed wave.
The phonon gas Is thrown out of e q u ilib riu m and
ir r e v e r s ib le processes occur which cause a tte n u a tio n losses;
i.e .
the thermal phonons continuously re la x towards a new e q u ilib riu m d is ­
tr ib u t io n
la rg e ly by phonon-phonon in te ra c tio n s .
For lo n g itu d in a l
waves th e re are local changes in the density and average tem perature of
the medium.
The re s u ltin g s p a tia l d iffu s io n of energy gives r is e to
th e rm o e la s tic losses.
waves wi l l
Both lo n g itu d in a l and transverse u ltra s o n ic
in general cause a separation in the e ffe c tiv e temperatures
44
o f th e d if f e r e n t branches o f the thermal phonon dispersion curve;
the
r e s u ltin g s p ec tra l d iffu s io n o f energy gives ris e to Akhieser losses,
c a lle d so because Akhieser proposed the f i r s t t h e o r y . T h e r e were
la t e r versions by Bommel and Dransfeld^ and Woodruff and E hrenreich,^^
A ll these d e riv a tio n s
lead to an equation o f the form:
8 .6 8
C
T Y*
wf
T
a =
(26)
2 P V * (l +
where
W^T^)
is the s p e c ific heat per u n it volume and the o th e r terms have
been defined p re v io u s ly .
The th e rm o e la s tic loss is im portant in metals but fo r semicon­
ductors and in s u la tin g c ry s ta ls the e f f e c t is s m a ll.
This has been
shown^^ using an equation derived by M a s o n . T h e th e rm o e la s tic mech­
anism depends on the thermal c o n d u c tiv ity and the expansion c o e f fic ­
ie n t o f th e c ry s ta l whereas the Akhieser mechanism depends on th e th e r ­
mal c o n d u c tiv ity and th e Gruneisen numbers o f the thermal modes which
in tu rn come from the t h ir d o rd er e la s t ic constants (see l a t e r ) .
The
Akhieser mechanism being much la rg e r (f o r t y times the th e rm o e la s tic
losses) Is a consequence o f th e d is tr ib u tio n o f th e Gruneisen numbers
about th e mean v a lu e .
Since th is is the most im portant mechanism
f o r th e c ry s ta ls studied i t wi l l now be considered in fu r th e r d e t a il .
The previous th e o rie s have been improved by Mason,
45
who calcu ­
lated the e f f e c t o f the s tr a in associated w ith a wave on phonons pro­
pagating in d if f e r e n t d ire c tio n s in the c r y s ta l.
to e lim in a te the a r b it r a r y constant of eqn. (2 6 ).
In th is way he hoped
The processess o f
e q u a lis in g th e d if f e r e n t tem peratures fo r d if f e r e n t phonon branches by
45
means of phonon-phonon c o llis io n s occurs w ith a re la x a tio n tim e x .
For an a lte r n a tin g stress the tem perature d iffe re n c e lags behind the
a p p lie d s tre s s causing an a tte n u a tio n governed by the usual re la x a tio n
e q u atio n .
So the In te rn a l f r i c t io n
q“ ‘
is given by:
=
}
^
(2 7 )
I t w*x*
where c is the low frequency modulus p e rta in in g to the wave under con­
s id e ra tio n and Ac is th e instantaneous increase in the modulus caused
by th e separation o f the phonon modes.
(where a
Therefore using a = 0
is th e a tte n u a tio n in nepers cm '
and B is the
phase s h i f t in
radians per c en tim e te r and is equal to w /v) and v the average
v e lo c ity
B/2
Debye
= /c /p , one fin d s
a=
J
s
(28)
il
2pv*(l+w*x*)
Hence the a tte n u a tio n is found by c a lc u la tin g how the no n -eq u ilib riu m
tem perature separation o f the phonon modes by the app lied s tra in
a ffe c ts the e la s t ic modulus.
This c a lc u la tio n has been c a rrie d ou t by
Mason^^ using th e Debye expression fo r in te rn a l energy, d if f e r e n t ia t in g
th is w ith respect to the s tr a in to obtain an expression r e la tin g stre ss
and s tr a in and th e re fo re g iv in g Ac.
For lo n g itu d in a l waves th is is
co rrec te d f o r the fa c t t h a t the e la s t ic modulus relaxes to the adiaba­
t i c value and th e re s u ltin g expression is :
6c
=
3 Z E (yJk)
I
_ y :p C T
(29)
where y (Gruneisen constant) is zero fo r shear waves since th e re is no
46
average r is e In tem perature.
rf
each d ir e c tio n and mode,
yj
E. Is the thermal energy associated w ith
are the Gruneisen numbers associated w ith
d if f e r e n t d ire c tio n s and modes, a ll designated by the s u b s c rip t I ,
caused by th e s tr a in
They are an expression of the change in
mode frequency due to
i.e .
W;
=
A general form ula f o r
“o { I - Zyjk
(30)
has been worked out by B ru g g e r^ and th is Is
(repeated s u f f ix in d ic a te s summation)
lu
-Y ,
where
and
N N
“
" j" k +
(Cjkpq + "r"s C jk p rq s '
are the d ire c tio n cosines fo r the propagation d ir e c tio n ,
Uj and u^ are the d ire c tio n cosines fo r the p a r t ic le displacem ents,
C .,
are th e second o rd e t e la s t ic constants, C.,
are the th ir d
jkpq
jkp rq s
o rd e r e la s t ic moduli and C is the
tio n in th e d ire c tio n N
p
B a t e m a n , u s i n g
, N
q
e la s t ic modulus c o n tr o llin g propaga­
w ith p a r t ic le motion u , u .Mason and
p
5
the t h i r t y nine pure modes o f the < I0 0 > , < IIO >
and < l l l > d ir e c tio n s , have constructed ta b le s fo r Gruneisen numbers fo r
various waves propagating in a number of d ire c tio n s and have applied
some o f these re s u lts to recen t a tte n u a tio n measurements w ith f a i r agree­
ment,
W ritin g eqn. (2 8 ) in the form:
E _(D /3)ufT
a
=
2-----------------
(32)
2 pv*(l+w*T%)
gives using eqn. (2 9 ):
D =
{3ZE,(yJk)2 _ yZpC^T}
(33)
47
and i t is Mason and Bateman’ s c a lc u la te d values fo r D which wi l l
be
used where possible to c on struct th e o re tic a l a tte n u a tio n curves fo r
th e p a r t ic u la r c ry s ta ls s tu d ie d .
o f eqn, (2 6 ) wi l l
be used.
frequency o f 9 GHz;
Otherwise a r e la tio n o f the form
Eqn. (2 1 ) Is v a lid fo r
> I i . e . fo r a
th is means temperatures above about 50°K.
W hile t h is form ula may e q u a lly w ell be applied to shear and longi­
tu d in a l waves care must be taken In the values of re la x a tio n time used
in th e two cases.
The acoustic wave, i f
i t is a shear one, can in ­
te r a c t d ir e c t ly w ith thermal phonons, see section 2 .
Hence i t is
reasonable to expect th a t the ra te a t which thermal energy is in te r ­
changed between the various modes wi l l be governed by the thermal
phonon re la x a tio n tim e , given by:
'T .
■
ÿ
where k is th e thermal c o n d u c tiv ity .
ment.
This is confirmed by e x p e ri­
However, although experiment in d ica te s th e shear and lo n g itu ­
d in a l re la x a tio n s to be s im ila r in magnitude, the l a t t e r waves can
only in te r a c t w ith lo n g itu d in a l and shear phonons of the same f r e ­
quency range and these have much longer re la x a tio n tim es.
lowing th e arguments of section
2
So, f o l ­
( i i ) , th e re is a b u ild -u p o f the popu­
la tio n o f low frequency modes which acc e le ra te s the ra te of energy
exchange, and the a tte n u a tio n o f the lo n g itu d in a l waves Is then gov­
erned by the ra te a t which energy is removed by the transverse waves
to th e high frequency thermal waves.
Since h a lf of the b u ilt-u p low
43
frequency phonons are shear and h a lf lo n g itu d in a l f t Is to be expected
t h a t th e re la x a tio n tim es o f the l a t t e r waves would be tw ice th a t fo r
shear waves.
Measurements have in d ica te d th is to be s o ,*^ but where
th e a tte n u a tio n o f lo n g itu d in a l waves is expected to vary as eqn. (25)
(low tem peratures) any fa c to r o f two would not be obvious because of
th e
4.
term .
Comparison o f th e o rie s fo r quartz
In th e lig h t o f what has been said In sections 2 and 3 the general
p ic tu re fo r q u a rtz wi l l
be o u tlin e d here.
F ig s . (25) and (26) show
th e form o f the a tte n u a tio n of shear and lo n g itu d in a l waves in q u a rtz .
Using eqn. (3 4 ) (re fe ren c e s fo r thermal c o n d u c tiv ity and s p e c ific heat
values given in r e f .
(4 5 )) good agreement is found fo r the m odified
Akhieser th e o ry , eqn. (3 2 ), down to wx = 2wXy^ = I fo r lo n g itu d in a l
waves (3 8 * f o r I GHz phonons) and the same Is so f o r shear waves except w ith wx = wXy^ = I and provided no peaks are p resent.
45
S ingle
c ry s ta l q u a rtz has a number o f re la x a tio n peaks associated w ith impur­
i t y content which have received extensive in v e s tig a tio n e .g . r e f .
(8 3 )
and are reviewed in r e f .
(4 5 );
quency and since thermal
losses increase ra p id ly w ith frequency these
l a t t e r wi l l
but these peaks are Independent o f f r e ­
predominate a t high frequencies.
For quartz th is is above
about 100 MHz.
Above a few GHz measurements are r e s tric te d to the wx »
I . e . th e th e o rie s o f section 2 a p p ly.
I region
Those measurements th a t have
been made In th is region agree w ell w ith th e o ry , e .g . r e f . (16) except
49
fo r th e
low tem perature plateaus o f f i g . (2 6 ).
Chapter
I I I on qu artz cover th is
m »
I region
The observations of
and wi l l be compared
In Chapter V w ith the re le v a n t th e o rie s given above.
s im ila r
low tem perature plateaus to f i g . (26) In
We have observed
a number o f impure
q u a rtz c ry s ta ls and these wi l l a lso be discussed in Chapter V, as wi l l
th e a tte n u a tio n measurements on spinel and YIG.
50
CHAPTER V
COMPARISON OF OBSERVATIONS WITH THEORY AND DISCUSSION
I.
Q uartz
(I)
Temperature dependence
The various th e o rie s o f a tte n u a tio n of waves In qu artz have been
described in Chapter IV.
For 9 GHz phonons the temperature range over
which measurements can be made is from about 35*K downwards.
out t h is range WT »
I and so the observations wi l l
the th e o rie s o f Chapter IV section 2 .
Through­
be compared w ith
In o rder to do th is the v a ria ­
tio n o f re la x a tio n tim e w ith tem perature has to be found.
This can
be estim ated from the thermal c o n d u c tiv ity re la tio n s h ip :
K
where
( 1 /3 )
V*
(I)
the thermal c o n d u c tiv ity , C^
the s p e c ific h e a t, v the aver­
age phonon
v e lo c ity and
phononre la x a tio n tim e . The
r e s u lt due
to M aris*^ Is shown in f i g .
k
is
=
th e thermal
(1 2 ).
For 9 GH^ waves arc * I
a t about 60® K.
In the case of slow transverse waves the normal Landau-Rumer term
varying as cjT** Is dominant w h ile the expression fo r the a tte n u a tio n o f
microwave phonons given by M arls*^ (Landau-Rumer mechanism plus the
col lin e a r in te r a c tio n ) appears to be the app ro p riate one fo r exp la in in g
a c c u ra te ly the a tte n u a tio n of lo n g itu d in a l phonons in x -c u t quartz and
s lig h t ly
less a cc u ra te ly the a tte n u a tio n of slow transverse phonons In
a c -c u t q u a rtz over a wide range of frequency and tem perature.
There-
51
fo re the re s u lts have been f i t t e d to M a rls 's expression;
4 .3 4 ‘ir*'h(oF, ^
4.34Tr4lwF^
a = / ------------------- + -----------------6 0 p \ '®
(2 )
6 0p ^ v '“
where the terms have been defined in Chapter IV eqns. (2 0 ), (2 2 ) and
(2 5 );
V
f o r which eqn. (2 ) o f th is chapter is a general statem ent.
is given th e value 5 .3 x 10® cm/sec and is here an average o f phonon
phase v e lo c itie s
74
o
and L ~ 5 A fo r q u a rtz.
Note th a t Fj = 0 fo r lo n g i­
tu d in a l phonons and th a t the Fj^ and F^* used are a fa c to r o f two
g re a te r than M aris used.
The procedure fo r computing Fj and F^ from
th e known e la s t ic constants o f q u a r t z ^ * i s
described In r e fs . (45)
and (4 6 ).
Using eqn. (2 ) and chosen values of Fj and F^ ( l i s t e d
in ta b le I I )
th e th e o r e tic a l curves fo r the slow transverse and lo n g itu d in a l waves
in f ig s .
(7 ),
q u ite good.
(9 ),
(1 0 ) and ( I I ) were computed.
The agreement is
This is p a r tic u la r ly s a tis fa c to r y in the case o f q u a sl-
lo n g itu d in a l phonons in a c -c u t quartz fo r which th e re Is only one
a d ju s ta b le parameter F^*.
This determines th e magnitude of the a tte n ­
u atio n but not th e slope o f the curve in f i g .
(7 ).
In the case o f
pure tran sverse waves in a c -c u t q u a rtz , the slope o f the measured
a tte n u a tio n is about the same as obtained using M aris*s values o f F j*
and Fj^ but to account fo r the magnitude of the a tte n u a tio n we must
take values of F j* and
about 0 .7 times M a ris ’ s values.
This d i f f e r ­
ence Is due to the fa c t th a t Mari s 's values were chosen as a best f i t
to th e re s u lts of various workers, including some on im perfect a c -c u t
q u a rtz.
52
For slow tra n s ve rs e waves on the ac- and x -c u t axes the s t r i c t l y
col I in e a r th re e phonon In te ra c tio n Is forbidden by s y m m e t r y . F ^
is a measure o f col lin e a r and near col lin e a r in te ra c tio n s and so Is
r e la t i v e l y small
in these cases.
I f any ad d itio n s o f consequence
were required to th e Landau-Rumer mechanism use would be made of the
non-col I in e a r mechanism o f S hiren.^ *
In these instances, because
th e e f f e c t is so s m a ll, no n o tic e ab le change in the th e o re tic a l curves
would r e s u lt.
The a tte n u a tio n o f fa s t transverse waves must be considered sep­
a r a t e ly .
In general these waves are expected to be attenuated by
th e M aris mechanism and the Landau-Rumer mechanism o f the type tra n s +
long
long (eqn. ( I 8 )a Chapter IV ), and are expected to have an
a tte n u a tio n In te rm ed ia te between th a t o f slow transverse and lo n g itu ­
d in a l waves.
However, fo r both the pure transverse waves in bc-cu t
q u a rtz and fa s t tran sverse waves In x -c u t q u a rtz , the s t r i c t l y c o llin e a r th re e phonon process Is forbidden by symmetry i . e .
F^^ - 0 ,3 5 *4 6 ,8 5
in eqn. (2 )
y ^ is presumably accounts fo r the very low values o f
th e a tte n u a tio n o f these modes, as w ith the slow transverse modes
s tu d ie d , but th e re is c le a r ly another process. In the fa s t transverse
wave case, In a d d itio n to any Landau-Rumer mechanism.
The only remain­
ing three-phonon process Is the non-col I inear mechanism f i r s t discussed
by S h iren .^ *
Therefore the measurements have been f i t t e d to an expres­
sion o f the form:
c
r
( k l) " +
60p’ 7 * “
^
(& '
60p’ v * “
(A i,
(3)
53
The f i r s t term In eqn, (3 ) Is a Landau-Rumer term and the second
is o f th e form derived in r e f .
(41) w ith
of r e f . (41) s e t equal to
zero since th e col I Inear in te ra c tio n vanishes.
f a s t tra n s ve rs e waves In x -c u t q u a rtz, f i g .
( 8 ) , used the values
A* = 5 .2
X
The s o lid curves fo r
( 9 ) , and b c -c u t q u a rtz , f i g ,
= 4 .8 x lof® and 5 .0 x 10** dyne*/cm** and
10*® and 3 .0 x 10*® dyne*/cm‘*, re s p e c tiv e ly .
While the
agreement w ith experim ent is f a i r l y s a tis fa c to r y , a t le a s t as good a
f i t can be made using eqn. ( 2 ) and th e values F j* *
10*® and 4 .4 x 10** dyne*/cm *, re s p e c tiv e ly .
0
and F^* = I . I x
Whatever the explana­
tio n o f the a tte n u a tio n o f fa s t transverse waves in q u a rtz . I t is c le a r
th a t th e c o n trib u tio n from the Landau-Rumer mechanism is very sm all.
This is also tru e of the a tte n u a tio n o f fa s t transverse waves in A I 2 O3
and L IF ,
24
23
f o r the Landau-Rumer mechanism causes an a tte n u a tio n varying
as T* a t th e lowest tem peratures.
Examination of r e fs . (23) and (24)
shows t h a t i f th is does occur, the amplitude is very sm all.
F in a lly , the s im ila r it y between the a tte n u a tio n s of slow tra n s ­
verse waves in x -c u t and a c -c u t quartz is not accidental since the
s tra in s involved are th e same, i . e . the slow transverse wave In the x
d ire c tio n Is p o la ris e d in the ac d ire c tio n and v ic e versa.
S im ila r
remarks apply to the fa s t transverse waves in the x and be d ire c tio n s .
(II)
Frequency dependence
A check on eqns. (2 ) and (3 ) can be made by comparing the attenua­
tio n measurements o f o th e r w o r k e r s ^ ' i n
measurements a t 9 GHz.
qu artz a t I GHz w ith our
The Landau-Rumer mechanism p re d icts an a tte n -
54
uatlon p ro p o rtio n a l to w, whereas the second terms In eqns. ( 2 ) and
(3 ) p re d ic t an a tte n u a tio n almost Independent of w.
Table I I I shows
t h a t these p re d ic tio n s are q u a lit a t iv e ly borne out by experim ent, but
th a t even the slow transverse waves do not show an a tte n u a tio n varying
q u ite as fa s t as w.
(ill)
Magnitude of the a tte n u a tio n
To compute
th e a tte n u a tio n of an u ltra s o n ic wave due to in te r ­
action s w ith a ll o th e r thermal phonons is very com plicated, even fo r
cubic c r y s t a l s . T h u s , to some e x te n t i t Is s a tis fa c to ry th a t the
coupling constants F | f , F^^ and
are the same order of magnitude as
th e e la s t ic constants of q u a r t z . H o w e v e r ,
fo r those modes
whose a tte n u a tio n Is dominated by col I inear in te ra c tio n s , the c a lc u la ­
tio n s s im p lify considerably and some values of the coupling constants
are lis te d
Fj
=
in ta b le I I .
For lo n g itu d in a l waves in x -c u t quartz
3C |I + C | 1 1 a 0 .4 8 X 10^* d y n e /c m * ^ ^ w h ic h gives an attenua­
tio n about 80 times s m alle r than measured ( I . e . F j is about 9 times
s m a lle r than measured from the a tte n u a tio n ).
This Is too large a fa c ­
t o r to be accounted fo r by experim ental e r r o r , so two a d d itio n a l mech­
anisms have been considered.
These were re c e n tly proposed to e xp la in
the T^ (m > 4) dependence o f the a tte n u a tio n .
The mechanism proposed
by K a le js , Marls and T ru e ll^ ® Is an In te ra c tio n o f the form:
long +
tra n s ■> tra n s (Eqn. (24)b Chapter IV) and gives an a tte n u a tio n whose
magnitude Is o f th e r ig h t o rd er but d i f f i c u l t to estim ate acc u ra te ly
because o f the anisotropy of the transverse phonon v e lo c lty ^ ^ which
55
e n ters to th e e ig h th power.
However, fo r wT »
I the a tte n u a tio n
v a rie s as T * / t = T®, which disagrees w ith the measured v a ria tio n of
about T®.
The second mechanism to be considered Is S h Ire n ’ s non-
col lin e a r mechanism.^*
For lo n g itu d in a l waves In x -c u t qu artz th is
mechanism is too weak to account fo r the discrepancy;
fo r example a t
20®K th e combined e ffe c ts o f M a rls ’ s col lin e a r process and S h iren ’ s
non-col I in e a r one is s t i l l about 40 times too small to account fo r the
experim ental ob servatio n s.
At present th e re is no explanation fo r
th is discrepancy.
For lo n g itu d in a l waves propagating on the z axis F j = 3 C j j +
C j 3 3 - 4 .9 X 10^* dyne/cm* and agrees with experiment to b e tte r than a
fa c to r of two.
S im ila r ly , fo r q u a s l-lo n g itu d in a l waves propagating
on an ac a x is , th e re is s u b s ta n tia l agreement (Table I I ) .
To s im p lify
the c a lc u la tio n the wave on th e ac axis was assumed to be purely lon­
g itu d in a l.
(iv )
R elaxation tim e
Using the values of re la x a tio n tim e deduced from thermal conduc­
t i v i t y , the th e o rie s of Maris and Shiren have been found to e xp la in
s u cc e ss fu lly the tem perature and frequency dependence o f the attenua­
tio n o f microwave phonons in q u a rtz.
used here (T «
However, a t the temperatures
O^^) th e re Is a considerable c o n trib u tio n to re la x a tio n
tim e from normal p r o c e s s e s ,w h ic h do not c o n trib u te d ir e c tly to th e r ­
mal r e s i s t a n c e . T h e
re la x a tio n tim e problem has been most f u l l y
tre a te d by Woodruff and Ehrenreich^^ and Orbach?^
The l a t t e r has used
56
the same equations to compute the microwave phonon a tte n u a tio n and the
thermal phonon re la x a tio n time due to a ll three-phonon processes, I . e .
normal and umklapp.
These are compared w ith th e thermal phonon re la x ­
a tio n tim es deduced from thermal c o n d u c tiv ity measurements In f i g .
of re f.
(7 0 ).
(13)
F o rtu n a te ly the two curves are not w idely separated In
the region o f In te r e s t.
However, the use o f re la x a tio n times deduced
from thermal c o n d u c tiv ity measurements would be more j u s t i f i a b l e i f
these were known to be dominated by im perfection o r isotope s c a tte rin g
in q u a rtz .
Some evidence fo r Im perfection s c a tte rin g comes from the
observation th a t the thermal c o n d u c tiv ity peak a t about IO*K is not as
g re a t as one would expect from umklapp processes and boundary s c a tte r ­
ing alone (see r e f .
It
tio n
( 8 8 ) and references c ite d t h e r e in ) .
is noteworthy th a t i f normal processes produce a modest reduc­
in re la x a tio n tim e (such th a t wr »
I s t i l l holds) then the micro­
wave phonon a tte n u a tio n mechanism of Marls should be s lig h t ly stronger
than c a lc u la te d above and should produce an a tte n u a tio n varying as wT*
in the range of the experim ents.
This Is probably the explanation of
th e behaviour of lo n g itu d in a l microwave phonons In MgO,
18
fo r Shiren
77
has shovm t h a t normal processes involving th re e col I inear lo n g itu d in a l
phonons occur more fre q u e n tly in MgO than in q u a rtz.
2
.
Im perfect quartz
In several specimens o f im perfect quartz we found steps in the
a tte n u a tio n p lo tte d as a functio n o f tem perature.
These steps always
occur a t tem peratures w ith in a few degrees of 16 and 24®K, regardless
57
of the o r ig in of the c ry s ta l o r of the phonon mode concerned.
The
existen ce o f th e steps Is not consistent w ith a reduction in the th e r ­
mal phonon r e la x a tio n tim e by s c a tte rin g , and so must be caused by a
d ir e c t In te ra c tio n o f th e 9 GHz microwave phonons w ith th e imperfec­
tio n s .
In th e megacycle range th e re Is abundant evidence fo r peaks
in the a tte n u a tio n o f im perfect quartz and fused s i l i c a a t various temp e ra tu re s ,
45 90.91
92
'
*
and re c e n tly Jones e t a l.
have observed large and
very broad peaks in th e a tte n u a tio n of I GHz phonons In fused s il i c a
a t tem peratures o f about 10 and 60*K.
I t seems probable, th e re fo re ,
t h a t th e steps observed are r e a lly peaks superimposed on a large
phonon-phonon a tte n u a tio n varying as T* to T * .
I t is not always
po ssible to v e r if y th is from the graphs by s u b tra ctin g a p e rfe c t curve
from an Im perfect curve to obtain the excess a tte n u a tio n because the
excess a tte n u a tio n o fte n continues a t temperatures above about 24°K,
probably due to th e t a i l o f a 60®K peak (see f ig s . ( 7 ) , (10) and
(ID ).
T h e re fo re , from the im perfect curve is subtracted a curve
w ith the same shape as the p e rfe c t one, b u t, when necessary, s h ifte d
along the tem perature axis to meet th e Im perfect curve a t th e highest
tem peratures.
shown in f i g .
An example o f the excess a tte n u a tio n so derived Is
(1 3 );
t h is v/as deduced from curve
th e o r e tic a l curve of f i g .
(1 0 ).
0
and the dashed
The re s u lts are c le a r ly c o n sis te n t
w ith a tte n u a tio n peaks a t about 16 and 24®K.
A possible exp lan atio n o f these peaks Is th a t they are caused by
a s tru c tu ra l re la x a tio n mechanism Involving displaced oxygen atoms,
'
The sharpness of th e peaks in d ica te s a narrow d is tr ib u tio n of s ilic o n -
58
oxygen bond angles such as might be caused by s p e c ific Im p u ritie s d is ­
to r t in g the l a t t i c e
In a w e ll-d e fin e d manner,^*
An a lte r n a tiv e
e xp la n a tio n o f the a tte n u a tio n peaks Is as fo llo w s .
Jones e t a l .
92
have noticed a possible c o rre la tio n between the tem peratures of the
a tte n u a tio n peaks In fused s i l i c a and the Einstein temperatures used by
Flubacher e t a l .
s ilic a .
93
to account fo r the anomalous s p e c ific heat o f fused
They suggest th a t the a tte n u a tio n is caused by in te ra c tio n s
w ith displaced oxygen atoms, low frequency sideways v ib ra tio n s of
which are believed to be responsible fo r th e excess s p e c ific heat of
fused s i l i c a .
93
Presumably the in te ra c tio n is of the form:
[u ltr a s o n ic phonon] + [lo w frequency o p tic a l phonon]
-► [some o th e r (th erm a l) phonon]
This is not norm ally an allowed process
94
(4 )
because the o p tic a l phonon
branch norm ally lie s above the acoustic branch, but im perfect quartz
and fused s i l i c a are exceptions.
This was d e f in it e ly estab lis h ed in
th e Raman-spectrum experiments described in r e f . (9 3 ).
This accounts
fo r r e la t iv e maxima In the u ltra s o n ic a tte n u a tio n in the region o f the
E in s te in tem peratures (1 3 , 32 and 58®K) although the highest tempera­
tu re peak Is a t le a s t p a r tly caused by a s tru c tu ra l re la x a tio n mechan­
ism.
This suggests th a t Im perfect qu artz behaves lik e ’’d ilu te " fused
s i l i c a , and t h a t the low tem perature a tte n u a tio n peak of r e f .
resolved In to two peaks a t about 16 and 24®K.
Evidence fo r the obser­
v atio n o f the t a i l o f a 60®K peak has already been mentioned.
width o f the peaks in fused s i l i c a
(92) is
The
in d ica te s a d is tr ib u tio n o f bond
59
angles as expected from the random-network theory o f the glassy s ta te .
The average bond angle Is about the same as In Im perfect q u a rtz.^ *
On the basis o f an In te ra c tio n of the form o f eqn. (4 ) I t
Is evident
th a t th e re Is a remarkable c o rre la tio n between the E in s te in tempera­
tu re s used to e x p la in th e s p e c ific heat (1 3 , 32 and 58°K) and the tem­
p eratures of th e a tte n u a tio n peaks (1 6 , 24 and 'v 6 0 °K ).
however, d i f f i c u l t i e s w ith th is exp lan atio n ;
There a re ,
fo r example i t
Is not
c le a r why th e a tte n u a tio n should peak near the E in s te in temperatures
ra th e r th an , say, f l a t t e n o f f .
I t has not been possible to choose between a s tru c tu ra l re la x a tio n
mechanism and an o p tic a l phonon-uItrasonic phonon in te r a c tio n .
The
accuracy is probably in s u ffic ie n t to d e te c t th e change in tem perature
o f th e peaks w ith frequency th a t is expected i f a s tru c tu ra l re la x a tio n
mechanism is o p e r a t i v e . C h e m i c a l an alysis o f the samples used re ­
v eals no c o r re la tio n between the excess a tte n u a tio n and the concentra­
tio n o f AI o r Na, but th e re is a very rough c o rre la tio n w ith th e iron
co n ten t.
I t is also q u ite possible th a t the growth ra te Is as impor-
ta n t as the im purity con ten t.
3.
96
Spinel
(I)
Temperature dependence
In o rd e r to compare the measurements w ith the th e o rie s of Chapter
IV the working region must be determined I . e .
wt
> or < I.
The re la x ­
a tio n tim e T can be estim ated from the thermal c o n d u c tiv ity measurements
o f Slack^^ using th e approximate expression eqn. ( I ) .
Following
60
O liv e r and Slack^^ v Is found from the expression:
1000
^
<5,
o
Here
9^ is In
has th e
Kand v^ Is the average volume per atom In cubic A and
value 9 ,4 2 fo r s p in e l.
This gives
"v = 6 .4 x 10®
cm/s.
From
the values fo r thermal c o n d u c tiv ity in r e f . (97) and the s p e c ific heat
deduced from th e Debye expression, w ith
v a r ia tio n o f
t
8^ ~
900*K, the tem perature
Is found to be as shown in f i g .
WT =
I a t IOO*K
WT =
0 .034 a t 300®K
(1 7 ),
Now:
)
)
)
(6 )
Thus the measurements were n e arly a l l taken In the Interm ediate to
high tem perature range.
q u a n titie s F in eqn.
I t Is not possible to o b ta in values o f the
(2 ) from measurements a t the
since the a tte n u a tio n Is
too
lowesttemperatures
lav to bemeasured in the region
v a r ia tio n o f T ", n >- 4 , is expected.
where a
For comparison some ty p ic a l
a tte n u a tio n curves obtained fo r quartz are shown In f i g . (1 4 ).
Now
the arguments o f Chapter IV section 3 ( 1 ) , fo llo w in g Mason and
Bateman^^'^* have shown th a t eqn. (2 6 ) of Chapter IV Is o ften a rea­
sonable f i t
in the Interm ediate region (comparison w ith the Improved
r e la tio n eqn. (3 2 ) Chapter IV is not possible because no values o f D
are a v a ila b le fo r s p in e l, th e re being no measured th ir d order e la s t ic
constants) so the re s u lts have been compared w ith th is equation.
s t i t u t in g eqn. ( I )
In dB/cm:
in th is
Sub­
leads to an average a tte n u a tio n c o e f fic ie n t
61
2 pv®()+W*T*)
Using eqn. (7 ) and the value y = 0 .5 6 the a tte n u a tio n a t room tempera­
tu re Is about 25 dB/cm, which value Is ty p ic a l o f the re s u lts obtained.
O liv e r and Slack
49
I
have found th a t fo r most m a te ria ls / 2 '< y '< I so
th a t th e average behaviour of spinel
th a t of o th e r m a te ria ls .
is not grossly d iffe r e n t from
However, an o rd e r o f magnitude c a lc u la tio n
such as t h is can be considerably in e r ro r in e stim a tin g the attenua­
tio n o f in d iv id u a l modes.
tu re is p lo tte d In f i g .
The v a ria tio n o f a tte n u a tio n w ith tempera­
(1 8 ).
The agreement w ith experiment Is not
p a r t ic u la r ly good, the c a lc u la te d a tte n u a tio n dropping to h a lf It s
room tem perature value a t about IOO*K as opposed to the observed
values of about I60°K , see fig s . (1 4 ), (15) and (1 6 ).
The reason
fo r t h is could be th a t the specimens o f spinel used had higher th e r­
mal c o n d u c tiv itie s than S lac k ’ s Impure natural specimens.
In order
to account fo r th e discrepancy the values o f re la x a tio n tim e would
need to be about fo u r times higher a t I60®K In the spinel used than in
S la c k ’ s specimen.
Such a large d iffe re n c e seems u n lik e ly a t the
tem peratures involved.
For completeness the tem perature v a r ia tio n o f
a tte n u a tio n is a ls o p lo tte d in f i g .
(18) using the v a r ia tio n o f eqn.
(26) Chapter IV due to Mason and Bateman
in which
is replaced by
T
E = /
CdT.
This is seen to give s lig h t ly improved agreement with
o
experim ent in the region wT < I .
In the case o f lo n g itu d in a l waves a
fu r th e r small improvement between theory and experiment can be obtained
by using in eqn. (2 6 ) Chapter IV a re la x a tio n tim e equal to tw ice the
62
thermal re la x a tio n as suggested in r e f.
(46) and discussed in Chapter
I Y o f th is th e s is .
Thus th e re s u lts are only In q u a lit a t iv e agreement w ith th e ex­
pression fo r a re la x a tio n mechanism when wT < I , but are tending
towards the T* v a r ia tio n expected from the Landau-Rumer Mechanism as
OJT becomes »
(il)
I.
Dependence o f the room tem perature a tte n u a tio n on the
u ltra s o n ic mode
Id e a lly , the c a lc u la tio n o f the a tte n u a tio n of In d iv id u a l u l t r a ­
sonic modes would be made through the t h ir d order e la s t ic constants.
However, In the absence o f any measured th ir d order e la s t ic constants
o f s p in e l, a s u ita b le comparison o f re s u lts w ith theory can be made
with an e f f e c t iv e v is c o s ity damping approach.
approach i t
By adopting th is
is possible to check th a t the measurements on the seven
pure mode u ltra s o n ic waves are co n sisten t with th e symmetry of the
e f f e c t iv e v is c o s ity m a trix .
Lamb and R ic h te r
26
showed th a t (on the
assumption th a t th e a tte n u a tio n is caused by a re la x a tio n mechanism)
th e a tte n u a tio n of I GHz phonons in quartz and s ilic o n a t room tem­
p eratu re Is given by;
a
=
8 .6 8
2 p V*
(wT < I )
(8 )
where a Is the a tte n u a tio n in dB/cm, q is an e ffe c tiv e v is c o s ity (see
a ls o r e f .
(1 6 )) and v is the v e lo c ity o f the wave under consid eratio n .
63
n .ji^ j
is th e general v is c o s ity te n s o r and Is r e la te d to th e e f f e c ­
t i v e v is c o s it y q , f o r
n
mode m, by th e r e la t io n :
=
where r Is th e wave normal and th e sum Is o v er I , j ,
(no sum o v er m),
(9 )
q j j k j cos ( r , J ) cos ( r j )
k,
I = 1,2,3
u. and u^ are th e d ir e c tio n cosines f o r p a r t i c l e d is ­
placem ent.
T h is eq u atio n is taken from th e Lamb and R ic h te r pap er,
re f.
Symmetry o f th e v is c o s it y m a trix is determ ined by th e c ry s ­
(2 6 ).
t a l c la s s and Is th e same as t h a t o f th e e l a s t i c c o n stan t m a tr ix .
Thus, s in ce s p in e l
is c u b ic th e r e a re o n ly th re e independent components
In th e v is c o s it y m a tr ix ,
V o ig t n o ta tio n ) .
i.e . qj j ,
q^^ and q^^ ( i n th e c o n tra c te d
The form o f th e m a trix is :
ze ro component
n on-zero component
equaI components
Using eqn.
(9 ) th e e f f e c t i v e q valu es f o r th e seven pure modes s tu d ie d
here may be c a lc u la te d .
They a re lis t e d
in t a b le
IV to g e th e r w ith
o th e r p e r tin e n t in fo rm a tio n .
The f i r s t o b s e rv a tio n to note Is t h a t th e a tte n u a tio n s o f th e two
modes w ith very Iow a tte n u a tio n s ,
(b ) and ( f ) ,
a re both c o n tr o lle d by
64
o n ly .
044
A 11 th e o th e r modes have h ig h e r a tte n u a tio n s and a l l these
c o n ta in q | |
q jj
and some n j 2 ‘
is c o n s id e ra b ly
(b ),
(f)
I "I" Is concluded t h a t
la r g e r .
Is very sm all and
In f a c t from th e a tte n u a tio n s o f modes
and (d ) th e fo llo w in g have been deduced (cP = c e n tip o is e )
=
r ) |, - n , 2
Although I t
0 .0 7 cP - \o%
( 10)
= 0 .1 8 cP i 15:?
(II)
is n o t p o s s ib le to deduce very a c c u ra te values o f q^ ^ and
q j 2 * reasonable e x tr a p o la tio n s o f th e lo n g itu d in a l wave a tte n u a tio n
curves to room te m p e ratu re g iv e c o n s is te n t r e s u lts
if:
q, ^
=
0 .6 1 cP
(1 2 )
q ,2
= 0 .4 5 cP
(1 3 )
W hile th e s e values are reasonably c o n s is te n t,
in view o f th e d i f f i c u l ­
t i e s encountered w ith th e te m p eratu re dependence o f th e a tte n u a tio n ,
it
Is not j u s t i f i a b l e to use eqn. ( 8 ) unless I t can be v e r i f i e d t h a t
th e a tte n u a tio n v a r ie s as w* a t room te m p e ratu re (see eqns.
(8 )).
t h is
(7 ) and
The o n ly o th e r equipment a v a ila b le o p e rates a t I GHz.
At
frequency th e ap p aren t a tte n u a tio n o f th e lo n g itu d in a l modes
(u sin g CdS tra n s d u c e rs ) is dominated by phase c a n c e lla tio n e f f e c t s * ^
(see f i g .
( 3 ) a ) and
is to o sm all to be measured b u t does not c o n f l i c t
w ith th e v alu e s o f a tte n u a tio n
( 0 .2 to 0 .4 dB/cm) expected from th e
r e s u lts a t 9 GHz assuming an w* dependence.
The measurements a re
n ot c o n s is te n t w ith
ano> dependence o f th e a tte n u a tio n which would
g iv e an a tte n u a tio n
o f 2 to 4 dB/cm f o r th e lo n g itu d in a l modes.
Is th e r e fo r e j u s t i f i e d to use eqn.
( 8 ) b u t measurements a t 3 GHz
It
65
would be u s efu l to v e r i f y th e frequency dependence more a c c u ra te ly .
A few general
s p in e l.
remarks w i l l
now be made on th e a tte n u a tio n
F i r s t l y th e o b s e rv a tio n s in f i g s .
(1 4 ),
in
(1 5 ) and (1 6 ) on th e
a tte n u a tio n s o f v a rio u s phonon modes are in broad agreement w ith th e
p r e d ic tio n s o f C ic c a r e llo and D ra n s fe ld *^ and
< I0 0 > a x is th e a tte n u a tio n o f tra n s v e rs e waves Is
lo n g itu d in a l waves, w h ile on a < l l l >
a x is
Thus on a
D ra n s fe ld ^ ^ ,
less than t h a t o f
(which lacks even fo ld sym­
m e try) th e a tte n u a tio n o f tra n s v e rs e waves Is s im ila r to t h a t o f lo n g i­
tu d in a l waves.
Secondly, th e
low u ltr a s o n ic a tte n u a tio n
not e n t i r e l y unexpected from th e work o f O liv e r and S lack
workers suggested t h a t In th e
pected in m a te ria ls w ith
(a )
wt
(1 4 ),
These
is ex­
these accounts
low a tte n u a tio n a t room tem­
i f any, th e Cr^^ im p u r itie s
is s tre ss e d t h a t th e measurements in
(1 5 ) and (1 6 ) were taken on d i f f e r e n t c r y s t a ls , but a l l
w ith Cr^^ c o n c e n tra tio n s o f < 0.1% .
it
On a l l
is no t known what in flu e n c e ,
have on th e measurements but I t
fig s .
.
is
l i g h t mass atoms, (b ) high Debye tempera­
s p in e l would be expected to have a f a i r l y
It
49
< I regio n a low a tte n u a tio n
tu r e s , and (c ) complex c r y s ta l s tr u c tu r e s .
p e ra tu re .
in s p in e l
According to O liv e r and S lack*^
is p o s s ib le t h a t any Im p u r itie s would reduce th e tem peratu re-d ep en ­
dent u ltr a s o n ic a tte n u a tio n by reducing th e r e la x a tio n tim e .
66
4.
Y ttr iu m
(i)
Iro n g a rn e t
Tem perature dependence
A gain, f o r purposes o f comparison o f measurements w ith th e th e o r ie s
o f C h ap ter IV , th e r e la x a tio n tim e was determ ined from th e c o n d u c tiv ity
r e la t i o n eqn. ( I ) .
0p = 560®K, as computed from th e e l a s t i c c o n s ta n ts ,
was used to d eterm ine th e s p e c if ic h e a t from th e Debye e x p re s s io n .
The
average therm al phonon (group) v e lo c it y v is g iv en by:
V
=
{ 1 (— -----------L _) } - 1 / 3
(14)
I /O
and has th e v alu e 4 .3 . 10® cm/s f o r Y IG , where v^ = (C ^ ^ /p )
v^ = ( C | | / p )
in Y IG .
I /2
and
a re th e v e lo c it ie s o f tra n s v e rs e and lo n g itu d in a l waves
Values o f th e therm al c o n d u c tiv ity a re taken from th e work o f
O liv e r and S lack
49
and th e r e s u ltin g v a r ia t io n o f r e la x a tio n tim e w ith
tem p eratu res is p lo tte d
in f i g .
(2 2 ).
The c o n d itio n tax = I occurs a t
about 67®K f o r 9155 MHz phonons.
B efore comparing th e computed and measured a tte n u a tio n curves fo r
lo n g itu d in a l waves, allow ance must be made f o r th e low tem peratu re
a tte n u a tio n peak, which can be seen in f i g .
phonon-phonon a tte n u a tio n .
(1 9 ) superimposed on th e
T h e re fo re , th e peak B has been s u b tra c te d
from th e curve A, a s im ila r procedure having been used by Spencer e t
a l.^ ^ to o b ta in th e i n t r i n s i c fe r rim a g n e tic resonance lin e w idth In Y IG .
For tem p eratu res above about 50®K th e c o rre c te d curve C is n o t p a r tic u ­
l a r l y s e n s itiv e to d e t a ils o f curve B (which Is a r b i t r a r i l y drawn as
sym m etrical on a lo g -lo g p lo t ) p ro v id in g th e peak does n o t occur a t a
57
te m p e ratu re h ig h e r than about 40®K.
curve A suggests t h a t t h is
Is
The form o f th e e xp e rim en tal
l i k e l y t o be th e case, th e most probable
lo c a tio n o f th e maximum being about 30®K.
The low te m p e ratu re t h e o r e t ic a l curve D In f i g .
(1 9 )
Is a p lo t o f
eqn. (2 ) w ith Fj = 0 (no Landau-Rumer mechanism) and F j = 3 C jj t C | | |
~ - 1 5 .2 X 10** dyne/cm *, v = 7.21 x 10® cm/s and n e g le c tin g th e e f f e c t s
o f d is p e rs io n
i . e . w ith th e term in square b ra ck e ts s e t equal to n / 2 .
The curve deduced from th e f u l l e xpression in eqn.
( 2 ) , using th e v alu e
o
L = 12.4 A f o r YIG and th e r e la x a tio n tim es in f i g .
(2 2 ),
lie s even
c lo s e r to curve C than does curve D but Is o m itte d from f i g .
avoid c o n fu s io n .
(1 9 ) to
In e i t h e r case th e agreement Is q u ite t o le r a b le .
The high te m p e ratu re t h e o r e t ic a l curve E Is d e riv e d from th e work o f
Mason and Bateman
46
I.e .
a
=
8 . 6 8 E (D /3 )w *2 T
---------
(1 5 )
2 p v ^ * ( I + 4 w* t * >
T h is is eqn. (3 2 ) o f C hapter IV and a is in dB/cm.
Twice th e therm al
phonon r e la x a t io n tim e Is used s in c e th e wave under c o n s id e ra tio n Is
lo n g itu d in a l.
D is g iv en th e v a lu e 5 .9 6 and Is an average f o r i n t e r ­
a c tio n s w ith therm al phonons o f d i f f e r e n t wave v e c to rs and p o la r iz a ­
tio n s
in t a b le V o f r e f .
(4 6 ).
The agreement between th e o ry and ex­
p e rim e n t is to w ith in a fa c t o r o f about tw o.
The e xp e rim en tal and t h e o r e t ic a l curves f o r tra n s v e rs e waves pro­
p ag atin g on a < I0 0 > a x is in YIG a re shown in f i g .
(2 1 ).
The low tem -
p e ra tu re th e o r e tic a l curve Is d e riv e d from th e work o f Pomerantz
21
using th e fo llo w in g exp re s sio n d e riv e d from th e Landau-Rumer mechanism:
68
01
=
.l* -34ïïk—
80pfi*v^®
y 2 [| -
(v ./v .)* ]w T *
^
where a Is in dB/cm and y = ( C jj + C jg +
(1 6 )
+ 4 C |g g ) /2 C ||.
C le a r ly
th e th e o ry u n derestim ates th e a tte n u a tio n by an o rd e r o f m agnitude.
A s im ila r discrepancy between th e o ry and experim ent has been found f o r
Si and Ge ( r e f .
in f i g ,
(2 1 )
(2 1 ) t a b le
I).
The high te m p e ratu re t h e o r e t ic a l curve
is d e riv e d from th e work o f Mason and Bateman using eqn.
(1 5 ) w ith 2 t in th e numerator and denom inator rep laced by th e o r ig in a l
T and v^ re p la c ed by v^.
The th e o r e tic a l valu e o f D is 0 .2 6 ( r e f .
(4 6 ) ta b le V ) .
A t high
tem p eratu res th e th e o ry underestim ates th e a tte n u a tio n by a fa c t o r of
two to th r e e .
T h is
Is th e
la r g e s t discrepancy in th e high tem p eratu re
reg io n t h a t has been noted .
The p o s s i b i l i t y cannot be c o m p lete ly ru le d
o u t t h a t some spurious e f f e c t s add t o th e phonon-phonon a tte n u a tio n , but
th e general form o f th e e xp e rim en tal curve In f i g .
usual fe a tu r e s .
For exam ple.
( 2 1 ) shows no un­
I f th e a tte n u a tio n curve were dominated
by losses in a small spin wave adm ixture In to th e e l a s t i c wave I t m ight
be expected t o v ary lin e a r l y w ith te m p e ra tu re .
99
T h is suggests t h a t
th e i n t r i n s i c a tte n u a tio n has been measured.
(il)
Frequency dependence
Concerning th e frequency dependence o f th e a tte n u a tio n a t room
te m p e ra tu re , th e o ry p re d ic ts t h a t t h is
(7 ),
( 8 ) and ( 1 5 ) .
Is p ro p o rtio n a l to w *, see eqns.
The upper l i m i t o f 0 ,3 4 dB/cm a t I GHz*^ Is rough I
in agreement w ith th e measurements made I f a is p ro p o rtio n a l to w *.
In
69
th e case o f
lo n g itu d in a l waves, a comparison o f f i g s .
(1 9 ) and (2 0 )
shows t h a t th e a tte n u a tio n a t low tem peratures (4 0 -5 0 *K ) v a r ie s as
w ith n = I .
T h is Is expected from eqn, (2 ) w hether th e e f f e c t s o f d is ­
p e rs io n a re included o r n e g le c te d .
tio n
is about 1.7 dB/cm.
A t room tem p eratu re th e a tte n u a ­
The 9 GHz curve e x tra p o la te d to room tem­
p e ra tu re g iv es an a tte n u a tio n o f about 50 dB/cm, which in d ic a te s t h a t
th e a tte n u a tio n v a rie s as w" w ith n = 1.5 in stead o f 2 .
The d e p a rtu re
from an w* dependence a t room tem p eratu re has been observed in a few
o th e r m a te r ia ls , see f o r example r e f .
(
100),
but is not understood a t
p re s e n t.
5.
The I GHz a tte n u a tio n peak in YIG
The behaviour o f th e sharp peak in th e
a tte n u a tio n o f I GHz lo n g i­
tu d in a l waves as a fu n c tio n o f te m p e ra tu re ,
observed in two < I0 0 >
two < l l l > YIG specimens, has been described
in C hapter I I I and is sum­
m arised in f i g .
(2 4 ).
and
To e x p la in th e a tte n u a tio n peak v a rio u s mech­
anisms have been t r i e d many o f which a re review ed by Le Craw and Corns to c k .^ ^
The o n ly mechanism which i t
Is b e lie v e d can account f o r th e
o b s e rv a tio n s is in te r a c tio n s w ith domain w a lls I . e .
because o f th e
m a g n e to s tric tiv e e f f e c t th e phonon d riv e s th e domain w a lls whose
motion is v is c o u s ly d a m p e d . Q u a l i t a t i v e l y
many o f th e e f f e c t s
observed in n ic k e l s in g le c r y s ta l a re s im ila r to those observed here
in YIG and these a re u s u a lly a t t r ib u t e d to domain w a lls .
peak in f i g .
(2 3 )
Thus, th e
is q u a l i t a t i v e l y s im i l a r t o , bu t much s h arp er th a n ,
t h a t observed f o r n ic k e l a t lower fre q u e n c ie s .*^ ^
On a p p ly in g a mag-
70
n e tic f i e l d th e excess a tte n u a tio n sometimes decreases m onotonica 11 y
105
and sometimes peaks one o r more tim es b e fo re d e c r e a s i n g , d e p e n d ­
ing on th e te m p eratu re and m agnetic f i e l d o r ie n t a t io n , see f i g .
(2 4 ).
Taborov’ s s u ggestion*^^ t h a t fe rro m a g n e tic resonance m ight be respon­
s ib le
is n o t v a lid s in ce th e fe rro m a g n e tic a tte n u a tio n peak has been
re so lv ed a t a l l
Schlomann.
'
tem p eratu res in agreement w ith th e th e o rie s o f
(See a ls o pp. 2 9 - 3 0 .)
The p o s s i b i l i t y has been considered t h a t th e a tte n u a tio n peak is
caused by a r e la x a tio n mechanism w ith a v a ry in g as w * t / ( I+ ü j* t* ) , where
T
=
exp (A ’ /k T )
is some r e la x a tio n tim e e .g . t h a t f o r e le c tro n s to
hop between Fe^* and im p u rity Fe^^ io n s .*^ *
f o r th e shape o f th e peak in f i g .
assume th e u n u s u a lly
(2 3 )
However, to account
i t would be necessary f o r A’ to
larg e valu e o f about 2eV.
In a d d itio n one would
3+
2+
exp e c t a d i r e c t in te r a c tio n between th e phonons and th e Fe /F e
I QO
io n s .
T h e re fo re , i t seems probable t h a t th e a tte n u a tio n peak is
caused by in te r a c tio n w ith domain w a lls , but th e mechanism is not a t
a l I c le a r .
71
CHAPTER VI
AMPLIFICATION OF 9 GHz WAVES IN GaAs
I.
In tro d u c tio n and th e o ry
In
1961 Hutson, McFee and W hite
50
observed s u b s ta n tia l a m p lif ic a ­
t io n o f u ltr a s o n ic waves in photoconductIve CdS.
a p p lyin g a d i r e c t c u rre n t e l e c t r i c f i e l d
p a g a tio n .
T h is was produced by
in th e d ir e c tio n o f wave pro­
T h e ir r e s u lts show g ains o f 26 dB/cm a t 15 MHz and 54 dB/
cm a t 45 MHz.
L ig h t s e n s itiv e u ltr a s o n ic a tte n u a tio n r e s u lts from th e
In te r a c tio n o f m obile charge c a r r ie r s w ith th e stro n g
lo n g itu d in a l
e l e c t r i c f i e l d s o f p ie z o e le c t r ic o r ig in accompanying c e r t a in a c o u s tic
waves in c r y s t a ls .
Thus a c o u s tic g a in could be achieved by causing
th e in te r a c tin g charge c a r r ie r s to d r i f t
in th e d ir e c tio n o f wave pro­
p ag atio n f a s t e r than th e sound v e lo c it y in th e c r y s t a l . F i g .
(2 7 )
i l l u s t r a t e s t h e i r r e s u lts which agree semi q u a n t it a t iv e I y w ith th e
th e o r ie s o f Hutson and W hite*^^ and W h i t e . T h e s e
th e o r ie s o b ta in
an exp ressio n f o r th e a m p lif ic a tio n o f an u ltr a s o n ic wave by s t a r t in g
from th e Gauss e q u a tio n , th e e q u atio n o f c u r r e n t c o n t in u it y , and th e
e q u a tio n f o r p ropagation o f an e l a s t i c wave in a p ie z o e le c t r ic medium.
B e tte r agreement has been o b ta in e d by ta k in g
in to account th e e f f e c t s
o f c a r r i e r tra p p in g on th e in t e r a c t i o n . * * *
A q u a l i t a t i v e p ic tu r e o f th e e f f e c t may be o b ta in e d In th e f o llo w ing manner.
1 12
Assume t h a t in th e absence o f an e l e c t r i c f i e l d e le c ­
tro n s absorb per u n it tim e an amount o f a c o u s tic energy A^ given by:
A^
o
=
n -tr Ü)
(I)
72
where n is th e number o f phonons absorbed per u n it tim e .
A bsorption
o f a phonon means t h a t i t s momentum hq is tr a n s fe r r e d to th e e le c tr o n .
The v e lo c it y o f th e e le c tr o n
is a lt e r e d by an amount-trq/m where m is
th e e f f e c t i v e mass o f th e e le c tr o n .
c o llis io n s
(d u rin g tim e
t
The mean displacem ent A between
) is equal to (tiq /m )x .
S ince d u rin g u n it
tim e n a b s o rp tio n s ta k e p la ce th e c u rre n t g enerated is :
J
=
- nei
=
where e is th e e le c t r o n ic charge.
(2 )
m
I f a steady e l e c t r i c f i e l d E is
imposed th e a c o u s tic energy absorbed per u n it tim e by an e le c tr o n w i l l
be:
A
=
A - JE
o
-
(3 )
Thus:
A
=
nftu { I + S S lI}
mo)
The e le c tr o n d r i f t v e lo c it y
=
A
{l+ S iH }
o
( 4)
mw
in th e e x te rn a l f i e l d E Is given by:
. (S'
So eqn.
(4 ) becomes:
v .q
A
It
=
A
{I
O
is c le a r from t h i s t h a t i f
-
(6 )
Ü)
is p a r a lle l to q and
> v (v = w/q
th e v e lo c it y o f sound) th e energy o f e l a s t i c v ib ra tio n s o f th e l a t t i c e
w ill
in c re a s e .
There a re many e f f e c t s r e la te d to t h i s phenomenon,**^
and th e s u b je c t has been review ed by Chaban
n .H .B .N .C .
114
amongst o th e rs .
113
73
Although exp erim en tal
in v e s tig a tio n s o f th e a m p lif ic a tio n phenomena
have been p r im a r ily r e s t r ic t e d to group l l - V I
in te r a c tio n
in l l l - V
compounds such as CdS th e
compounds has a ls o been In v e s tig a te d .
T h is has
shown th e a c o u s tic g a in p o te n tia l o f GaAs to be good d e s p ite I t s
p ie z o e le c t r ic c o u p lin g c o n s ta n t, and gains o f
in t h is m a te ria l a t 90 M H z.**^
low
10 dB/cm have been achieved
In a d d itio n GaAs re p res e n ts a commer­
c i a l l y a v a ila b le m a te ria l t h a t could be o b ta in e d in la rg e s in g le c r y s ta l
form w ith a p p a re n tly good e le c tr o n ic p ro p e rtie s f o r in v e s tig a tio n o f th e
e le c tr o a c o u s tic g a in in te r a c tio n a t 9 G H z.**^
e le c tr o n s
*
In p a r t i c u l a r th e
In GaAs can have a very high m o b ilit y a t low te m p e ra tu re s .
T h is high m o b ilit y
Is p a r t ly due to th e
low e f f e c t i v e mass o f th e e le c ­
tro n s but th e moderate p ie z o e le c t r ic co u p lin g a s s is ts as w e ll s in c e .
If
la r g e r , t h i s would decrease th e m o b ility by in c re a s in g p ie z o e le c t r ic
s c a t t e r in g .
Also th e photoconductive p r o p e rtie s o f GaAs a llo w tr a n s ­
m ission o f microwaves in th e absence o f illu m in a t io n .
Thus th e i n i t i a l
adjustm ents In any pulse echo e xperim ent a re f a c i l i t a t e d a f t e r which
illu m in a tio n may be used to a s s is t a m p lif ic a t io n .
For th e above re a ­
sons i t was decided to a tte m p t a m p lif ic a tio n o f 9 GHz waves in GaAs.
2,
E xperim ental d e t a ils and r e s u lts
The GaAs used In t h i s e xperim ent was an in g o t o b ta in e d from th e
Monsanto Chemical C o rp o ra tio n possessing according to t h e i r s p e c if ic a ­
tio n s a r e s i s t i v i t y o f th e o rd e r o f 9 0cm, a m o b ilit y o f 9 0 0 0 cm */V .s
and a c a r r i e r c o n c e n tra tio n o f about
p e r a tu r e .
T h is was c u t f o r
1 0 **
atoms/cm®, a l l a t room tem­
lo n g itu d in a l wave p ropagation along th e
74
< 1 1 l> d ir e c tio n to w ith in and about
8
mm by
(A /IO o f v i s i b l e
8
1 /2 ° and th e ends o f a p ie ce 19.3 mm long
mm in cross s e c tio n were p o lis h e d o p t i c a l l y f l a t
l i g h t ) and p a r a l l e l to b e t t e r than 4 sec. o f a r c .
From t h i s specimen f u r t h e r specimens were c u t having th e same length
and 1.5 mm by 3 .2 mm in cross s e c tio n .
Although i t would be p o s s ib le to use a s in g le 9 GHz c a v ity appara­
tu s f o r th e e x p e rim e n t, s in ce any gain in one d ir e c tio n would no t nec­
e s s a r ily be e x a c tly n u l l i f i e d by loss in th e o p p o s ite d ir e c t io n , a
tra n s m is s io n ty p e apparatus was c o n stru cted f o r ease o f o b s e rv a tio n .
T h is is i l l u s t r a t e d
in f i g .
( I ) d , th e g e n e ra tio n and d e te c tio n o f th e
u ltr a s o n ic waves being as described in C hapter 11 f o r 9 GHz waves.
I n i t i a l l y experim ents were c a r r ie d o u t to see i f th e GaAs specimen
could be used as i t s own tra n s d u c e rs .
fig .
However, using th e s e t up o f
( I ) a o n ly one very sm all echo could be observed a t helium tem pera­
tu re s so q u a rtz rods ( I cm long and 3 mm in d ia m e te r) placed In th e
microwave c a v it ie s were used as tra n s d u c e rs .
These were bonded to
th e GaAs specimen w ith a r a l d i t e adhesive and th e assembly mounted as
in f i g .
(I)d .
In th e regio n 4 -3 5 °K an u ltr a s o n ic wave e x c ite d a t th e
s u rfa c e o f one o f th e q u a rtz rods is tra n s m itte d through th e assembly
(q u a rtz rod-G aA s-quartz r o d ), d e te cted as a v o lta g e pulse in th e
re c e iv in g c a v ity and a f t e r a m p lif ic a tio n d is p la y e d on an o s c illo s c o p e .
F ig s .
fig .
(2 8 ) and (2 9 ) show one o f th e c a v it ie s w orking in r e f l e c t io n and
(3 0 ) what is observed in tra n s m is s io n .
guished below each photograph.
The echoes a re d i s t i n ­
In general a one p s . pulse was used
bu t t h i s could be reduced i f b e t t e r r e s o lu tio n was re q u ire d .
In f i g .
75
(3 0 )
i t was f a i r l y c e r t a in t h a t some echoes t h a t had tra v e rs e d q u a r tz -
G aA s-quartz could be seen as w e ll as e x tr a echoes due to p ic k up.
A ll th ese echoes p e rs is te d up to about 3 5°K , a t which tem p eratu re
a tte n u a tio n
GaAs.
in th e q u a rtz was to o high to a llo w tra n s m is s io n in to th e
Since th e v e lo c it y o f u ltr a s o n ic waves in x -c u t q u a rtz Is
5 .7 X 10® cm /sec, th e v e lo c it y o f u ltr a s o n ic waves in GaAs can be c a l­
c u la te d from f i g s .
(2 8 ),
(2 9 ) and ( 3 0 ) .
The r e s u lt is 5 .3 x 10® cm/
sec, which agrees w e ll w ith p re v io u s ly measured v a lu e s .
GaAs is photoconducting so,
120
in o rd e r t h a t more m obile c a r r ie r s
m ight be produced to enhance o r make p o s s ib le a m p lif ic a tio n when a
d r if t fie ld
sample.
is a p p lie d , p ro v is io n was made f o r illu m in a tio n o f th e
T h is was done by means o f a l ig h t p ip e made o f q u a rtz
ing from th e sample to th e e x t e r i o r o f th e a p p a ra tu s .
o r a sm all to rc h were used as sources.
In itia lly
lead­
A 100 W. bulb
th e e f f e c t s o f i l l u ­
m in a tio n a t 4.2®K w ith th e apparatus w orking In tra n s m is s io n and re ­
f l e c t io n were observed.
of fig .
<28)a and s tro n g ly a tte n u a te th e second GaAs echo and a re
shown in f i g .
th e
100
These were t o a tte n u a te th e f i r s t GaAs echo
(2 8 )b .
The a tte n u a tio n produced was about I dB/cm w ith
W. bulb and about tw ic e t h is w ith th e sm all to rc h more accur­
a t e ly p la c e d .
W ith th e tra n s m is s io n arrangem ents th e
l i g h t a tte n u a te d
th e q u a rtz-G a A s -q u a rtz echoes as expected b u t sometimes th e p ic k -u p
echoes were a ffe c te d as w e l l , p o s s ib ly due to reduced e le c tro m a g n e tic
co u p lin g between th e c a v it ie s when th e GaAs becomes c onducting.
During th e above t e s ts an unusual e f f e c t was observed;
t h is was
t h a t th e a tte n u a tio n produced by Illu m in a tio n appeared t o decrease w ith
76
tim e .
In o rd e r to
In v e s tig a te t h is
o f Illu m in a tio n was used.
In some measure a s tro n g e r source
T h is was a sm all
th e apparatus n e xt to th e sample.
lig h t bulb placed in s id e
The r e s u lt a t 4.2®K was a marked
In crease in th e a tte n u a tio n o f any echoes due in p a r t to passage
through th e GaAs, b u t th e e f f e c t was h e a v ily tim e dependent,
a p p ro xim a te ly th r e e m inutes.
la s tin g
I f th e sample was re tu rn e d to room tem­
p e ra tu re and then back to 4 .2 *K th e e f f e c t o f any illu m in a tio n was
re s to re d bu t s t i l l
w ith a tim e dependence.
These o b s e rv a tio n s were
th o ug ht to be due to th e presence o f tra p s in th e GaAs but s in c e , a t
le a s t f o r some p e rio d o f tim e , an a tte n u a tio n
increase could be achieved
by illu m in a t io n , th e p o s s i b i l i t y o f a m p lif ic a tio n by use o f a d r i f t
f i e l d was not ru le d o u t.
The d r i f t f i e l d was a p p lie d to th e sample v ia ohmic c o n ta c ts a t
each end.
gold t i n
These were made by vacuum d e p o s itin g
a llo y
(r a tio
I mm wide s t r ip s o f a
19:1) about each end o f th e GaAs specimen a t a
te m p e ratu re o f 300°C , then h e a tin g th e specimen in an i n e r t atmosphere
to 550°C , causing d iffu s io n o f th e a llo y
in to th e specimen.
Gold
w ires were connected to th e depo sited su rfa ce s using g o ld -lo a d e d a r a l d it e and th e d r i f t pulse source in tu rn Joined to th e s e .
was a 3 ys pulse a t a r e p e t it io n
The source
r a te o f 1000 per s . and i t s v o lta g e
(maximum o f 5 kV) and tim e d e lay w ith re s p e c t to th e i n i t i a l microwave
p ulse could be s e p a ra te ly c o n tr o lle d .
P re lim in a ry measurements o f th e
r e s i s t i v i t y o f th e specimen now in d ic a te d i t to be much h ig h e r than
s p e c if ie d , p o s s ib ly
c u lt.
10,000
J^cm which would make a m p lif ic a tio n more d i f f i ­
77
The tem p eratu re o f th e sample was lowered to 4.2®K and then a!low ed
to r is e t o about 30®K a t which te m p e ratu re th e c o n d u c tiv ity should be
g r e a t e s t.
121
However, although Illu m in a tio n o f th e sample y ie ld e d
Increased a tte n u a tio n f o r a p e rio d of tim e , a p p lic a tio n o f th e d r i f t
fie ld
pulse during t h i s tim e , such t h a t I t overlapped th e t r a n s i t tim e
o f th e f i r s t microwave pulse in th e GaAs, produced no v i s i b l e e f f e c t
upon th e echo.
T h is procedure was c a r r ie d o u t a t a l l
tem peratu res a t
which t h i s echo e x is te d and w ith a l l v alu e s o f d r i f t f i e l d pulse up to
th e maximum b u t no changes were observed.
A f u r t h e r s e rie s o f experim ents were c a r r ie d o u t on GaAs specimens
c u t from d i f f e r e n t p a rts o f th e o r ig in a l
n e g a tiv e r e s u lt s .
In g o t, bu t s t i l l
Another specimen o f GaAs, o b ta in e d from th e C e n tra l
E l e c t r i c i t y Research L a b o ra to rie s , was a ls o used.
t h e i r sample as fo llo w s :
c a r r i e r d e n s ity
1.55 x 10^® atoms/cm.
ta in e d .
(0 . 5
They s p e c ifie d
r e s i s t i v i t y 0.091 S^cm, m o b ilit y 4 ,4 0 0 cm^/Vs,
Although th ese v alu es appear
more u s efu l as regards a c h ie v in g a m p lif ic a t io n ,
was very small
w ith th e same
119
th e specimen I t s e l f
cm long) and ag ain no p o s itiv e r e s u lts were ob­
Because of I t s high c o n d u c tiv ity no echoes were e v e r observed
when using t h i s sample.
Attem pts to make f u r t h e r measurements on th e
p r o p e rtie s o f th e o r ig in a l sample have had to be postponed because o f
damage to th e i n i t i a l
3.
p ie c e o f GaAs c u t from th e In g o t.
Conclusions
The p r in c ip a l
reason f o r f a i l u r e to a m p lify 9 GHz waves In GaAs
appears t o be th e sample used.
In p a r t i c u l a r th e d r i f t m o b ilit y may
78
be much less than th e H a ll m o b ilit y due to th e presence o f tr a p s .
tim e-d ep en d en t Illu m in a tio n e f f e c t s
The
in d ic a te t h a t tra p p in g could a l t e r
d r a s t ic a l ly any r e s u lts expected even though the m obile c a r r i e r d e n s ity
could be Increased a t le a s t f o r a s h o rt space o f tim e .
The e v id e n t
low c o n d u c tiv ity o f th e GaAs bears t h i s o u t,
in a d d itio n th e d r i f t f i e l d
e f f e c t i v e manner,
pulse was not a p p lie d in th e most
i . e . c o n ta c ts to th e end faces r a th e r than th e sid es
would be p r e fe r a b le .
I t was w h ile c o n ta c ts were being evaporated to
th e end faces t h a t one o f these was damaged.
The general conclusion Is t h a t th e GaAs used was f a r from s a t is ­
fa c to r y .
In o rd e r to
in v e s tig a te fu r t h e r i t was intended to a tte m p t
a m p lific a tio n a t a lower frequency (about 60 MHz) but th e apparatus f o r
t h is
is no longer aval lab le .
79
CHAPTER Vi I
CONCLUSIONS
The tem p eratu re-d ep en d en t a tte n u a tio n o f 9 GHz phonons o f v a rio u s
wave v e c to rs and p o la r iz a tio n s has been measured in some p e r fe c t q u a rtz
specimens.
The fin d in g s a re t h a t th e a tte n u a tio n o f slow tra n s v e rs e
phonons Is determ ined by th e Landau-Rumer mechanism, whereas th e a tte n ­
u a tio n o f f a s t tra n s v e rs e and lo n g itu d in a l phonons is determ ined by
three-phonon processes which a re o n ly allow ed as a r e s u lt o f th e f i n i t e
l i f e t i m e of th e therm al phonons.
tim e w ith te m p e ratu re r e s u lts
The ra p id v a r ia t io n o f r e la x a tio n
in an a tte n u a tio n v a ry in g f a s t e r than T**
in th e range 4 to 30®K a t 9 GHz.
The r e s u lts a re f i t t e d to th e
th e o r ie s of M a ris and S h iren and a p p ro p ria te c o u p lin g constants a re
lis te d .
The use o f r e la x a t io n tim es from therm al c o n d u c tiv ity meas­
urements appears v a l i d probably because th e l a t t e r a re a t le a s t in p a r t
determ ined by Im p e rfe c tio n s .
th e r e s t i l l
For lo n g itu d in a l waves on th e x -a x ls
remains a la rg e discrepancy In th e magnitude o f th e th e o re ­
t i c a l and e xp e rim en tal a tte n u a tio n s which cannot be accounted f o r .
Steps have been observed In th e a tte n u a tio n as a fu n c tio n o f tem­
p e ra tu re In q u a rtz specimens c o n ta in in g v a rio u s degrees o f im perfec­
tio n .
The steps occur a t 16 - 3®K and 24 - 3®K superimposed on th e
phonon-phonon a tte n u a tio n found In p e r fe c t specimens.
There Is a
p o s s ib le c o r r e la tio n between th e Iro n c o n te n t o f th e Im p e rfe c t s p e c i­
mens and th e In t e n s it y o f th e peaks.
peaks is probably a s tr u c tu r a l
The o r ig in o f th e a tte n u a tio n
r e la x a t io n mechanism o r an u ltr a s o n ic
p ho n o n -o p tical phonon in te r a c tio n .
80
S im ila r a tte n u a tio n measurements have been made on th e seven pure
u ltr a s o n ic modes which can be propagated In MgAUOi».
T h is m a te ria l
possesses lower u ltr a s o n ic losses In c e r t a in modes than in any o th e r
m a te ria l measured.
The a tte n u a tio n a t 9 GHz and room te m p e ratu re has
been found c o n s is te n t w ith a phonon v is c o s ity damping mechanism,but
th e a tte n u a tio n drops to h a lf I t s room te m p e ratu re v alu e a t I60®K
r a th e r than IOO®K as c a lc u la te d from therm al c o n d u c tiv ity measurements.
T h is may be due to th e specimen used having a h ig h e r c o n d u c tiv ity than
th e n a tu ra l s p in e l on which th e therm al c o n d u c tiv ity measurements were
made.
A t low tem p eratu res ( o)T > I ) th e a tte n u a tio n v a r ie s as T® which
is approaching th e T** v a r ia t io n exp e c te d , a t le a s t f o r tra n s v e rs e modes,
when OJT »
I.
Because o f th e very
low u ltr a s o n ic
losses in t h is
m a te ria l th e r e a l i z a t io n o f microwave d e lay lin e s a t x-band fre q u e n c ie s
and room tem p eratu res becomes p o s s ib le and u ltr a s o n ic s tu d ie s on
liq u id s can now be extended to much h ig h e r fre q u e n c ie s ,p o s s ib ly
10 GHz.
The a tte n u a tio n as a fu n c tio n o f te m p eratu re has a ls o been meas­
ured o f 9 GHz lo n g itu d in a l and tra n s v e rs e u ltr a s o n ic waves p ropagating
in a < I00 > a x is
in Y IG .
The r e s u lts agree w ith a th e o ry based on In ­
te r a c tio n s w ith therm al phonons to w ith in a f a c t o r 'O f two o r th re e a t
high te m p e ra tu re s .
T h is discrepancy is s l i g h t l y g r e a t e r than in most
m a te ria ls bu t th e shape o f th e a tte n u a tio n curve and th e frequency
dependence suggest t h a t th e I n t r i n s i c a tte n u a tio n has been measured.
Good agreement between th e o ry and e xp erim en t Is found a t low tem pera­
tu re s f o r lo n g itu d in a l waves, but th e th e o ry underestim ates th e a tte n ­
u a tio n o f tra n s v e rs e waves by an o rd e r o f m agnitude.
The agreement a t high tem p eratu res Is q u ite s a t is fa c to r y s in c e :
(I)
th e r e are no a d ju s ta b le p aram eters,
(2 )
c e r t a in approxim ations
have to be made In e v a lu a tin g com plicated averag es, (3 )
r e la x a tio n
tim es a re taken from therm al c o n d u c tiv ity measurements which vary from
specimen to specimen, and (4 )
th e Debye e x p re s s io n .
th e s p e c if ic h e at is c a lc u la te d from
I t would be u n r e a l i s t i c to e xp e c t b e t t e r a g ree ­
ment In view o f th e above, unless s e p a ra te measurements o f a l l th e
q u a n t it ie s
in v o lv e d a re made on th e specimen used f o r a tte n u a tio n
measurements.
The la r g e r d is c re p a n c ie s a t low te m p e ra tu re s , p a r t i ­
c u la r ly f o r tra n s v e rs e waves, a re d i f f i c u l t to account f o r ;
a c a lc u la t io n
c e r t a in ly
li k e t h a t o f Simons f o r a l l WT, but w ith no a d ju s ta b le
p a ram eters, should be a tte m p te d , s in ce th e tra n s v e rs e th e o r ie s a t high
and low tem p eratu res a re in c o m p a tib le .
W hile checking th e frequency dependence in th e <iOO> d ir e c tio n
in
YIG by measurements a t I GHz an a tte n u a tio n peak was observed a t t h is
fre q u e n cy .
and two < l l l >
T h is was found to e x i s t In two < I 0 0 > specimens a t 227®K,
specimens a t 260®K.
The peak has th e same shape In th e
two cases and could be removed by a p p lic a tio n o f a m agnetic f i e l d .
In th e re g io n o f th e peak th e a tte n u a tio n as a fu n c tio n o f f i e l d was
specimen dependent perhaps peaking s ev e ra l tim es b e fo re d e cre a s in g ,
depending on th e te m p e ratu re and f i e l d o r ie n t a t io n .
T h is peak Is
tho u g h t t o be caused by in te r a c tio n w ith domain w a lls b u t th e mechanism
remains obscure.
Attem pts were made to a m p lify 9 GHz waves in GaAs.
The reasons
f o r o b ta in in g no p o s itiv e r e s u lts w ith t h i s a re a t t r ib u t e d to th e p ro -
82
p e r t îes o f th e GaAs used being f a r d if f e r e n t from expected.
I t can be seen c le a r l y t h a t th e tre a tm e n t o f th e a b so rp tio n o f
u ltra s o u n d In d i e le c t r i c s given In C hapter IV e x p la in s very w e ll th e
e xp e rim en tal o b s e rv a tio n s although
in some cases not enough is known
about th e m a te ria l t o e s tim a te b e t t e r than o rd e rs o f m agnitude.
s o lid s ta te d e la y
s u lte d
The
lin e s o p e ra tin g a t microwave fre q u e n c ie s t h a t have r e ­
from such e x p e rim e n ta tio n have many advantages o v er co n ven tio n al
d elay lin e s , p a r t i c u l a r l y
in s iz e and perform ance.
Such a device is
being p aten ted by th e G eneral E l e c t r ic Co. as a r e s u lt o f th e measure­
ments re p o rte d here t h a t were made on s p in e l.
Microwave measurements p ro v id e an e x c e lle n t to o l
f o r th e in v e s t i­
g a tio n o f p r o p e rtie s o f m a te ria ls p a r t i c u l a r l y w ith regard to normal
processes and a c le a r p ic tu r e
m a te ria l f o r low u ltr a s o n ic
is emerging o f th e n a tu re re q u ire d o f a
losses (see C hapter V ).
wave a m p lif ie r such as t h a t discussed in C hapter V I ,
A tr a v e llin g
I f t h i s could be
s u c c e s s fu lly operated a t GHz fre q u e n c ie s , would o b v io u s ly prove v e ry
u s e fu l.
o f th e
These developments appear p o s s ib le In th e near fu tu r e because
in te n s iv e research a t p re s e n t being conducted In to c ry s ta l grow­
ing te c h n iq u e s .
83
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39.
The o n ly s e rio u s e xc e p tio n noted Is th e measurements a t 500 MHz
o f Nava e t a l .
(R e f.
1 7 ), v/ho fin d
8
, 354 (1 9 63 )
2913 (1 9 64 )
1055 (1929)
( F ig . 5 o f r e f .
below IO*K In Im p e rfe c t a c -c u t q u a r tz .
It
17) a step
Is q u ite p o s s ib le
t h a t th e r e a re steps a t tem peratu res below I5*K b u t a t 9 GHz
th e accuracy Is I n s u f f i c ie n t to p o s it iv e ly
Id e n t if y th e s e .
However, In some samples th e re is evidence o f a shallow In th e
a tte n u a tio n a t about
8 *K;
th is
Is In d ic a tiv e o f an o th er step o r
peak a t a low er te m p e ra tu re .
90.
ANDERSON, Û .L . and
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103. WANAS, M.A.
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140 (1 9 53 )
1573 (1 9 54 )
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Zh. EKSP. I THEOR F IZ . 4 3 , 4 (1 0 )
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IEEE TRANS. S .U . J ^ , 73 (1 9 66 )
117. BEALE, J .R .A .
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M B . BEALE, J .R .A . and
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»PHYSICS OF l l l - V
WILEY (1964)
1548 (1 9 6 3 )
Ml
(1 9 6 4 )
COMPOUNDS’ NEW YORK,
90
INDEX OF SYMBOLS EMPLOYED
D e f in it io n
SymboI
Page
P e r io d ic it y v e c to r o f a c r y s ta l
32
a(q)
D e s tru c tio n o p e ra to r
33
a * (q )
C re a tio n o p e ra to r
33
A
A c o u s tic energy
72
»,
Ao
A c o u s tic energy absorbed by an e le c tr o n per
u n it tim e
71
b
In v ers e
32
b(q,j)
Normal c o o rd in a te
31
B
Phase s h i f t
45
c
E la s t ic modulus
45
Second o rd e r e l a s t i c c o n stan t
46
T h ird o rd e r e la s t i c c o n stan t
46
S p e c ific h eat p e r u n it volume
44
D
N o n - lin e a r it y c o n stan t
46
E
E le c tr ic f ie ld
72
E(q,j)
Energy
32
e
P o la r iz a tio n
31
e
E le c tr o n ic charge
72
F
Average o f second and t h i r d o rd e r e l a s t i c
^jkpq
^Jkprqs
l a t t i c e v e c to r
38
co n stan ts
G
Number o f
l a t t i c e s it e s
31
91
Symbol
D e fin itio n
Page
h
P la n c k ’ s c o n s ta n t
H
H a m ilto n ian
33
H’
P e rtu rb a tio n H a m ilto n ian
35
H
M agnetic f i e l d
28
l,J ,k
D ir e c tio n
31
J
C u rre n t
k
Boltzm ann’ s c o n s ta n t
9
Z
Mean f r e e path o f therm al phonons
9
L
L a t t ic e dimension
39
m
E f f e c t iv e mass o f an e le c tr o n
72
M
Mass o f an atom o r u n it c e ll
32
n
Number o f
phonons absorbed p e r u n it
N
Number o f
phonons
33
D ir e c tio n
cosines f o r p ropagation d ir e c tio n
46
N
P
)
)
9
in d ic e s
72
tim e
72
q
Phonon wave v e c to r
31
Q
Q u a lity fa c t o r
16
r
Wave normal
63
S ..
S tr a in
46
t
Time
31
T
A bsolute te m p eratu re
U(X)
Displacem ent a t a l a t t i c e s i t e
31
•
D ir e c tio n cosines f o r p a r t i c l e displacem ent
46
u.
)
)
9
92
Symbol
v,v^yV^
D e fin itio n
V e lo c ity o f microwave phonons
Page
*
39
V
Average therm al phonon v e lo c it y
38
Vj
E le c tro n d r i f t v e lo c it y
72
v^
Average volume per atom
60
W (t)
T r a n s itio n p r o b a b ilit y
35
a
A tte n u a tio n In dB/cm
• 22
Y*
Gyromagnetic r a t i o
28
Y
Grunelsen c o n s ta n t
36
Y (q )
Anharmonic c o e f f i c i e n t
36
Grunelsen number
46
A
Mean fr e e path f o r e le c tro n s
72
A»(X)
D ila t io n
36
n
E f f e c t iv e v is c o s it y
62
General v is c o s it y te n s o r
63
0
M agnetic f i e l d o r ie n t a t io n
30
0»
Angle between wave v e c to r and d ir e c tio n o f energy
flo w
23
0p
Debye tem p eratu re
42
K
Thermal c o n d u c tiv ity
47
X
Wavelength
p
D e n s ity
T
R e la x a tio n tim e
9
w
Frequency
9
9
38
93
LIST OF TABLES
T a b le
I
II
Page
Phonon modes In a c -c u t q u a r tz .
94
Measured and computed v alu es o f F j^ and
95
f o r v a rio u s phonons In q u a r tz .
Ill
Comparison o f th e a tte n u a tio n o f I GHz and
96
9 GHz phonons In q u a rtz a t 20*K .
IV
L i s t o f th e p r o p e r tie s o f th e seven p u re mode u ltr a s o n ic waves s tu d ie d in s p in e l.
97
94
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98
LIST OF FIGURES
F ig u re
I .a
Page
V e r t ic a l 9 GHz c a v ity
102
b
H o riz o n ta l 9 GHz c a v ity
102
c
I GHz c a v ity
103
d
Transm ission s e t up
104
105/106
2 . a and b
Q uartz echo p a tte rn s a t 4.2®K and 9 GHz
3 . a and b
CdS and Ni f i l m tra n s d u c e r echo p a tte rn s
107
4.
Block diagram o f apparatus
108
5.
D e ta ils o f 9 GHz apparatus
109
D e te c tin g system
I 10
6
.
7.
A tte n u a tio n o f
9155 MHz phonons in a c -c u t q u a rtz
111
8.
A tte n u a tio n o f
9155 MHz phonons In b c -c u t q u a rtz
112
9.
A tte n u a tio n o f
9155 MHz phonons In x -c u t q u a rtz
113
10.
A tte n u a tio n o f
9155 MHz phonons in im p e rfe c t x -c u t
114
q u a rtz
11.
A tte n u a tio n o f 9155 MHz phonons in p e r fe c t and
im p e rfe c t z - c u t q u a rtz
12.
115
C a lc u la te d v a r ia t io n o f th e therm al phonon r e la x a ­
t io n tim e versus te m p e ratu re f o r q u a rtz taken from
M a ris r e f .
13.
(1 6 )
116
Excess a tte n u a tio n o f 9155 MHz phonons In an
im p e rfe c t q u a rtz specimen
117
99
Figure
14.
Page
A tte n u a tio n o f 9155 MHz
s p in e l.
phonons In < I00 > c u t
For comparison some t y p ic a l measurements
made on q u a rtz a re Included
118
15.
A tte n u a tio n o f 9155 MHz
phonons In < l l l >
c u t s p in e l
119
16.
A tte n u a tio n o f 9155 MHz
phonons In < IIO > c u t s p in e l
120
17.
C a lc u la te d v a r ia t io n o f
th e therm al phonon r e la x a ­
tio n tim e versus te m p eratu re In s p in e l.
Values
were computed from th e therm al c o n d u c tiv ity meas­
urements o f r e f .
18.
(9 7 )
121
C a lc u la te d v a r ia t io n o f
th e a tte n u a tio n o f 9155
MHz phonons In s p in e l on th e b a sis o f a r e la x a tio n
mechanism.
Curve A Is deduced from eqn.
Ch. V w ith Y = 0 .5 6 .
Curve B Is deduced from th e
work o f Mason and Bateman r e f .
w ith f i g s .
19.
(1 4 ),
(7 )
(4 6 ).
Compare
(1 5 ) and (1 6 ) above about 100®K
122
E xperim ental and t h e o r e t ic a l a tte n u a tio n a t 9155
MHz f o r
lo n g itu d in a l waves on a < I00 > a x is in Y IG .
A.
E xperim ental curve
B.
Approxim ate form o f th e low te m p e ratu re a tte n ­
u a tio n peak
C.
Approxim ate form o f th e i n t r i n s i c a tte n u a tio n
in YIG
D.
Low tem p eratu re th e o r e tic a l curve
E.
High te m p eratu re th e o r e tic a l curve
123
100
Figure
20.
Page
E xperim ental a tte n u a tio n curve a t I GHz f o r lo n g i­
tu d in a l waves on
21.
a< I0 0 > a x is
In YIG
124
E xperim ental and th e o r e tic a l a tte n u a tio n curves
f o r tra n s v e rs e waves on a < I00 > a x is In YIG a t
9155 MHz
22.
125
Thermal phonon r e la x a t io n tim e versus tem p eratu re
In YIG computed from th e therm al c o n d u c tiv ity data
of r e f .
23.
(4 9 )
126
D iffe r e n c e In th e a tte n u a tio n In th e s a tu ra te d and
u n s atu rated s ta te s f o r I GHz lo n g itu d in a l phonons
pro p ag atin g on
24
a < I00 > a x is In YIG
127
A ngular p lo t o f a tte n u a tio n peaks as a fu n c tio n o f
m agnetic f i e l d
fo r a I cm long 3 mm d ia m e te r < I00 >
YIG ro d , w ith m agnetic f i e l d moving In th e < 0 0 1>
p la n e .
The fo u r s e c tio n s o f th e curve A show th e
o nset o f th e a tte n u a tio n peak due to fe rro m a g n e tic
resonance.
Curves B and 0 show approxim ate p o s i­
tio n s o f th e dominant a d d itio n a l peaks observed
j u s t above and j u s t below 227®K.
128
25.
A tte n u a tio n of
129
26.
A tte n u a tio n o f tra n s v e rs e phonons In a c -c u t q u a rtz
27.
Observed a tte n u a tio n as a fu n c tio n o f d r i f t f i e l d
lo n g itu d in a l phonons In x -c u t q u a rtz
In CdS a f t e r r e f .
(5 0 )
130
131
101
Figure
28.
a
Page
Echoes observed a t 9 GHz and 4.2®K in r e f le c t io n
from a GaAs specimen bonded to a q u a rtz ro d .
b
132
Echoes observed a t 9 GHz and 4.2®K in r e f le c t io n
from an illu m in a te d GaAs specimen bonded to a
q u a rtz rod.
2 9.
Echoes observed a t 9 GHz and 4.2®K In r e f l e c t i o n
from a GaAs specimen bonded to a q u a rtz rod.
30.
132
133
Echoes observed a t 9 GHz and 4.2®K In tra n s m is s io n
through a q u a rtz-G a A s -q u a rtz system.
134
102
Coupling hole
JTuning mechanism
Fig. l a
9GHz cavity (sample vertical)
Tuning mechanism
11
Reduced size wave guide
|X
Dimensions
in mm
upling hole
Samole
Fig. lb
9GHz cavities (samples horizontal)
r|Tuning mechanism
Worm .& wheel
mechanism
,Wav9 guide 16
^Coupling hole
sample
HKROWAVE
SUPEKHET
PULSE
RECEIVER
GENERATOR
C.R.O.
COAXIAL
CABLES
N
MICROWAVE ELECTRIC FIELD
czzrzzs MICROWAVE MAGNETIC FIELD
FIG .Ic
OF CAVITY ( F re q u e n cy IGH 2 )
lolt
Wave guide 16
Quartz
Quart
k / / /
Fig. I d
Apparatus for transmission experiments
105
Fig 2a
Photograph o f
lo n g itu d in a l wave echoes a t 9155 MHz
and 4.2®K in x - c u t q u a r t z .
s c a le d i v i s i o n .
Time base 500 y s / l a r g e
Time increases t o the
le ft.
106
f!
Fig
2b
Photograph of
lo n g itu d in a l wave echoes a t 9155 MHz
and 4.2®K in x - c u t q u a r t z .
s c a le d i v i s i o n .
Time base 50 y s / l a r g e
Time increases t o th e
le ft.
107
I
Fig 3a
Photograph o f lo n g itu d in a l v.'ave echoes a t I GHz and room
tem peratu re in < l l i > c u t s p i n e l .
Time base 5 y s / l a r g e
s ca le d i v i s i o n .
Time increases to th e r i g h t .
Fig 3b
Photograph o f f a s t tra n s v e r s e wave echoes a t 9155 MHz and
room tem perature in <I I O> c u t s p i n e l .
Time base I y s / i a r g e
s c a le d i v i s i o n .
Time increases t o the l e f t .
108
A
L
l- l
Fir. i; Block diagram o f the PGHz apparatus
109
Wave guide 16
Tube for liquid *
helium syphon
_
To rotary
idiffus- ,
Lon pumps
manometer
Helium gas
inlet
Liquid nitrogen
r\
Tuning
mechanism
Fig. ^ Detail of 9GHz system
Tube for evacuating
can
Cavity
Leads to carbon resist­
ance and thermocouple
-Sample
Liquid helium
110
•H
I
I
•H
.cd
A
S
0
iH
1
©
I
©
I
•H
ra
«M
o
3
s
©
o
vo
M
id
3
•H
p;
a g
Io
'1
T
r
10 r.
j
!
1
"K
1,
.//A
111
i
i
1
PUAE TPANSVEPSE WAVES
IN AC - C U T OUAPTZ
X JACObSEN'S
RESULTS
REF.(ii:)
a PRESENT AUTHCM3' RESULTS
FOP NATURAL QUARTZ
— THEORETICAL ATTENUATION
IN PERFECT SPECIN'IEN
/Ü
ACCORDING TO :VAP)S REF.Ü6)/'^[p
E
u
cn
-o
Z
O
K-
<
D
Z
uJ
; i
0 '2
H
IS
—I
O'
0 0 5
QUASI LONGITUDINAL WAVES
IN A C - C U T OUASTZ
O EXPSR I.M6N T
S O L ID
CURVE, t h e o r y
REF. (16)
0-02
O 'O I
1
5
lO
20
TEMPERATURE r x l
50
Kg 7
112
CUT QUARTZ
G
EXPERTYCNT
NATURAL QUARTZ
SOLID CURVE; THEORY, PCF. ( U )
0-5
0-05
0*02
0 -0 1
20
50
ICO
113
10
T’ ~:
d
V
X
NEARLY PLPFECT X-OJT
LO NG ITUD INA L WAV ES
O'JARTZ
^
SLOW TRANSVERSE WAVES
.
S OL ID CURVES, THEORY P E F (]6 )_ f
FAST TRANSVERSE WAVES
V;
SOLID CURVES; THEORY P E F .( m ) /
/
/
O
~
i
--
J
V
1
0
z
O
<
z
LÜ
02
h-
G
0-05
002
O 'O I
5
.1
1
lO
20
TEMPERATURE (°K )
J
IC O
50
Pig 9
!t"r:
24°%
llli
o ,
—
£
d 'i
0-5
Z
o
h<
:>
z
LU
C
0 2
<L
O
I
0 0 5
LONGITUDINAL WAVES IN X-CUT
.
QUARTZ
'
X NATURAL BRAZILIAN 0UAP1Z (A) f
G SYNTHETIC QUARTZ
002
'
(s)
G DOPED SYNTHETIC QUARTZ ( ( )
— — THEORETICAL ATTENUATION IN PERFECT SPECIMEN
ACCORDING TO M A R I S
R E F .( l6 )
O 'O I
1
J.
5
.10
TEMPERATURE
20
50
lOO
Tig 10
i f-
i U
iC
r.
''
T.
J' c
115
t
E
u
X]
x?
z
0
5
O
<
D
Z
LU
0-2
G
l o n g it u d in a l
O 'O
WAVES IN
O
DOPED
SYNTHETIC
G
NATURAL
QUARTZ
(Z,)
X
n a tu ra l
QUARTZ
( 2 2)
2-CUT
OUAPTZ
QUARTZ
(c )
SOLID CURVE THROUGH Z ^ P O I N T S :
THEORY FOR PERFECT SPEClLiEN, REE. ( 16)
O 01 ^
- 1 _________
2
10
TEA4PERATUPE
20
50
100
Tig 11
116
10000
1000
I
o
o
0)
to
•H
<D
100
•p
S
G
I
0
g
53
"50
Temperatiire (°K)
117
CM
CM
CM
CM
rH
CM
O
iH
oo
(luo/gp)
uoxq.'BTiueq.q.v
o
o
CM
I—
I
ssaoxg;
\A
o
118
lOO
X CUT QUARTZ
SLOV/ TRANSVERSE
LONGITUDINAL
(RER 3 5 )
10
<IO O > SPINEL
/
O - LONGITUDINAL WAVES
X
20
50
lOO
TRANSVERSE WAVES
200
TEMPERATURE (°K.)
500
lOOC
119
OO
■o
•O
< |II> SPINEL
G LONGITUDINAL WAVES
X TRANSVERSE WAVES
THESE TWO SETS OF
MEASUREMENTS WERE
TAKEN ON DIFFERENT .
SPECIMENS.
O'
20
50
iOO
200
TEMPERATURE (^K)
500
ooo
120
IOO
<1 IO > SPINEL
lO
20
50
O
LONGITUDINAL WAVES
X
FAST TRANSVERSE WAVES
Q
SLOW TRANSVERSE WAVES
IOO
200
TEMPERATURE ( ° k )'
I OOO
500
F Is
16
121
lOOO
tO
Û
z
500 -
8
uJ
to
fO
I
o
200
o
tO
»-
Z
3
10 0
Z
uT
SO
H-
z
O
1-
<
<
X
20
uJ
ÙL
Z
o
z
o
X
a
SPINEL
10
-J
<
2
5
Û1
UJ
X
»Û
LÜ
»
—
3
a
2
2
o
u
50
ICO
temperature
200
C’ k )
500
Fig 17
lOOC
122
IOO
B
: CALCULATED
USING
cx=
ATTENUATION
8 . 6 8
2J5
WITH ■^ = 1
Eo
W ^ T
------ 3 :-% ---------——
( l + CO ^T^)
(^SEE REF-Uô)
CALCULATED
USING Dl =
ATTENUATION
8.68 3 K T
ÿ
x>
WITH '5 = 0 . 5 6
LÜ
SPINEL
10
20
50
I OO
200
T E M P E R A T U R E ( ° k)
500
Fig. 18
1000:
30
lO
E
u
cû
/o ,
z
O
I-
<
3
Z
uJ
►5
G
YIG
(g.EC.
L O N G IT U D IN A L
0.1
< IO O >
.
6 9
s
p u ritÿ )
WAVES
AXI S
O QUARTZ TRANSDUCER
O CdS
TH IN
FILM
TRANSDUCER
FREQUENCY : 9 155 M H z
0 .0 3
lO
30
TEMPERATURE
300:
OO
Fis 19
J2h
10
G-E.C. 6 . 9 s
YIG
l o n g it u d in a l
<100>
II
waves
AXI S
II
II
II
II
CdS T R A N S D U C E R
FREQUENCY : l O O O M H z
3 .0
II
II
SOLID
e
o
CURVE ; SAMPLE
SATURATED
' i
DASHED CURVE ; SAMPLE
UN SATURATED
S'
fl.O
<
3
Z
ixJ
O. 3
0.1
o
30
100
TEMPERATURE ( ° k)
P ig 20
300
ooc:
0 .3
YIG
(g.e.c.
6 . 9 's p u r i t y )
TRANSVERSE
0 .0 3
WAVES
< 1 0 0 AXIS
- RELAXATION
VALID
FOR
THEORY
CJRr<!
• LANDAU-RUMER
FOR
v a lid
THEORY
CJT » l
0.01
10
30
I00
TEMPERATURE
300
^
k
)
1 0 0 0
Hg
21
lOOO
cO
:
O
500
cO
fO
200
lOO
50
20
lO
YIG
20
50
lOO
TEMPERATURE
200
500
OOO
CM
CM
CM
CO
in
CO
CM
126
ïîg
2
U
129
10
i
I
100
20
Fig 2S
Temperature
L o n g itu d in a l ^ a v e s
v
an x-cut q
130
10
2.0GHz
8
l.lGHz
6
c
0
1
s
h
2
0
100
Temperature
îlg 26
(°K)
Shear waves in ac-cut quartz
031
•H
Drift field volts/cm
-2 9 0
-
^ 0
0,2
1900
120
0 .2
Normalized drift field
CdS
Resistivity
-liO
Mobility
300 cm^/volt s
Frequency
20MHz
-60
fig. 27
i|. x lO^ohra cm
Theoretical variation of ultrasonic gain with
drift field at constant frequency in Cd3
M a in p u ls e
32
Q u a rtz
IQ u a r tz -G a ^ s
1 1 11 I I I I
I I I I
i w i m é
Fig 28a
Photograph o f lo n g itu d in a l wave echoes a t 9155 MHz and
4 .2 °K in GaAs bonded to x -c u t q u a r tz .
Time base 5 y s /
la rg e s c a le d iv is io n .
Time increases to th e l e f t .
I 11 M I 1 I I
F ig 28b
I I 1I
Photograph o f lo n g itu d in a l wave echoes a t 9155 MHz and
4 .2 °K in illu m in a te d GaAs bonded to x -c u t q u a rtz .
Time
base 5 y s /la r g e s c a le d iv is io n .
Time increases to th e
le ft.
33
Main pulse
Quartz
iQuartz-GaAs
M
F ig 29
Photograph o f lo n g itu d in a l wave echoes a t 9155 MHz and 4 .2 *K
in GaAs bonded to x -c u t q u a rtz .
Time base 2 y s /la r g e s c a le
d iv is io n .
Time increases to th e l e f t .
34
Main pulse
Quartz-GaAs-quartz
Fig 30
Photograph o f
lo n g itu d in a l wave echoes in th e regio n 4 to
30®K and 9155 MHz in tra n s m is s io n through q u artz-G aA sq u a r tz .
Time base
creases to th e
le ft.
2
y s /la r g e s c a le d iv is io n .
Time In ­
R ep rinte d fro m T
he
P hysical R
eview,
V o l. 159, N o . 3, 703-711, 15 J u ly 1967
Printed in U . S. A.
Microwave Phonon-Attenuation Measurements in Quartz
M . F. L e w i s
and
E. P a t t e r s o n *
The General Electric Company Lim ited, Central Research Laboratories,
H irs t Research Center, Wembley, England
(Received 20 December 1966; revised m anuscript received 13 F ebruary 1967)
Measurements have been made of the temperature dependence of the attenuation at 9000 M c/sec of
tw o slow transverse, two fast transverse, and three longitudinal phonon modes in “ perfect” natural-quartz
specimens. The attenuation of the slow transverse waves follows the theory of Landau and Rumer. The
attenuation of the lo n g itu dina l and fast transverse waves follow the theories of M aris and Shiren, which
take in to account the fin ite life tim e of the therm al phonons w ith which the microwave phonons interact,
b u t some discrepancies s till exist in the magnitude of the attenuation. The steps observed in the attenuation
of im perfect quartz are identified as tw o peaks at 1 6 ± 3 ° K and 2 4 ± 3 ° K , sumperimposed on a phononphonon background. Possible mechanisms fo r these attenuation peaks are discussed.
I. INTRODUCTION
S IN C E the development of microwave phonon trans­
ducers/-^ several workers have measured the v a ri­
ation of the phonon attenuation as a function of tem­
perature in various dielectric and semiconducting
crystals.®” ^^ In most materials the attenuation of trans­
versely polarized phonons at low temperatures is due
to microscopic interactions w ith thermal phonons by
three-phonon processes, and is found to va ry as ooT^,
where co is the phonon frequency and T is the absolute
temperature. Landau and Rumer^^ originally predicted
such behavior at temperatures low enough to satisfy
the condition WT>1, where r is the thermal phonon re­
laxation time. T ypically, this condition is satisfied
below about 30°K fo r 9000 Mc/sec phonons, i.e.,
roughly throughout the range in which attenuation
measurements can be made.
The basic attenuation mechanism (s) for longitudinal
microwave phonons remains in some doubt. O riginally
i t was thought th a t three-phonon processes could only
produce a small attenuation (varying as
and i t
was necessary to consider four-phonon processes. H ow ­
ever, these were also calculated to produce a small at­
tenuation, b u t varying^® as
In all the materials
studied so far, these mechanisms can be ruled out from
the experimental observations th a t the attenuation of
longitudinal phonons is of the same order of magnitude
* Permanent address: Chelsea College of Science and Tech­
nology, U n ive rsity of London, London, England.
^ H . E. Bom mel and K . Dransfeld, Phys. Rev. Letters 1, 234
(1958).
* E. H . Jacobsen, Phys. Rev. Letters 2, 249 (1959).
®H . E. Bommel and K . Dransfeld, Phys. Rev. 117, 1245 (1960).
^ E. H . Jacobsen, Quantum Electronics, edited by C. H . Townes
(Columbia U n ive rsity Press, New Y o rk, 1960), pp. 468-484.
®R . Nava, R. A z rt, I . Ciccarello, and K . Dransfeld, Phys. Rev.
134, A581 (1964).
®I. S. Ciccarello and K . Dransfeld, Phys. Rev. 134, A1517
(1964).
’’ H . J. M aris, P h il. M ag. 9, 901 (1964).
®M . Pomerantz, Phys. Rev. 139, A501 (1965).
9 J .d e K le rk , Phys. Rev. 139, A1635 (1965).
19 J. de K le rk and P. G. Klemens, Phys. Rev. 147, 585 (1966).
u j . d e K le rk , J. A p p l. Phys. 37, 4527 (1966).
19 L . Landau and G. Rum er, Physik Z. Sowjetunion 11, 18
(1937).
191. J. Pomeranchuk, J. Phys. (USSR) 6, 237 (1942).
159
as th a t of transverse phonons, varies w ith frequency as
cü" where 0 < w < l , and varies w ith temperature as
T ” , where
Herring^^ has shown th a t in
real crystals, where anisotropy in general splits the
transverse phonon velocities, the attenuation of longi­
tudinal microwave phonons can proceed by a small
number of three-phonon processes involving transverse
thermal phonons on different branches of the dispersion
curve. However, the attenuation from the Herring
mechanism is small and varies as
where n is an
integer which depends on the crystal symmetry b u t is
never less than two ; again, this conflicts w ith experiment.
Recently i t has been pointed out^®-^® th a t because the
thermal phonons have a finite lifetim e (c±^T) their fre­
quency is uncertain to the extent of A c o ~ l/r. This is
often sufi&cient to reinstate three-phonon nearly collinear attenuation processes for longitudinal microwave
phonons, and several workers have given expressions for
the resulting attenuation.®-’^-^'^ These expressions predict
an attenuation of roughly the rig h t magnitude, while
the expression given b y M aris’^ also seems capable of
explaining the shape of the attenuation curves in quartz.
In general, the attenuation of fast transverse waves
is intermediate between th a t of slow transverse waves
and longitudinal waves®“ ^^ and is expected to be caused
b y the Landau-Rumer mechanism and the M aris
mechanism. Even if interactions involving three
s tric tly coUinear phonons are small or vanishing, noncollinear interactions can proceed because of the aniso­
tropic lattice anharmonicity as discussed b y Shiren.^®
We have extended the attenuation measurements of
previous workers to include the attenuation a t 9000
M c/sec of several new phonon modes in quartz, two of
which are neither “ slow transverse” nor “ fast longitu­
dinal” in the notation of- Ref. 5. In both cases the at­
tenuation varies in the manner expected fo r longitudi­
nal waves, i.e., noticeably faster than T^, the average
variation in the range &-30°R being about T®, and,
99 C. Herring, Phys. Rev. 95, 954 (1954).
“ S. Simons, Proc. Phys. Soc. (London) 82, 401 (1963).
99 N . S. Shiren, Phys. Rev. Letters 11, 3 (1963).
See Ref. 21 of Ref. 8.
^9 N . S. Shiren, Phys. Letters 20, 10 (1966).
703
704
M.
F.
LEWIS
AND
rOdb DIRECTIONAL
E.
PATTERSON
159
in the sample due to transmission into the bonds, is
found to be negligible. The former is reasonable since
our bonds are only about 5000 Â th ic k ; the la tte r indi­
cates th a t the acoustic impedances of crystals and
bonding material are not temperature dependent in this
region.^°
A precision rotary-vane attenuator is used to compare
the intensities of several pairs of echoes at various tem­
peratures, e.g., the power d B ’s necessary to reduce the
height of the first echo to th a t of the tenth echo is
measured as a function of temperature. This procedure
eliminates any nonlinearities in the detection system.
The accuracy is of the order of 0.1 dB/cm , b u t depends
F ig . 1. The apparatus (C.R.O. indicates cathode-ray
oscilloscope).
when compared w ith the results of Refs. 3, 7, and 11,
noticeably slower than co. We have interpreted these
and all other measurements on “ perfect” specimens (see
below) in terms of the modifications b y Maris^ and
Shiren^® of the three-phonon attenuation process of
Landau and Rumer.^^
The presence of imperfections in quartz has been
found to cause “ steps” in the attenuation as a function
of temperature^-® b u t so far no explanation of this has
been given. We have measured the attenuation of
various phonon modes as a function of temperature in a
number of imperfect quartzes. In specimens of various
origins, the steps are always found to occur at the same
temperatures, which are close to the Einstein tempera­
tures used b y Flubacher et al}^ to account fo r the
anomalous specific heat of fused silica. This suggests
th a t the additional phonon attenuation in imperfect
quartz is caused b y regions whose structure resembles
th a t of fused silica.
n . EXPERIMENT
The apparatus is shown in Fig. 1. A 2J51A pulsed
magnetron excites a tunable re-entrant microwave
cavity containing a quartz transducer. The operating
frequency is 9155 Mc/sec. Echoes are observed in re­
flection. The quartz specimens are in the form of rods
typically 1 cm long and 3 mm in diam, w ith end faces
optically flat, and parallel to a few seconds of arc. When
2-cut samples are used they are bonded to a;-cut trans­
ducers w ith Araldite. The temperature dependence of
any attenuation in the bonds, and of the attenuation
" P. Flubacher, A. J. Leadbetter, J. A. M orrison, and B. P.
Stoicheff, J. Phys. Chem. Solids 12, 53 (1959).
0.5
< 0 .2
0.1
0.05
0.02
0.01
20
50
lO O
TEMPERATURE (« k)
F ig . 2. M icrowave phonon attenuation in ac-cut quartz. Pure
transverse waves: X, Jacobsen’s results (Ref. 4 ); □ , present
authors’ results for na tura l quartz; dashed curve indicates theo­
retical attenuation in perfect specimen according to M a ris (Ref. 7).
Quasilongitudinal waves: O , experiment; solid curve indicates
theory (Ref. 7).
considerably on the intensity and separation of the
echoes. Measurements are taken as the apparatus
warms up from 4.2 to about 35°K. A piece of lead at“ These conclusions were tested b y A raldite-bonding to *-cu t
quartz transducers, samples of [ 100] cut silicon and [ 111] cut
semi-insulating GaAs. N o change in the attenuation of either
specimen could be detected in warm ing up from 4.2 to '^ 2 5 ° K
because both m aterials have a negligible attenuation up to this
temperature [R e f. 8 and K . R. K e lle r and B. Abeles, J. Appl.
Phys. 37,1937 (1966)] ; th is verifies the fa ct th a t there is no change
in the attenuation due to the bond. I n addition, our results were
reproducible from bond to bond,
159
MICROWAVE
PHONON-ATTENUATION
tached to the cavity increases its thermal capacity and
prolongs this process to several hours, perm itting many
readings to be taken. A calibrated carbon resistance
thermometer is used to measure the temperature to
The quartz samples used are natural Brazilian quartz
and synthetic quartzes grown hydrotherm ally at The
General Electric Co. L td ., Wembley, England. The
la tte r contain various concentrations of impurities.
Although some of these are paramagnetic, no extra at­
tenuation was observed when they were brought to
resonance at the phonon frequency b y the application
of a magnetic field.
m . RESULTS
A. Observations on N atural Quartz
This section deals w ith observations on specimens
free from imperfections. In Sec. I I I B , we discuss im ­
perfect specimens. A ll the measurements reported are
a rb itra rily normalized to zero at 4.2°K. Our first meas­
urements were made on a 1-cm-long, 3-mm-diam ac-cut
natural quartz crystal. The predominant phonon mode
excited w ith the geometry of Fig. 1 is the pure-trans­
verse mode w ith velocity ~3.3X10® cm/sec. The at­
tenuation is shown in Fig. 2, and varies as T^-^, in fa ir
agreement w ith previous workers.® These results are a
T
able
IN
QUARTZ
705
I . Phonon modes in ac-cut quartz.
Polarization
Pure transverse
Quasilongitudinal
Quasitransverse
Calculated*
angle 6
between
Measured
wave vector
velocity
and direction
in units
Calculated of energy
of 10®cm/sec velocity*
flow
3.37
7.04
3.75
3.33
6.98
3.69
< 1°
6°
4°
“ In th e calculations, th e ac direction w as assum ed to m ake an angle of
ex a ctly 3 1 ° w ith the y axis.
useful check on our measuring technique and tempera­
ture scale. The absence of steps in the attenuation
indicates th a t this sample is relatively free from
imperfections.^'®
We have also observed echoes due to extra phonon
modes in this sample. The velocities of these were most
easily measured using nickel film transducers, for by
varying the orientation of the applied magnetic field i t
is possible to enhance and suppress wanted and un­
wanted modes, respectively.^^ For completeness we
include a lis t (Table I) of measured velocities, and ve­
locities calculated from the elastic constants of quartz,^^
together w ith the computed angle 6 between the wave
vector and direction of energy flow. The agreement
between measured and computed velocities is con­
sidered satisfactory in view of errors in cutting the
crystals ( ~ | ° ) , and the use of room-temperature elastic
constants. The density of quartz was taken as 2.65
g/cm®.
We also measured the attenuation of several other
phonon modes in quartz. These are :
(1) The quasilongitudinal mode in ac-cut quartz.
From the results shown in Fig. 2 i t is clear th at the
attenuation is small and varies faster than
over a
considerable temperature interval.
(2) The pure transverse mode in ôc-cut quartz, ve­
lo c ity ~5.0X10® cm/sec. The attenuation is shown in
Fig. 3 and is very close in magnitude and slope to th at
of (1) above. The low value of the attenuation in 6c-cut
quartz has already been noticed b y Bommel and
Dransfeld.® N o extra modes were observed inour25-mmlong, 3-mm-diam specimen.
(3) The longitudinal mode in rr-cut quartz, velocity
"^5.7X10® cm/sec. Our most perfect specimen gave the
attenuation shown in Fig. 4. The agreement w ith the
theory of Maris^ is seen to be excellent. The 25-mm-long
specimen used is considered perfect because i t gave no
less than 1000 echoes at 4.2°K, and showed no steps in
the attenuation versus temperature curve. Other im -
1.0
UJ 0 .2
0.1
0 .0 5
0.02
0.01
MEASUREMENTS
i
lO
20
50
lOO
TEMPERATURE(
®
k)
F i g . 3. M icrow ave phonon attenuation in &c-cut quartz. Pure
transverse waves: O , experiment (natural quartz); solid curve
indicates theory (Ref. 18).
21M . F. Lewis, T . G. Phillips, and H . M . Rosenberg, Phys.
Letters 1, 198 (1962).
22 W . p . Mason, Piezoelectric Crystals and their Application to
Ultrasonics (D. Van N ostrand Company, Inc., New Y o rk, 1949),
p. 84,
706
M.
F.
LEWIS
T a b le I I . Measured and computed values of
for various phonons in quartz.
Wave
vector
X axis"
■Taxis
ac axis'*
ac axis
ac axis
z axis
and
AND
F3^
Calculated
values
F32
of
Polarization
dyn^/cm^ d}'n2/cm‘‘ dyn2/ cm4
1.9X1026 2.3X10=
Longitudinal
0
1.25X1025 ^2X1025
Slow transverse
1.25X1026 1.35X1026
Pure transverse
8.8X1024 9.4X1022
Pure transverse
5.3X1026 9X1024
Quasilongitudinal
0
3.8X1026 2.4X1026
Longitudinal
0
M aris and present authors.
M aris.
perfect specimens show different behaviors as discussed
below.
(4) The fast transverse mode in :r-cut quartz, ve­
lo city ~5.1X10® cm/sec. This can be excited, w ith the
arrangement of Fig. 1, b y tiltin g the specimen at a few
degrees to the post of the cavity. The attenuation in our
most perfect specimen is shown in Fig. 4, and, sur­
prisingly, is lower than th at of the longitudinal wave.
(5) The slow transverse mode in az-cut quartz, ve­
locity ~3.3X10® cm/sec. The attenuation is shown in
Fig. 4 and varies as
over the range of measure­
ments. This behavior is expected for slow transverse
phonons on the Landau-Rumer theory.
(6) The longitudinal mode in z-cut quartz, velocity
~6.4X10® cm/sec. The samples are bonded to %-cut
transducers. Inefficient bonds make the results less
accurate than for other cuts. The attenuation of our
most perfect specimen is shown in the lowest curve of
Fig. 5, labelled Z2.
E.
PATTERSON
159
crystal turned black and opaque, the effect on the tem­
perature-dependent p art of the attenuation was
minimal.
The attenuation of longitudinal waves in two im ­
perfect z-cut quartz crystals have also been measured.
One of these was a synthetic quartz rod from the same
crystal as (C) above. The other (Zi) was cut from a
natural crystal. The attenuation curves are shown in
Fig. 5, and again the steps occur at about 16 and 24°K.
Annealing the natural crystal at 500°C for five days
produced no change in the temperature-dependent
attenuation.
As previously mentioned, all results are normalized
to zero at 4.2°K. This procedure is reasonable for the
perfect specimens, b u t some of the imperfect specimens
show a rapid rise of attenuation (on a log-log plot) just
above 4.2°K. This indicates th a t there is s till a tem­
perature-dependent attenuation a r (4.2°K) at 4.2°K,
in addition to any residual attenuation ar, e.g., due to
scattering processes. The total attenuation at tempera­
ture T i is then given b y a{T-i) = ar-\-aT (4.2°K )+o:ri
(graph). A p lo t of the temperature-dependent attenua­
tion should therefore be corrected for a r (4.2°K). How­
ever, i t is most unlikely th a t a r (4.2°K) exceeds
a(8.4°K)—a(4.2°K) and i t is easily verified th a t the
addition of ar(4.2°K) ^a(8.4°K)-a(4.2°K) to all the
B. Observations on Im perfect Quartz
Measurements of the attenuation of longitudinal
phonons were made in the following imperfect a;-cut
quartz crystals; (A) natural Brazilian quartz crystal,
(B) a synthetic crystal grown hydrotherm ally perpen­
dicular to the basal plane, and (C) a synthetic crystal
grown hydrotherm ally perpendicular to the basal plane.
(This last crystal was deliberately doped w ith ~0.015%
iron and was slightly blue in color.)
The attenuation curves for (A), (B), and (C) are
shown in Fig. 6 together w ith the attenuation in a
perfect specimen. The steps in the attenuation in the
imperfect quartzes always occur at temperatures of
1 6 ± 3 °K and 2 4 ± 3 °K . In general i t is not possible to
determine the positions of the peaks to better than
± 3 ° K . I t is interesting to note th a t the first observation
of such steps was made b y Jacobsen^ using an ac-cut
specimen. We have replotted Jacobsen’s results in
Fig. 2, from which i t can be seen th a t the steps occur at
about 16 and 22°K , i.e., in the ranges 1 6 ± 3 ° K and
2 4 ± 3 °K . We have attempted to produce imperfections
in one of our natural ac-cut rods by irradiating i t w ith
3, Co®° source to the extent of ~10^ rad. Although the
1.0
O.s
0 .2
0 .0 5
0.02
i
ID
20
TEM PERATURE ( ° k )
50
100
F i g . 4 . M icrowave phonon attenuation in nearly perfect %-cut
quartz. O , lo ngitudinal waves: V» slow transverse waves; solid
curves indicate theory (Ref. 7): X, fast transverse waves; solid
curve indicates theory (Ref. 18).
159
MICROWAVE
PHONON-ATTENUATION
MEASUREMENTS
points in Figs. 2, 5, and 6 w ill not significantly affect
the positions of the peaks/steps. There is no evidence
from the echo patterns of an excessive attenuation at
4.2°K in our imperfect specimens,
IN
I6°K
QUARTZ
707
24 “K
IV. ATTENUATION IN NATURAL QUARTZ
A. Temperature Dependence
Several workers have taken account of the finite life­
time of thermal phonons in deriving expressions fo r the
attenuation of microwave phonons.®-^*^^
In the case of slow transverse waves, the normal
Landau-Rumer term varying as
is dominant. The
0 .5
0 .2
0.1
0 .0 5
0.02
0.01
o.s
5
10
20
50
lO O
TEM PERATURE ( “k)
F i g . 6 . M icrowave phonon attenuation in imperfect .r-cut
quartz (longitudinal waves). X , n atural B razilian quartz (A ):
O , synthetic quartz (B) : □ , doped synthetic quartz (C) : dashed
curve indicates theoretical attenuation in perfect specimen ac­
cording to M aris (Ref. 7).
0 .2
0.1
expression,
0 .0 5
4.347T%wf1^/^
3/10
60pV
002
kT \ ^
X I^^TT—tan-i^0.01
S
lO
20
SO
—\
(1)
lO O
T EM PER A TU R E
F i g . 5 . M icrow ave phonon attenuation in perfect and imperfect
z-cut quartz (longitudinal waves). O , doped synthetic quaitz
(C) : □ , na tura l quartz (Zj) : X , n atural quartz (Zg) ; solid curve
through Zg points indicate theory fo r perfect specimen (Ref. 7).
only term in the expression for the attenuation of longi­
tudinal microwave phonons contains the thermal
phonon relaxation tim e r. The variation of r w ith tem­
perature can be estimated from the thermal conduc­
tiv ity . The expression fo r the attenuation of microwave
phonons given b y Maris^ appears to be capable of ex­
plaining accurately the attenuation of longitudinal
phonons in ic-cut quartz and, slightly less accurately,
the attenuation of slow transverse phonons in ac-cut
quartz over a wide range of frequency and temperature.
We have therefore attempted to fit our results to M aris’s
where a is the attenuation in dB/cm , p is the density of
quartz = 2.65 g/cm®, f is the velocity of microwave
phonons under consideration, s' is an average thermal
phonon velocity fo r quartz which is given the value
5.3X10® cm/sec, k is Boltzm ann’s constant, Z, is a
lattice dimension ~ 5 Â fo r quartz, F i and Fz are certain
averages of second- and third-order elastic constants of
quartz, and F i= 0 fo r longitudinal phonons. The com­
plicated procedure for computing F i and Fz from the
known elastic constants of quartz^^’^® is described in
Ref. 24.
Using Eq. (1) and chosen values of Fi^ and Fz^
(listed in Table I I ) , we have computed the theoretical
^ R. N . T hurston, H . J. M cSkim in, and P. Andreatch, Jr.,
J. A p p l. Phys. 37, 267 (1966).
3'! W . P. Mason and T . B. Bateman, J. Acoust. Soc. Am . 40,
852 (1966).
708
M.
F.
LEWIS
AND
curves fo r the slow transverse and longitudinal waves
in Figs. 2, 4, 5, and 6. The values of t are taken from
Fig. 1 of Ref. 7. The agreement w ith experiment is quite
good. This is particularly gratifying in the case of quasi­
longitudinal phonons in ac-cut quartz, fo r which there
is only one adjustable parameter Fz^. This determines
the magnitude of the attenuation, b u t not the slope of
the curve in Fig. 2. In the case of pure transverse waves
in ac-cut quartz, the slope of the measured attenuation
curve is about the same as obtained using M aris’s
values of Fi^ and Fz^, b u t to account for the magnitude
of the attenuation we m ust take values of Fi^ and Fz^
about 0.7 times M aris’s values. This difference is due to
the fact th a t M aris’s values were chosen as a best fit
to the results of various workers, including some on
imperfect ac-cut quartz.
The attenuation of fast transverse waves is w orthy
of particular attention. In general, these waves are ex­
pected to be attenuated b y the M aris mechanism and
the Landau-Rumer mechanism of the type T fast
+Zthermai ^ ^thermal, and are expected to have an
attenuation intermediate between th a t of slow trans­
verse and longitudinal waves. However, fo r both the
pure transverse waves in ôc-cut quartz, and fast trans­
verse waves in a:-cut quartz, the strictly collinear threephonon process is forbidden b y symmetry, i.e., in
Eq. (1), F 3^=0 since C555, Csss, Csee, Ceee, C15, Cie, Czs,
C26, C35, C46, and C46 all vanish.^^-^® This presumably
accounts for the very low values of the attenuation of
these modes, (and also fo r the low values of Fz^ for the
slow transverse modes studied since the same argu­
ments apply) b u t there is clearly another process in
addition to any Landau-Rumer mechanism. Since the
only three-phonon process not yet considered is the noncoUinear mechanism first discussed b y Shiren^® w ith
particular reference to longitudinal waves in isotropic
media, we have attempted to f it our measurements
w ith an expression of the form
60p V ^ ‘> \ ^ /
60p V ° \ h ) \c o r /
The first term in Eq. (2) is a Landau-Rumer term,
and the second is of the form derived in Ref. 18, w ith
A 111 of Ref. 18 set equal to zero since the collinear inter­
action vanishes. The solid curves fo r fast transverse
waves in a:-cut quartz (Fig. 4) and 6c-cut quartz (Fig.3)
used the values F i^= 4.8X10^® and 5.0X10^® dynYcm'^
and ^^=5.2X1CF® and 3.0X10^® dynVcm*, respec­
tively. W hile the agreement w ith experiment is fa irly
satisfactory, at least as good a fit can be made using
Eq. (1) and the values F i^ = 0 and ^ 3^= I.IX K P ® and
4.4X10^^ dynVcm^, respectively. Whatever the explana­
tion of the attenuation of fast transverse waves in
quartz, i t is clear th a t the contribution from the
Landau-Rumer mechanism is very small. This is also
“ K . Brugger, J. A p p l. Phys. 36, 759 (1965).
E.
PATTERSON
159
true of the attenuation of fast transverse waves in
A I2O3 ® and LiF,^° fo r the Landau-Rumer mechanism
causes an attenuation varying as T* at the lowest tem­
peratures. Exam ination of Refs. 9 and 10 shows th at if
this does occur, the amplitude is very small.
Finally, we should note th a t the sim ilarity between
the attenuations of slow transverse waves in x-cut and
ac-cut quartz is not accidental since the strains in ­
volved are the same, i.e., the slow transverse wave in
the X direction is polarized in the ac direction and vice
versa. Similar remarks apply to fast transverse waves in
the X and he directions.
B. Frequency Dependence
A check on Eqs. (1) and (2) can be made b y com­
paring the attenuation measurements of other
workers®'^'^^ in quartz at 1000 M c/sec w ith our measure­
ments at 9000 Mc/sec. The Landau-Rumer mechanism
predicts at attenuation proportional to w, whereas the
second terms in Eqs. (1) and (2) predict an attenuation
almost independent of w. Table I I I shows th a t these
predictions are qualitatively borne out b y experiment,
b u t th a t even the slow transverse waves do n ot show
an attenuation varying quite as fast as w.
C. Magnitude of the Attenuation
The problem of computing the attenuation of an
ultrasonic wave due to interaction w ith all other thermal
phonons is very complicated, even fo r cubic crystals.^^
Thus to some extent we are content th a t the coupling
constants
F 3®, and
are of the same order as the
elastic constants of quartz.^^-^® However, fo r those modes
whose attenuation is dominated b y collinear inter­
actions, the calculations sim plify considerably and some
values of the calculated coupling constants are listed in
Table I I . For longitudinal waves in x-cut quartz,
F3=3C ii-1-C ui— 0.48X10^2 dyn/cm^^®-^^ which gives
an attenuation ~ 8 0 times smaller than measured (i.e.,
F 3 is ~ 9 times smaller than measured from the attenu­
ation). This is too large a factor to be accounted fo r by
T
able
I I I . Comparison of the attenuation of 1000 M c/sec
and 9000 M c/sec phonons in quartz at 20°K.
Phonon mode
Æ-axis lo n g itu dina l waves
x-axis fast transverse
x-axis slow transverse
ac-axis slow transverse
6c-axis fast transverse
2-axis longitudinal waves
®R eference 3.
« R eference 7.
R atio of
attenuaAttenua­ tions at
9000
tio n
M c/sec
d B /cm
A tte n u a tion
a t 9000 and 1000
d B /cm
M c/sec
a t 1000 M c/sec M c/sec
0.1,“ 0.12,b 0.14“
0.19b
0.45d
0.59,“ 0.36»
0.14“
0.1“
0.25
0.12
2.5
1.6
0.12
0.18
b R eference 11.
d R eference 11 extrapolated.
2.5, 2.1, 1.8
0.63
5.6
2.7, 4.4
0.86
1.8
159
MICROWAVE
PHONON-ATTENUATION
experimental error, so we briefly consider two additional
attenuation mechanisms recently proposed to explain
the
(w > 4 ) dependence of the attenuation. The
mechanism proposed b y Kalejs, Maris, and Truell®® is
an interaction of the form T+Ttherm —> Ttherm and
gives an attenuation whose magnitude is of the rig h t
order, b u t difficult to estimate accurately because of
the anisotropy of the transverse phonon velocity^^
which enters to the eighth power. However, for
the attenuation varies as ~ r y T ~ r ® , which disagrees
w ith the measured variation of
The second
mechanism to be considered is Shiren’s noncollinear
mechanism.^® For longitudinal waves in x-cut quartz,
this mechanism is too weak to account fo r the dis­
crepancy ; for example, at 20°K the combined effects of
M aris’s collinear process and Shiren’s noncollinear
process is still ~ 4 0 times too small to account fo r the
experimental observations. A t present we have no ex­
planation for this discrepancy.
For longitudinal waves propagating on the z axis,
F 3= 3C33+ C 333— 4.9X10^2 dyn/cm^ and agrees w ith
experiment to better than a factor of 2. Sim ilarly, fo r
quasilongitudinal waves propagating on the ac axis,
there is substantial agreement between theory and ex­
periment (Table I I ) . T o sim plify the calculation, the
wave on the ac axis was assumed to be pure longitudinal.
D. Role of Normal Processes in Determining x
We have seen th a t w ith the values of t deduced from
the thermal conductivity, the theories of M aris and
Shiren are successful in explaining the temperature and
frequency dependence of the attenuation of microwave
phonons in quartz. However, a t the temperatures used
here (F<<C0 Debye), one m ight expect a considerable con­
trib u tio n to T from normal processes, which do n ot con­
tribute directly to thermal resistance.^® I n fact, very
little is known about normal processes except from
microwave ultrasonic attenuation measurements. In
principle, therefore, the problem is to find a self-con­
sistent solution,
a[w,r]=o:[w,r,T(T)],
where
r ( T ) ~ ------------------------------------- .
( 3)
5o:[c0thermphon(T),r,T(T)]
This is a formidable problem, which is most fu lly treated
b y W oodruff and Ehrenreich^® and Orbach.®° Orbach
has used the same equations to compute the microwave
phonon attenuation, and the thermal phonon relaxation
time due to all three-phonon processes, i.e., normal and
*®J. Kalejs, H . M aris, and R. T ru e ll, Phys. Letters 23, 299
(1966).
G. W . Farnell, Can. J. Phys. 39, 65 (1960), especially Fig. 2.
R. Peierls, Ann. Physik 3, 1055 (1929).
^ T . 0 . W oodruff and H . Ehrenreich, Phys. Rev. 123, 1553
(1961).
“ R. Orbach, P h.D . thesis, U n ive rsity of California, 1959
(unpublished).
MEASUREMENTS
IN
QUARTZ
709
umklapp. The la tte r are compared w ith the thermal
phonon relaxation times deduced from thermal con­
d u c tiv ity measurements in Fig. 13 of Ref. 30. For­
tunately, the two curves are not widely separated in the
region of interest. However, one would feel more con­
fident about using values of t deduced from thermalconductivity measurements if the latter were known to
be dominated b y imperfection or isotope scattering.
Some evidence fo r imperfection scattering in quartz
comes from the observation th at the thermal conduc­
tiv ity maximum at ~ 10° K is not as great as one would
expect from umklapp processes and boundary scattering
alone (see Ref. 31 and references cited therein.) I t
should be pointed out, however, that i f normal processes
produce a modest reduction in t (such th a t WT)$>1 still
holds), then the microwave phonon-attenuation mecha­
nism discussed b y M aris should be slightly stronger
than calculated above and should produce an attenua­
tion varying as
in the range of our experiments.
This is probably the explanation of the behavior of longi­
tudinal microwave phonons in MgO,® fo r Shiren^® has
shown th a t normal processes involving three collinear
longitudinal phonons occur more frequently in MgO
than in quartz.
V. ATTENUATION IN IMPERFECT QUARTZ
In several specimens of imperfect quartz we found
steps in the attenuation plotted as a function of tem­
perature. These steps always occur a t temperatures
w ith in a few degrees of 16 and 24°K, regardless of the
origin of the crystal, or of the phonon mode con­
cerned.®® The existence of the steps is not consistent
w ith a reduction in the thermal phonon relaxation time
T b y scattering, and so must be caused b y a direct inter­
action of the microwave phonons w ith the imperfec­
tions. In the megacycle range there is abundant evi­
dence fo r peaks in the attenuation of imperfect quartz
and fused silica at various temperatures,®®"®® and re­
cently Jones ei aZ.®® have observed large and very broad
peaks in the attenuation of ~1000 M c/sec phonons in
fused silica at temperatures of ~ 1 0 and ~ 6 0 ° K . I t
seems probable, therefore, th a t the steps we observe
are really peaks superimposed on a large phononphonon attenuation varying as
to T®. I t is not always
P. G. Klemens, Solid State Phys. 7, 56 (1958).
® The only serious exception we have come across is the meas­
urement at 500 M c/sec of N ava et al. (Ref. 5), who find (Fig. 5 of
Ref. 5) a step below 10°K in im perfect ac-cut quartz. Of course,
i t is quite possible th a t there are steps at temperatures below 15°K,
b u t at 9000 M c/sec our accuracy is insufficient to positively
id e n tify these. However, in some of our samples there is evidence
of a shallow m inim um in the attenuation at ~ 8° K ; this is indica­
tive of another step or peak at a lower temperature.
W . P. Mason, Physical Acoustics (Academic Press In c., New
Y o rk, 1965), Vol. I I I B , p. 235, and references cited therein.
^ O. L . Anderson and H . E. Bommel, J. Am . Ceram. Soc. 38,
125 (1955).
H . E. Bommel, W . P. Mason, and A. W . W arner, Phys. Rev.
102, 64 (1956).
C. K . Jones, P. G. Klemens, and J. A. Rayne, Phys. Letters
8, 31 (1964).
M
710
LEWIS
AND
2 .0
lO
30
T E M P E R A T U R E (fK )
F ig . 7. Excess attenuation in an im perfect quartz specimen,
possible to ve rify this from our curves b y subtracting a
perfect curve from an imperfect curve to obtain the
excess attenuation because the excess attenuation often
continues at temperatures above ~ 2 4 °K , probably due
to the ta il of a ~ 6 0 ° K peak (see Figs. 2, 5, and 6). We
therefore subtract from the imperfect curve a curve
w ith the same shape as the perfect one, but, when neces­
sary, shifted along the temperature axis to meet the im ­
perfect curve at the highest temperatures. This seems
reasonable since the regions between the steps all have
about the same slope. A n example of the excess attenua­
tion so derived is shown in Fig. 7; this was deduced
from curve C and the dashed theoretical curve of Fig. 6.
The results are clearly consistent w ith attenuation peaks
at ~ 1 6 and 24°K.
A possible explanation of these peaks is th at they are
caused b y a structural relaxation mechanism involving
displaced oxygen atoms.®^*®® The sharpness of the peaks
indicates a narrow distribution of silicon-oxygen bond
angles such as m ight be caused b y specific impurities
distorting the lattice in a well-defined manner.*® A n
alternative explanation of the attenuation peaks is as
follows. Jones et aZ.*® have noticed a possible correlation
between the temperatures of the attenuation peaks in
fused silica, and the Einstein temperatures used b y
Flubacher et al}^ to account for the anomalous specific
heat of fused silica. They suggest th a t the attenuation
is caused b y interactions w ith displaced oxygen atoms,
low frequency sideways vibrations of which are believed
to be responsible fo r the excess specific heat of fused
silica.^® Presumably the interaction is of the form
[ultrasonic phonon]
-|-[lo w frequency optical phonon]
[some other (thermal) phonon]
(4)
This is not norm ally an allowed process*'^ because the
optical phonon branch norm ally lies above the acoustic
branch, b u t imperfect quartz and fused silica are excep­
tions. This was definitely established in the RamanJ. M . Ziman, Electrons and Phonons (Oxford U niversity Press,
New Y o rk, 1958), p. 114.
E.
PATTERSON
159
spectrum experiments described in Ref. 19. This
accounts fo r relative maxima in the ultrasonic attenua­
tion in the region of the Einstein temperatures (13, 32,
and 58°K) although the highest temperature peak is
at least p a rtly caused b y a structural relaxation mecha­
nism. We suggest th a t our imperfect quartz behaves
like “ dilute” fused silica, and th at we are able to resolve
the low-temperature attenuation peak of Ref. 36 into
two peaks at ~ 1 6 and ~ 2 4 ° K . We have already men­
tioned evidence for our observation of the ta il of a
~ 6 0 ° K peak. The w idth of the peaks in fused silica
indicates a distribution of bond angles as expected from
the random-network theory of the glassy state.*® The
average bond angle is about the same as in imperfect
quartz.*® On the basis of an interaction of the form of
Eq. (4), we see th a t there is a remarkable correlation
between the Einstein temperatures used to explain the
specific heat (13, 32, and 58°K), and the temperatures
of the attenuation peaks (16, 24, and ~ 6 0 ° K ). There
are, however, difficulties w ith this explanation; for
example, i t is n ot clear w hy the attenuation should
peak near the Einstein temperature rather than, say,
flatten off.
We have not been able to choose between a structural
relaxation mechanism and an optical phonon-ultrasonic
phonon interaction. The accuracy is probably insuffi­
cient to detect the change in the temperature of the
peaks w ith frequency th a t is expected if a structural
relaxation mechanism is operative.*^ Chemical analysis
of the samples used reveals no correlation between the
excess attenuation and the concentration of A1 or Na,
b u t there is a very rough correlation w ith the iron
content. I t is also quite possible th a t the growth rate is
as im portant as the im p u rity content.*®
VI. CONCLUSION
We have measured the attenuation of microwave
phonons of various wave vectors and polarizations in
perfect quartz specimens. We find th at the attenuation
of slow transverse phonons is determined b y the
Landau-Rumer mechanism, whereas the attenuation of
fast transverse and longitudinal phonons is determined
b y three-phonon processes which are only allowed as a
result of the finite lifetim e of the thermal phonons. The
rapid variation of t w ith temperature results in an
attenuation varying noticeably faster than
in the
range 4 -3 0°K at a frequency of 9000 Mc/sec. The
results are fitte d to the theories of M aris and Shiren,
and the appropriate coupling constants are listed for
several modes in quartz. I t appears to be valid to use
the thermal phonon lifetim e deduced from thermalconductivity measurements, probably because the latter
are at least p a rtly determined b y imperfections. A t
least in the case of longitudinal waves propagating on
B . E. W arren, J. A p p l. Phys. 8, 645 (1937).
C. S. Brown. Proc. Phys. Soc. (London) 75, 459 (1960).
159
MICROWAVE
PHONON-ATTENUATION
the X axis in quartz, there still remains a rather large
discrepancy in the magnitudes of the theoretical and
experimental attenuations.
We have found steps in the attenuation as a function
of temperature in quartz specimens containing various
degrees of imprefection. These steps are consistent w ith
attenuation peaks at 16dh3°K and 2 4 ± 3 ° K super­
imposed on the phonon-phonon attenuation found in
perfect specimens. There is a possible correlation
MEASUREMENTS
IN
QUARTZ
711
between the iron content of the imperfect specimens and
the intensity of the peaks. The origin of the attenuation
peaks is probably a structural relaxation mechanism or
an ultrasonic-phonon-optical-phonon interaction.
ACKNOWLEDGMENT
The authors are grateful to Professor E. H . Jacobsen
for permission to reproduce some of his measurements.
R eprinted fro m Jo u r n a l o f A pplied P h y s i c s , V o l. 39, N o. 7, 3420-3425, June 1968
C o p y rig h t 1968 by the A m e rican In s titu te o f Physics
Printed in U . S. A .
Microwave Phonon Attenuation in Magnesium Aluminate Spinel
M . F. L e w i s
and
E. P a t t e r s o n *
The General Electric Company Lim ited, Central Research Laboratories, H irs t Research Centre, Wembley, England
(Received 18 December 1967)
Measurements are reported of the attenuation of microwave phonons of various modes in magnesium
alum inate spinel (M gAla04) . The attenuation of certain modes is lower than in any other m aterial measured
and perm its the observation of 9000 M H z echo patterns a t room temperature. Some possible applications of
this discovery are discussed. The attenuation a t room temperature is interpreted in terms of an effective
viscosity damping mechanism and approximate values of the components of the viscosity m a trix are given.
I. INTRODUCTION
W ith in the past few years many measurements have
been reported of the temperature dependence of the
attenuation of microwave phonons in various dielectric
and semiconducting m a t e r ia l s T h e interest in these
measurements arises p a rtly from the ligh t they have
begun to shed on the interactions of ultrasonic and
thermal phonons/*-^® b u t also because an understanding
of these attenuation processes could lead to the identi­
fication or development of new materials exhibiting
lower phonon attenuation. Such materials could find
apphcation as microwave delay lines. So far, ultrasonic
delay lines at microwave frequencies have only been
practicable for frequencies below about 3000 M H z
at room temperature, although Van de Vaart et a lP
have extended the range by employing narrow-band
parametric amplification of hybrid spin-elastic waves in
y ttriu m iron garnet. The results on magnesium alumi­
nate spinel described in this paper indicate th at passive
ultrasonic delay lines should now be practicable up
to 9000 M H z at room temperature. However, in order
* Permanent address: Chelsea College of Science and Technol­
ogy, U n ive rsity of London, London, England.
1H . E. Bom mel and K . Dransfeld, Phys. Rev. Letters 2, 298
(1959).
®H . E. Bom mel and K . Dransfeld, Phys. Rev. 117,1245 (1960).
®E. H . Jacobsen, Quantum Electronics C. H . Townes, Ed.
(Columbia U n ive rsity Press, New Y ork, 1960), p. 468.
* E. G. Spencer, R. T . Denton, and R. P. Chambers, Phys. Rev.
125, 1950 (1962).
« H . J. M aris, P hil. M ag. 9, 901 (1964).
®R. Nava, R. A zrt, I. Ciccarello, and K . Dransfeld, Phys; Rev.
134, A 581 (1964).
’ I. S. Ciccarello and K . Dransfeld, Phys. Rev. 134, A1517
(1964).
®H . J. Shaw, D . K . Winslow, A. K arp, and R. A . Wilson, Appl.
Phys. Letters 4, 28 (1964).
®T . M . Fitzgerald, B. B. Chick, and R. T ru e ll, J. Appl. Phys.
35, 2647 (1964).
M . Pomerantz, Phys. Rev. 139, A 501 (1965).
" R. A. Wilson, H . J. Shaw, and D . K . W inslow, J. A ppl. Phys.
36, 3269 (1965).
.
J. de K le rk , Phys. Rev. 139, A1635 (1965).
J. de K le rk and P. G. Klemens, Phys. Rev. 147, 585 (1966).
J. de K le rk , J. A ppl. Phys. 37, 4527 (1966). ■
J. Lam b and J. Richter, Proc. Roy. Soc. A293, 479 (1966).
^®M. F. Lewis and E. Patterson, Phys. Rev. 159, 703 (1967).
” M . G. H olland, A. E. Paladino, and R. W . Bierig, J. Appl.
Phys. 38, 4100 (1967).
M . Pomerantz, Phys. Letters 24A, 81 (1967).
H . Van de Vaart, D . H . Lyons, and R. W . Damon, J. Appl.
Phys. 38, 360 (1967).
to reduce the insertion loss of about 80 dB in our
experiments, some refinement of the transducer designs
would be required, possibly along the lines described in
Refs. 11 and 20.
The discovery of a suitable ultrasonic delay medium
should also perm it the extension of characteristic
impedance measurements on liquids,®^ to X-band
frequencies at room temperature.
II. EXPERIMENTAL PROCEDURE AND RESULTS
The apparatus for microwave ultrasonic generation
and detection is basically the same as th at described
elsewhere,^® except th a t the pulselength is reduced to
0.2 /isec in order to resolve echoes in the short spinel
crystals. The operating frequency is 9155 M H z. For
transverse wave generation, th in-film nickel trans­
ducers were used®^“ ®^ and for longitudinal wave gener­
ation, thin-film CdS transducers were used.-®
The technique we used for longitudinal waves was
th at normally employed in the measurement of the
temperature dependence of the attenuation, i.e., to
compare the intensities of various pairs of echoes as a
function of temperature.^® The graph of attenuation vs
temperature is conventionally normalized to zero at
4.2°K, thereby subtracting from the total attenuation
any temperature independent attenuation, e.g.,
Rayleigh scattering from point defects in the lattice.
Transverse waves presented more of a problem. As
well as the uniform precession mode, th in ferromagnetic
films display a spectrum of spin-wave modes which also
couple magnetostrictively to the lattice3*-^^ Overlap,
and interference between these modes then lead to an
echo pattern which changes drastically w ith magnetic
.'field. Now the demagnetizing and strain-induced
anisotropy fields in the nickel films change w ith
temperature due to the variation of the saturation
magnetization, and differential contraction between
the film and substrate, respectively. These effects are
fa irly small from 4.2° to ~ 6 0 ° K , and Pomerantz^®
has used magnetic films to measure the attenuation of
J. de K le rk and E. F. K e lly , A ppl. Phys. Letters 5 , 2 (1964).
J. Lam b and H . Seguin, J. Acoust. Soc. Amer. 39, 519 (1966).
H.
E. Bommel and K . Dransfeld, Phys. Rev. Letters 3, 83
(1959).
®®M . Pomerantz, Phys. Rev. Letters 7, 312 (1961).
®<M. F. Lewis, T . G. Phillips, and H . M . Rosenberg, Phys,
Letters 1, 198 (1962).
3420
3421
MICROWAVE
PHONON
ATTENUATION
microwave phonons in many materials at temperatures
up to, and occasionally above, liquid-nitrogen temper­
atures. However, between '~ 6 0 °K and room temper­
ature, things change considerably. In the experiments
reported here, this difficulty was accentuated by the
smallness of the available crystals, and in some cases,
b y the smallness of the attenuation being measured.
A fter tryin g several measuring techniques (e.g., using
a fixed magnetic fie ld ), we found one th at gives reason­
ably consistent results; by consistent we mean that the
indicated attenuation is approximately the same for
any pair of echoes, The entirely empirical procedure is
as follows: We adjust the specimen orientation w ith
respect to the dc and microwave magnetic fields u n til
the echo pattern is as nearly as possible exponentially
decaying at room temperature (see Fig. 1), and where
the echo pattern peaks up as a whole at one particular
field at the high-field end of the spin-wave spectrum,
i.e., near the uniform precession mode. W ith patience,
these conditions can usually be satisfied simultane­
ously. As the temperature is slowly changed, the
magnetic field is adjusted so as to follow the peak in the
echo pattern. Typically, the applied field decreases by
500 Oe as the temperature is reduced from room
temperature to 4.2°K. This indicates that the change in
the strain-induced anisotropy field is more im portant
than the change in demagnetizing field. The attenuation
curves for the transverse waves shown in Figs. 2-4 are
each the averaged results of several runs. We regret
th a t at present we cannot be sure th a t the attenuation
indicated at the lower temperatures is accurate to better
than ± 5 0 % . The accuracy at the higher temperatures
is approximately ± 1 5 % , b u t i t is difficult to allow for
any remaining systematic errors which arise from the
transducer properties. A further difficulty w ith nickelfilm transducers is mode conversion of linearly polarized
Drj ving
p u ls e
II
in K J t in
F i g . 1. Photograph of fast transverse wave echoes a t 9155
M H z and room temperature in (110) cu t spinel. Tim e base:
1 /tsec/large scale division. Tim e increases to the le ft.
IN
M gA I 2 O4
100
X C UT QUAR TZ
SLO W TRANSVERSE
L O N G IT U D IN A L
(RER 16)
< 1 0 0 S P IN E L
01
20
50
O
L O N G IT U D IN A L W AVES
X
T R A N S V E R S E WAVES
100
200
500
1000
T E M P E R A TU R E (“k .)
F i g . 2. A tte n u a tion of 9155 M H z phonons in
(100) cut
spinel. F or comparison we include some typical measurements
fo r quartz taken from Ref. 16.
transverse waves on reflection from the nickel film.^®
This gives rise to an additional attenuation (e.g., for
transverse waves in the ( 110) direction), which we
have assumed to be independent of temperature. Our
attempts to extend the measurements on transverse
waves to temperatures above room temperature were
largely foiled by a decline in the efficiency of the trans­
ducers, probably due to increasing spin-wave losses.^®
Nevertheless, echoes have been observed up to 200°C.
The specimens used were s tric tly stoichiometric
single crystals of MgAlg04, grown b y the fluxed melt
technique at The General Electric Co., L td ., Wembley,
England. Some contained small concentrations of
Cr*+ (< 0 .1 % ) and were slightly pin k in color. No
extra phonon attenuation was observed at any temper­
ature when the magnetic field brought the Cr®+ spins
to resonance at the phonon frequency. Samples were
typ ically 4-5 mm long and 3 mm in diameter. They
were oriented in the ( 100), (110), and ( 111) directions
“ M . F. Lewis and T. G. P hillips, Proc. I.E .E .E . 56,343 (1968) ;
see also T . G. P hillips, thesis, U n iv. Oxford, England (1964)
(unpublished).
See Ref. 25 ( b ) . Other possible reasons fo r the reduced effici­
ency of N i film transducers above room temperature are the
reduction in m agnetostriction constants and increased phonon
losses in the films.
M.
F.
LEWIS
AND
E.
PATTERSON
3422
extended by several w o r k e r s I t has been pointed
out b y Mason®® th at all these treatments lead to an
attenuation of the form
a = [C T fw 4 /2 p F » ( l+ w V ) ]
L O N G IT U D IN A L WAVES
X TR A NSVERSE WAVES
TH E S E TW O SETS O F
M E A S U R E M E N T S WERE
TAKEN O N D IF FE R E N T
S P E C IM E N S .
( w r < l) ,
( 1)
where C is the specific heat per u n it volume, T the
absolute temperature, p the density, V the velocity of
the ultrasonic mode considered, and y a Griineisen
constant (of order u n ity ), which is some measure of the
anharmonicity of the lattice forces. A similar equation
to (1) has been employed successfully by Mason and
Bateman®^ in which the uncertainty in the y is removed
b y evaluating the y in terms of the measured thirdorder elastic constants of MgO, Y IG , Si, Ge, NaCl,
and K C l. In the case of transverse ultrasonic waves, the
value of T is closely th a t determined from the thermal
conductivity, b u t for longitudinal waves, better agree­
ment is obtained using twice the thermal conductivity
relaxation time.®^
In the low-temperature region, the interaction of
ultrasonic phonons w ith thermal phonons proceeds by
discrete, energy-conserving, wavevector conserving
three-phonon processes. The theory for the attenuation
I
too
lO
SO
lO O
200
T E M P E R A T U R E (^K)
F i g . 3. A ttenuation of 9155 M H z phonons in
(111) cut spinel
to
b y the x-ray Laue back-reflection technique.
The faces perpendicular to these ultrasonic pure-mode
axes were polished fla t to about one fifth of a wave­
length of sodium light, and parallel to a few seconds of
arc. The CdS and N i film transducers were typ ically
3000-5000 Â th ick and were vacuum deposited w ith
substrate temperatures of 200° and 300°C, respectively.
The attenuation measurements fo r longitudinal and
transverse waves in the ( 100), ( 111), and ( 110) direc­
tions are shown in Figs. 2-4, respectively.
lO
1.0
III. DISCUSSION
A. Mechanism and Temperature Dependence of the
Attenuation
< I I O > S P IN E L
LO N G ITU D IN A L WAVES
The attenuation of microwave phonons in good single
crystals of dielectric materials is believed to be domi­
nated by interactions w ith thermal phonons (see, e.g..
Ref. 27, and references cited therein). A t high temper­
atures where w r < l (w is the ultrasonic frequency and t
is the thermal phonon relaxation time) the attenuation
is believed to occur b y a mechanism first described by
Akhiezer.2® Subsequently, Akhiezer’s ideas have been
^ P. G. Klemens, in Physical Acoustics W . P. Mason, E d.
(Academic Press Inc., New Y prk, 1965), Vol. 3b.
A . Akhiezer, J. Phys. Moscow I , 277 (1939).
FAST TR A N S V E R S E WAVES
SLOW
T R A N S V E R S E WAVES
T E M P E R A TU R E (» K )
F ig . 4. A ttenuation of 9155 M H z phonons in (110) cu t spinel.
“ T . 0 . W o o d ru ff and H . Ehrenreich, Phys. Rev. 123, 1553
(1961).
W . P. Mason, in Ref. 27, Chap. 6.
P. Mason and T . B. Bateman, J. Acoust. Soc. Amer.
40 ,85 2 (1966).
3423
MICROWAVE
PHONON
ATTENUATION
where s' is an average thermal phonon phase velocity
and F i is a complicated average of second- and thirdorder elastic constants. The actual form of at in Eq.
(2) is taken from Ref. 5. The attenuation mechanism
for longitudinal ultrasonic waves at low temperatures
is p rim arily through interactions w ith collinear
thermal phonons.®'®'^'®^ Provided the temperature is
not too low, the attenuation takes the form of Eq. (2),
b ut at very low temperatures, the attenuation varies
more rapidly than
explained b y Maris,®
Nava et al.,^ and Simons.®^
In order to compare our measurements on spinel w ith
the theories discussed above, we must determine which
region we are working in, i.e., w r> 1 or c o r< l. We can
estimate t from the thermal conductivity measurements
of Slack®® on natural spinel crystals, using the ap­
proximate expression
M g A 1 %0 &
lO O O
of transverse waves was worked out by Landau and
Rumer^ who obtain a result of the form
(2)
IN
Z
500
200
3
lO O
50
20
SPINEL
(3)
where C =specific heat per u n it volume, V is an average
thermal phonon (group) velocity in the Debye sense,
and r is an average thermal phonon relaxation time.
Following Oliver and Slack®® we compute V from the
expression
V = 1OOO0debye(F o)'/V2.98.
(4)
Here 0 d e b y e is in °K , and Fo is the average volume per
atom in cubic  and has the value 9.42 for spinel. This
gives F~6.4X10® cm/sec. From the values of K in
Ref. 35 and values of C deduced from the Debye ex­
pression w ith 0 d e b y e ~ 9 O O ° K , we deduce the temper­
ature variation of r shown in Fig. 5, and note that
cdT— 1.0
at
100°K
ûj7^0.034
at
300°K.
(5)
Thus, our measurements were nearly all taken in the
intermediate to high-temperature region. I t is not
possible to obtain values of the quantities F in F q. (2)
from measurements at the lowest temperatures, since
the attenuation is too low to be measured in the region
where a T* variation is expected. For comparison, we
show some typical attenuation curves for quartz in
Fig. 2; these are taken from Ref. 16. Now Mason and
Bateman®^ have shown th a t F q. (1) is often a reason­
able f it in the intermediate region, so we attem pt to
compare our results w ith Fq. (1). We substitute Fq.
(3) into F q. (1) and obtain, for some average ultra32 L . Landau and G. Rumer, Phys. Z. Sowjetunion 11, 18
(1937).
33 N . S. Shiren, Phys. L e tt. 20, 10 (1966).
34 S. Simons, Proc. Phys. Soc. (London) 83, 749 (1964).
33 G. A. Slack, Phys. Rev. 126, 427 (1962).
36 D . W . O liver and G. A. Slack, J. Appl. Phys. 37,1542 (1966).
20
50
IC O
200
500
IO O O
TEM PERATURE C k )
F i g . 5 . Com puted va riatio n of the therm al phonon relaxation
tim e, T, ys temperature. Values were computed from the therm al
conductivity data of Ref. 3 5 .
sonic attenuation coefficient (in d B /c m ),
a = 8.68 (3A:rcoV)/2pF® ( 1 +co®r2).
(6)
The simple (unweighted) average of the roomtemperature attenuation coefficients listed in Table I
is a ~ 2 5 dB/cm . Using this value in F q. (6) gives
7 = 0.56. Oliver and Slack®® have found th at for most
materials 0.6 < 7 ' < 1.2, where 7 '= (f)^ ^ ^ 7, so th a t the
average behavior of spinel is not grossly different from
th at of other materials. However, i t is evident th at an
order-of-magnitude calculation like this can be con­
siderably in error in estimating the attenuation of
individual modes. In Fig. 6 we have plotted the v a ri­
ation of a w ith temperature calculated from F q. (6) .
The agreement w ith experiment is not particularly
good, the calculated attenuation dropping to half its
room-temperature value at ^ 1 0 0 ° K as opposed to the
observed values of '^ 1 6 0 °K (see Figs. 2 -4 ). The reason
for this could be th at our specimens had higher thermal
conductivities than Slack’s impure natural specimen.
In order to account for the discrepancy, the values of r
would need to be ~ 4 times higher at 160°K in our
specimen than in Slack’s specimen. Such a large d if­
ference seems unlikely at the temperatures involved.
For completeness, we also plot in Fig. 6 the variation of
a w ith T using a variation of Fq. (1) due to Mason and
M.
F.
LEWIS
AND
E.
PATTERSON
3424
T a b l e I . L is t of the properties of the seven pure-mode ultrasonic waves studied in MgAlzO*.
W avevector
M ode
Velocity*
in units of
10®cm/sec.
Polarization
Measured a tt. Calculated att.**
a t room temp, a t room temp.
(d B /cm )
(d B /cm )
Effective
viscosity
m
( 100)
longitudinal
8.83
'-'3 6
36
b
( 100)
transverse
6.54
10
10
Vii
Vu
20
K ’lu+2»7ii-l-477«)
c
( 111)
longitudinal
d
(111)
transverse
10.6
e
( 110)
longitudinal
f
( 110)
fast
transverse
6.54
g
( 110)
slow
transverse
4.19
5.1
10.15
“ M . F . Lew is, J. A coust. Soc. A m . 4 0 , 728 (1 9 6 6 ).
CdT.
This is seen to give slightly improved agreement w ith
experiment in the region w r < l. In the case of longi­
tudinal ultrasonic waves, a further small improvement
100
B: C A L C U L A T E D
i (%l4-«744—?iz)
23
I ( ’7i i + t?12+2j74«)
9
10
Vu
~42
49
see Fig. 4
1712)
jji2=0.43
c P and 1744= 0 .0 7 c P .
between theory and experiment can be obtained by
using in Eq. (1) a relaxation time equal to twice the
thermal relaxation time, as suggested in Ref. 31.
In concluding this section on the temperature de­
pendence of the attenuation, we can say th at the results
are only in qualitative agreement w ith the expression
for a relaxation mechanism when w r < l, b u t are tending
toward the T* variation expected from the LandauRumer mechanisms as wr becomes )$>!.
B. Dependence of the Room-Teniperature Attenuation
on the Ultrasonic Mode
A T T E N U A T IO N
8 .6 8 Ep
U S IN G
25
25
^ C om puted from E q . (7) using th e valu es i;u = 0 .6 1 cP ;
Bateman®^ in which C T is replaced by
Eo,= f
see Fig. 3
tx =
2 J > V * ( l + C O *T *)
W IT H
y =1
(S E E
Ideally, the calculation of the attenuation of in ­
dividual ultrasonic modes would be made through the
third-order elastic constants.®^ However, in the absence
of any measured values of the third-order elastic
constants of spinel, we believe th a t the most suitable
comparison of our results w ith theory can be made w ith
an effective viscosity damping approach. B y adopting
this approach, we are able to check th a t our measure­
ments on the seven pure-mode ultrasonic waves are
consistent w ith the symmetry of the effective viscosity
m atrix. Lamb and Richter^® showed th at (on the
REF. 3 j)
C A LC U LA TE D
A T T E N U A T IO N
8 .6 8 3 K T
w it h
ÿ
■3 = 0 . 5 6
SPINEL
0.1
20
50
100
200
500
1000
T E M P E R A T U R E ( “ k)
F ig . 6. Calculated variation of the attenuation of 9155 M H z
phonons in spinel on the basis of a relaxation mechanism. Curve
A is deduced from Eq. (6) w ith 7=0.56. Curve B is deduced
from the work of Mason and Bateman.®^ Compare w ith Figs.
2-A a t temperatures above 100'’K .
F ig . 7. Photograph of longitudinal wave echoes at 1000 M H z
and room temperature in (111 ) cut spinel. Tim e base: 5 /xsec/large
scale division. Tim e increases to the right. The echoes overlap as
this particular specimen is only 3 m m long.
3425
MICROWAVE
PHONON
ATTENUATION
assumption th at the attenuation is caused by a relaxa­
tion mechanism) the attenuation of <^1000 M H z
phonons in quartz and silicon at room temperatures is
given by
a = 8.68(coV 2pP)
(w r < l),
(7)
where a = attenuation in dB/cm , rj is an effective vis­
cosity,^ F = velocity of the wave under consideration.
Since MgAla04 is cubic and 17 is a fourth-rank tensor,
there are only three independent components, viz.,
7711, 7712, and 1744 (in the contracted Voigt n o ta tio n ). The
effective 77values for the seven pure modes studied here
are listed in Table I, together w ith other pertinent
inform ation. From the room-temperature attenuations
of modes ( b ) , ( f) , and ( d ) , we deduce the following
(cP = centipoise) :
7744= 0.07 cP ± 1 0 % ;
(8)
7711—7712= 0.18 cP ± 1 5 % .
(9)
The very small value of 7744 m ay be related to the
fact th at the modes whose attenuation i t controls, (b)
and ( f ) only interact w ith a small number of thermal
phonons (18 of the 39 pure mode thermal phonons
phonons which propagate in cubic crystals*®). Reason­
able extrapolations of the longitudinal wave attenuation
curves to room temperature give consistent results if
7711=^0.61 cP;
( 10)
77i 2~0.43 cP.
( 11)
W hile these values are reasonably consistent, in view
of the difficulties encountered w ith the temperature
dependence of the attenuation, i t is not justifiable to
use Eq. (7) unless i t can be verified th a t a varies as
(jp at room temperature [E qs. (1) and ( 7 ) ]. The only
other equipment we have available operates at 1000
M H z. A t this frequency, the apparent attenuation of
the longitudinal modes (using CdS transducers) is
dominated b y phase-cancellation effects,® (Fig. 7),
and is too small to be measured, b u t does not conflict
w ith the values a i'-'0.2 to 0.4 d B /cm expected from
our results at 9000 M H z, assuming a w® dependence.
The measurements are not consistent w ith a to de­
pendence of the attenuation which would give an
attenuation of 2-4 d B /cm for the longitudinal modes.
We are therefore supported in using Eq. (7), b ut
measurements at '—'3000 M H z would be useful to verify
the frequency dependence more accurately.
®7K. Dransfeld, J. de Physique, Colloque C l, Suppl. 28, C l
(1967).
IN
MgAl,0«
Finally, a few general remarks: F irs tly , the observa­
tions in Figs. 2-4 on the attenuations of various phonon
modes are in broad agreement w ith the predictions of
Ciccarello and Dransfeld^ and Dransfeld.*’ Thus, on a
(100) axis, the attenuation of transverse waves is less
than th a t of longitudinal waves, while on a (111)
axis (which lacks even-fold symmetry) the attenu­
ation of transverse waves is similar to th a t of longi­
tudinal waves. Secondly, we note th a t the low u ltra ­
sonic attenuation in spinel is not entirely unexpected
from the work of Oliver and Slack.®* These workers
suggested th at in the w r < l region, a low attenuation is
expected in materials w ith (a) ligh t mass atoms, (b)
high Debye temperatures, and (c) complex crystal
structures. On all three accounts, spinel would be
expected to have a fa irly low attenuation at room
temperature. We do not know what influence, if
any, the Cr®+ impurities have on our measurements,
b u t we repeat th a t the measurements in Figs. 2-4
were taken on different crystals, b u t all w ith Cr*+
concentrations of < 0 .1 % . According to Oliver and
Slack,** i t is possible th a t any im purities would reduce
the temperature-dependent ultrasonic attenuation by
reducing t . Further studies are in progress to determine
the effects of impurities on the room temperature
attenuation.
IV. CONCLUSION
Measurements are reported of the temperature
dependence of the attenuations of the seven pure-mode
ultrasonic waves which can be propagated in single­
crystal M gA b 04. I t is shown th at this material possesses
lower ultrasonic losses in certain modes than any
other material measured. The attenuation at 9000
M H z and room temperature is shown to be consistent
w ith a phonon viscosity damping mechanism. The
temperature at which the attenuation drops to half its
room temperature value is about 160°K, rather than
the value of 100°K calculated from the thermal con­
d u c tivity. This m ight in p art be due to the higher
thermal conductivity of our specimens than the
natural spinel on which the thermal conductivity
measurements were made. A t low temperatures,
(wT> 1) the average variation of the attenuation w ith
temperature is
F*, which is approaching the T*
variation expected (a t least for transverse waves)
when
The low ultrasonic losses in M g A l204
could lead to the development of microwave delay
lines operating at X-band frequencies and room
temperature, and should enable ultrasonic studies on
liquids to be extended to much higher (possibly X band) frequencies at room temperature.
R eprinted fro m J o u r n a l o f A p p l i e d P p i y s i c s , V o l. 39, No. 4 , 1932-1936, M a rc h 1968
C o p y rig h t 1968 by the A m e rican In s titu te o f Physics
Printed in U . S. A.
Microwave Phonon Attenuation in Yttrium Iron Garnet
M . F. L e w i s
and
E. P a t t e r s o n *
,
The General Electric Company Lim ited, Central Research Laboratories, H irs t Research Center, Wembley, England
(Received 7 September 1967; in final form 20 November 1967)
We report some microwave ultrasonic attenuation measurements a t 9155 M H z on pure single-crystal
y ttriu m iron garnet (Y IG ) which we believe determine the in trinsic attenuation in this m aterial. Previous
measurements on Y IG b y other workers were complicated by spurious effects. Our measurements are
compared w ith current theories of attenuation by phonon-phonon interactions using the recently measured
third-order elastic constants of Y IG . The agreement found is about the same as in m any other materials,
which suggests th a t the in trinsic attenuation in Y IG is understood and is caused by interactions w ith therm al
phonons.
1. INTRODUCTION
Recently there has been a considerable interest in the
attenuation of microwave phonons in dielectric and
semiconducting materials. We lis t here some of the
more recent m e a su re m e n ts,w h ile references to earlier
work can be found in Refs. 9 and 10. The material
y ttriu m iron garnet (Y IG ) is of particular interest be­
cause i t is known to exhibit very low ultrasonic" and
spin-wave losses, (see, for example. Ref. 12). These
properties make i t an attractive m aterial fo r such de­
vices as microwave delay lines.^® Despite this, there are
no reliable measurements of the intrinsic ultrasonic a t­
tenuation in Y IG . The earliest measurements at 500
and 1000 M H z were only able to set an upper lim it on
the attenuation of transverse waves propagating in a
(100) direction, because there is an unknown ultrasonic
attenuation due to magnetoelastic effects."’" Meas­
urements at low temperatures are usually confused by a
relaxation peak; for 1000 M H z phonons, this occurs at
12.5°K." The experiments reported here are an exten­
sion of these measurements to W-band frequencies, and
temperatures from 4.2°K to the highest temperature at
which the attenuation can be measured.. The high
* Permanent address: Chelsea College of Science and
Technology, U niversity of London, London, England.
^ R. B. H em phill, Appl. Phys, Letters 9, 34 (1966) ; (ZnO ),
* C. P. Wen and R , F. M ayo, Appl, Phys. Letters 9,135 (1966) ;
(LiNbOa).
®M , I . Grace, R. W . Kedzie, M . Kestigian, and A. B. Smith,
Appl. Phys, Letters 9, 155 (1966); (LiN bO s),
^ E, G, Spencer, P. V. Lenzo, and A, A. Ballm an, Appl, Phys.
Letters 9, 290 (1966) ; (Bi.;GeO%o).
®J. Ilu k o r and E. H . Jacobsen, Science 153, 1113 (1966);
(SiOa, 114 000 M H z ).
®M . G. B la ir and E. H . Jacobsen, Phys. Letters 23, 647 (1966) ;
’ R, M . A rzt, E. Salzmann, and K . Dransfeld, Appl. Phys.
Letters 10, 165 (1967) ; (SiOz).
®E. G. Spencer and P. V . Lenzo, J, Appl. Phys. 38, 423 (1967) ;
(LiNbOs and LiTaOa).
* M . Pomerantz, Phys. Rev. 139, A501 (1965).
“ M . F. Lewis and E. Patterson, Phys. Rev. 159, 703 (1967).
E. G. Spencer, R , T . Denton, and R. P. Chambers, Phys. Rev.
125, 1950 (1962).
^^T. Kasuya and R. C. LeCraw, Phys. Rev. Letters 6 , 223
(1961).
E. Schlomann, R. I. Joseph, and T . Kohane, Proc. IE E E 53,
1495(1965).
/
R. C. LeCraw and R. L . Comstock, in Physical Acoustics,
W . P. Mason, Ed. (Academic Press In c., New Y o rk, 1965), Vol.
3b, Chap. 4.
phonon-phonon attenuation in these circumstances re­
duces the relative effects of the low-temperature re­
laxation peak. Further, the use of th in-film CdS trans­
ducers for longitudinal wave generation" and thin-film
N i transducers for transverse wave generation" elim i­
nates any problems arising from magnetoelastic inter­
actions, as discussed later. We are therefore able to
present some measurements which we believe to be
characteristic of pure Y IG . The results are compared
w ith current theories of ultrasonic attenuation by inter­
actions w ith thermal phonons. Use is made of the th ird order elastic constant measurements of Eastm an" and
of the thermal conductivity measurements of Oliver and
Slack." The results are found to be in fa ir agreement
w ith theory, which suggests th a t the intrinsic attenua­
tion in Y IG is due to phonon-phonon interactions as in
most or all other nonmetallic single crystals studied.
2. EXPERIMENTAL PROCEDURE AND RESULTS
The experiments employ a pulse-echo technique
which has been fu lly described before." Basically, this
technique compares the intensities of various pairs of
echoes at different temperatures. As a result, we only
measure the temperature-dependent p art of the at­
tenuation, the residual attenuation at 4.2°K being
a rb itra rily set to zero.
The single-crystal Y IG specimens were the best
available, i.e., were grown from 99.9999% pure y ttria
and gave the lowest ultrasonic relaxation peaks at
9155 M H z and low temperatures, and also the lowest
magnetoelastic wave losses at 1000 M H z and room
temperature." Samples were in the form of rods, ty p ­
ically 3 mm in diameter and 7-10 mm long. They were
oriented in a (100) direction to an accuracy of
and
the end faces were polished optically fiat, and parallel
to a few seconds of arc. The CdS and N i film trans­
ducers were typ ically 2000-5000 A thick, and were
vacuum-deposited w ith substrate temperatures of 200°
and 300°C, respectively. The CdS films were never very
“ J. de K le rk and E. F. K e lly, Appl. Phys. Letters 5, 2 (1964).
1®H . E. Bommel and K . Dransfeld, Phys. Rev. Letters 3, 83
(1959).
:
" D . E. Eastman, J. Appl. Phys. 37, 2312 (1966).
“ D . W . O liver and G. A. Slack, J. Appl. Phys. 37, 1542 (1966),
1932
PHONON
1933
ATTENUATION
efficient transducers, and at the lowest temperatures it
was found more satisfactory to measure the attenuation
of longitudinal waves by bonding the samples to %-cut
quartz transducers w ith Araldite, in the manner de­
scribed in Ref. 19. The two sets of measurements (using
quartz and CdS transducers) joined smoothly together
as shown in Fig. 2 (a ). The accuracy of the measure­
ments can be estimated from the scatter of the points
in Fig. 2 (a ), taken on several different runs. To check
th at the residual attenuation at 4.2°K was essentially
temperature-independent, we reduced the temperature
to 2.2°K and observed no change in the attenuation to
w ith in the experimental accuracy of ± 0 .1 dB /cm . H ow ­
ever, a less pure ( lll) -o rie n te d specimen which had a
temperature-dependent attenuation of ^^5 d B /cm at
20°K (and too high to be measured above 20°K)
showed a reduction in attenuation by 0.7 d B /c m on
cooling from 4.2° to 2.2°K. This is presumably due to
the ta il of a very large attenuation peak at ^ 3 0 ° K . I t
was found that the attenuation of the 9155-MHz longi­
tudinal waves was not affected b y an applied magnetic
field, except for magnetic field values near
(co= phonon frequency, 7 = gyromagnetic ratio)
In
these circumstances there was an additional attenuation
when the field made an angle other than 0° or 90° w ith
YIG rod
Nickel film
z—f
2-L
IN
YIG
the rod axis (i.e., the ultrasonic wavevector). This be­
havior is expected from the work of Schlomann^^ and
has been observed previously a t lower ultrasonic frequencies.®^'^® The detailed attenuation measurements on
9155 M H z longitudinal waves reported here were taken
in zero magnetic field. In order to establish the fre­
quency dependence of the attenuation of longitudinal
waves in Y IG , we have made some measurements at
1000 M H z using CdS th in -film transducers. The re­
sults are shown in Fig. 2(b) where the accuracy varies
from ± 0 .1 d B /c m at low temperatures to ± 0 .3 d B /cm
at room temperature. The attenuation peak which oc­
curs a t ~ 2 3 0 ° K for 1000 M H z phonons in zero field
can be removed by the application of a large magnetic
field, and appears to be caused by/interactions w ith
domain walls.
Previous microwave ultrasonic attenuation measure­
ments on transverse waves relied on the magnetostric­
tive properties of Y IG its e lfH o w e v e r , when using
this technique i t is only possible to set an upper lim it on
the intrinsic attenuation as discussed in Ref. 14. In the
present experiments, these difficulties were overcome
b y using N i film transducers to generate transverse
waves. The applied magnetic field, typ ically 5 kOe, is
then sufficiently high th at the magnetic field throughout
the Y IG specimen exceeds w/7 (see Fig. 1). This is
shown in the Appendix from first-order demagnetizing
field calculations of the field on the rod axis.^^ The meas­
urements on transverse waves were taken w ith the ap­
plied magnetic field adjusted at each temperature for
the echo pattern to peak up as a whole. The procedure
has been described elsew h e r e . I n the present experi­
ments the applied field only varied b y one percent from
4.2° to 250°K. The results of measurements taken on
several runs are shown in Fig. 3. Between 4.2° and
about 50°K, the attenuation does not change measur­
ably. A n y peak in the attenuation in this temperature
range cannot exceed the accuracy of measurement
( ± 1 d B /c m ), or i t would be detected. Between 150°
and 250°K, the echoes are so small th at the accuracy is
only ± 2 dB /cm .
2nM i
3. DISCUSSION
F i g . 1. Approxim ate form of the internal magnetic-field
d istrib u tio n on the axis of the combined system, Y IG rod plus N i
film . The external field Ha is applied parallel to the axis of the
system. M \ and
are the saturation magnetizations of Y IG and
N i, respectively.
“ M . F. Lewis and A. M . Stoneham, Phys. Rev. 164, 271
(1967).
A ctually, the m aximum attenuation does not occur when the
b u lk of the sample is in an effective field H t = 0 / 7 , b u t in a field
H e = H t—Do^/V^, where D is the exchange energy constant and V
is the transverse phonon velocity, i.e., at the magnon-phonon
crossover p o in t discussed by C. K itte l, Phys. Rev. 110, 836 (1958).
F or Y IG at %-band frequencies, D w '/F ^ lO O Oe. A fu rth e r small
correction to H t arises aue to anisotropy fields.
The attenuation of microwave phonons of frequency
CÜdiffers markedly in the temperature regions fo r which
W T < 1 and W T > 1 . A t low temperatures ( w r ) ^ l) , the
attenuation mechanism is the microscopic three-phonon
interaction discussed b y Landau and Rumer^® and
“ E. Schlomann, J. Appl. Phys. 31, 1647 (1960).
“ B. L u th i, Phys. Letters 3, 285 (1963).
“ G. A. Smolenskii and A. N asyrov, Fiz. T ver. Tela 7, 3704
(1965): [E n g lish transi. : Soviet Phys.— Solid State 7, 3002
(1965)1
A. Sommerfeld, Electrodynamics (Academic Press In c., New
Y o rk, 1952), p. 82.
“ M . F. Lewis and E. Patterson, J. Appl. Phys. (to be pub­
lished) .
“ L . Landau and G. Rumer, Phys. Z. Sowjetunion 11, 18
(1937).
M.
F.
LEWIS
AND
E.
PATTERSON
1934
lO O
1
G.EC. 6 .9 ’s
30
1
l o n g it u d in a l
< IO O >
w aves
if
ii
A X IS
C dS t r a n s d u c e r
-
1
Y IG
FR EQ U EN C Y : lO O O M H z
SO LID
CURVE : SAMPLE SATURATED
%
DASHED CU RVE : SAM PLE
UN SATURATED
-
1
1
100
300
T EM PER A TU R E (” k)
0 .3
(b)
Y IG
(g .E C .
L O N G IT U D IN A L
0.1
< IO O >
6 . 9 's
P U R ITY^
F ig . 2. (a) Experim ental and theoretical attenuation curves fo r
longitudinal waves on a (100) axis in Y IG . A = experimental
curve, B = a p p ro xim a te form of the low-temperature attenuation
peak in Y IG , C = approximate form of the in trinsic attenuation in
Y IG , D = low-temperature theoretical curve, E = high-temperature
theoretical curve, (b) Experim ental attenuation curve fo r longi­
tu d in a l waves at 1000 M H z.
WAVES
A X IS
0
Q U A R TZ TR A N S D U C E R
•
CdS
T H IN
F IL M
TRANSDUCER
FREQU ENC Y : 9 1 5 5 M H z
0 .0 3
30
^ ^
TE M P E R A TU R E ( " k)
100
300
(a)
o th e r s . ^ A t high temperatures (c o r< l), the ultrasonic
wave is deemed to interact w ith all the thermal phonons
collectively. The attenuation is conventionally inter­
preted as due to a relaxation process.^®” ®®For the pur­
pose of comparing our measurements on Y IG w ith these
theories, we determine t from the approximate expres­
sion for the thermal conductivity K ,
K = \C Ÿ h .
Here C is the specific heat per u n it volume and can be
calculated from the Debye expression for the specific
heat using 0Debye=56O°K, as computed from the elastic
constants. The q ua ntity Ÿ is the Debye velocity and
has the value 4.3X10® cm/sec for Y IG . Values of K
are taken from the w ork of Oliver and S la c k ,a n d the
resulting variation of r w ith temperature is plotted in
Fig. 4. The condition cor= 1 occurs at about 67°K for
9155 M H z phonons. Before comparing the computed
and measured attenuation curves for longitudinal
27 H . J. M aris, Phil. Mag. 9, 901 (1964).
28 W . P. Mason, in Ref. 14, Chap. 6.
22 W . p. Mason and T . B. Bateman, J. Acoust. Soc. Am. 40,
852 (1966).
22 J. Lam b and J. Richter, Proc. Roy. Soc. (London) A293, 479
(1966).
waves, allowance must be made for the low-temperature
attenuation peak, which can be seen in Fig. 2(a) super­
imposed on the phonon-phonon attenuation. We have
therefore subtracted from the measured curve A, the
peak B, a similar procedure having been used by Spencer
et al.^^ to obtain the intrinsic ferrimagnetic resonance
line w idth in Y IG . For temperatures above about 50°K,
the corrected curve C is not particularly sensitive to de­
tails of curve B. The low-temperature theoretical curve
D in Fig. 2(a) is derived from the work of Pomerantz
[R ef. 9, Eq. (4 )] using the third-order elastic constants
of Eastman.^^ A curve very similar to D is obtained from
the work of Maris®^ using the relaxation times in Fig. 4.
The high-temperature theoretical curve E is derived
from the work of Mason and Bateman [R ef. 29, Eq.
(2 1 )], using the value D = 5.96 calculated in Table V of
Ref. 29. The agreement between theory and experiment
is to w ith in a factor of about two, which is tolerable.
The experimental and theoretical curves fo r trans­
verse waves propagating on a (100) axis in Y IG are
shown in Fig. 3. The low-temperature theoretical curve
is derived from the work of Pomerantz [R ef. 9, Eq, (3) ]
27 E. G. Spencer, R. C. LeGraw, and R. C. Linares, Jr., Phys.
Rev. 123, 1937 (1961).
PHONON
1935
ATTENUATION
and clearly underestimates the attenuation b y an order
of magnitude. A similar discrepancy between theory and
experiment has been found for Si and Ge (Ref. 9,
Table I ) . The high-temperature theoretical curve in
Fig. 3 is derived from the work of Mason and Bateman
using the value, Z)=0.26 (Ref. 29, Table V ). A t high
temperatures, the theory underestimates the attenua­
tion by a factor of two-three. This is the largest dis­
crepancy in the high-temperature region of which we
are aware. We cannot completely rule out the possi­
b ility th at some spurious effects add to the phononphonon attenuation, b u t the general form of the experi­
mental curve in Fig. 3 shows no unusual features. For
example, if the attenuation were dominated by losses in
a small spin-wave admixture into the elastic wave, it
m ight be expected to va ry as
This suggests that
we have measured the intrinsic attenuation. Concerning
the frequency dependence of the attenuation at room
temperature, theory predicts th a t a is proportional to
The upper lim it of 0.34 d B /cm at 1000 MHz^^ is
roughly in agreement w ith our measurements on trans­
verse waves if a is proportional to co^. In the case of
longitudinal waves, a comparison of Fig. 2(a) and (b)
shows th a t the attenuation at low temperatures (40 °-
IN
lO O O
O
500
200
so
20
YIG
20
30
1
I
o
1
YIG
so
lO O
200
SO O
lO O O
. fEM PÊR ATURE (°K^
F ig.4. Therm al phonon relaxation times in Y IG computed from
the therm al co n d u ctivity measurements of Ref. 18.
lO _
o
' -
To
/
3
/ /
i4
It
U
1
f
i /
-
y
4. CONCLUSION
/
I
^
0 .3
3
Z
5
0.1
/
/
YIG
(G .E.C. 6 .9 's p u r i t y )
TRANSVERSE WAVES
< IO O > A X IS
-
/
0 .0 3
;
/
/
,
0.01
10
50°K) varies as w" w ith n ^ \ , as expected. A t room
temperature the attenuation is ~ 1.7 dB /cm . Our
9155-MHz curve extrapolated to room temperature
gives an attenuation of ~ 5 0 dB /cm , which indicates
th a t the attenuation varies as to” w ith ;i~ 1 .5 instead
of 2. The departure from an co^-dependence at room tem­
perature has been observed in a few other materials
(see, for example Ref. 32, b u t is not understood at
present).
30
i
------ RELAXATION THEORY
VALID
FOR G J T < I
----- LANDAU-RUMER THEORY
VALID FOR C J T » 1
1
1
IC O
TEM PERATURE
300
lO O O
( '" k )
F i g . 3 . Experim ental and theoretical attenuation curves fo r trans­
verse waves on a (100 ) axis in Y IG .
We have measured the attenuation as a function of
temperature of 9155 M H z longitudinal and transverse
ultrasonic waves propagating on a (100) axis in Y IG .
The results agree w ith a theory based on interactions
w ith the thermal phonons to w ithin a factor of two or
three at high temperatures. Although the discrepancy
between theory and experiment is slightly greater than
in most materials, the shapes of the attenuation curves
and the frequency dependence of the attenuation sug­
gest th a t our measurements represent the intrinsic
attenuation in Y IG . A t low temperatures, there is good
agreement between theory and experiment for longi­
tudinal waves, b u t the theory underestimates the
attenuation of transverse waves by an order of magni­
tude.
R. A. Wilson, H . J. Shaw, and D . K . W inslow, J. Appl. Phys.
36,3269 (1965).
M.
F.
LEWIS
AND
APPENDIX: DEMAGNETIZING FIELD IN
COMBINED YIG ROD AND NICKEL FILM
We compute the demagnetizing field on the axis of
the combined system, Y IG rod plus nickel film . We as­
sume th a t both specimens are magnetically saturated.
Since V • B = 0 we have.
V ’H = —47rV’M.
( A l)
Since no currents flow, V x H = 0 and we can w rite
H = —V0, where 0 is the magnetostatic potential.
Thus,
V V = 47rV-M
(A2)
B y analogy w ith Poisson’s equation in electrostatics, the
solution is
(V 'M )r'dV
(A3)
In the case of Fig. 1,
V * M = iifi5 (0 ) fl- (Afg—M l) 5(/) —
,
where 5 is the D irac delta function. The only d ifficulty
in integrating Eq. (A3) concerns the modulus sign in
E.
PATTERSON
1936
the denominator. We find th a t w ithin the Y IG rod this
gives ( 0 < Z < / ) ,
-ci>= 27rMi(Z2-f p2)i/2+2x(M2-M:) [( /- Z ) 2 - f
-
2r
f 2[ ( L - Z ) 2+p 2]l /2_ 47TMiZ
+2TrMiZ+2ir%(L-Z).
Hence, on the axis,
(A4)
—d4>/dZ,
H = -4 7 rM i+ 2 7rM iZ (Z 2 -F p 2)-i/2
-
2 T(M g -M i)
(Z -Z )[(Z -Z ):+ p ^ }-i /2
+ 2 xM 2(L -Z )[(i:-Z )2 + p 2 ]-i/2 .
(A5)
Considering the effect of the terms involving
on
the Y IG , we find th at even the face in contact w ith the
nickel {Z -^ l) only experiences a field of
where t = ( L —l) is the thickness of the nickel film . In
our experiments this field is of order 5 Oe and is quite
negligible compared w ith the applied field of '^SkOe.
Similar calculations for other parts of the system yield a
magnetic field variation like th a t sketched qualitatively
in Fig. 1.
• ,
«-otnmunîeat/on No. ^ 2 2 9
R ep rinte d fro m J o u r n a l o f A p p l i e d P h y s i c s , V o l. 39, N o. 3, 1913-1914, IS F e b ru a ry 1968
C o p y rig h t 1968 by the A m e rica n In s titu te o f Physics
Printed in U . S. A.
Interaction of Microwave Phonons with Domain
Walls in Yttrium Iron Garnet ‘
M . F. L
e w is
and
E . Patterson*
The General Electric Company Lim ited, Central Research Laboratories, H irs t
Research Centre, Wembley, England
(R eceived 5 Septem ber 1967)
I n the course of some experiments on the propagation of 10(X)M H z phonons in y ttriu m iron garnet ( Y IG ), we have observed
a sharp peak in the attenuation of longitudinal waves measured
as a function of temperature.' In two (100) specimens, the peak
occurs at 2 2 7 °± 3 °K , and in two (111) specimens i t occurs at
2 6 0 °± 5 °K . A ll four specimens are of different origin and p u rity .
The attenuation peak can be removed by applying a magnetic
field sufficiently strong to saturate the sample. Except fo r the
temperature of the peak, the difference in the attenuations in
the saturated and unsaturated states has very nearly the same
shape and magnitude in all the specimens studied; this difference
fo r one of our (100) specimens is plotted in Fig. 1. I t is not known
if transverse waves are affected by the peak because the generation
of transverse waves usually requires a magnetic field.^
We have attem pted to explain the attenuation peak in terms
of various mechanisms, m any of which are reviewed by Le Craw
and Comstock.® The o n ly mechanism which we believe can account
fo r our observations is interactions w ith domain walls, i.e.,
through the m agnetostrictive effect, the phonon drives the walls
whose m otion is viscously damped.'"® Q ualitatively, we observe
m any of the effects observed in nickel single crystals which are
usually attrib u te d to domain walls. Thus, the peak in Fig. 1 is
q u a lita tive ly sim ilar to, b u t much sharper than, th a t observed fo r
nickel at lower frequencies.' On applying a magnetic field, the
excess attenuation sometimes decreases monotonically® and some­
times peaks once or more times before decreasing,®depending
on the temperature and magnetic field orientation. I n the case of
pIO O O Oe
-9 0 0
-8 0 0
-7 0 0
-SOO
X
H //< O IO >
\
-7 0 0
H //< IO O >
210
220
230
240
TEMPERATURE(
“k)
F ig . 1. D ifference in the atten u ation s in the saturated and unsaturated
sta tes for 1000-M H z longitudinal phonons propagating on a (100 ) axis
in Y IG .
- 800
F i g . 2. A ngular plot o f atten u ation peaks as a function of m agnetic
field H foi a 1-cm long, 3-m m diam (100 ) rod w ith H m ovin g in th e (001)
plane. T he four sections of curve A show the on set of the atten u ation peak
due to ferrom agnetic resonance. Curves B and C show approxim ate posi­
tion s of the dom inant additional peaks observed ju st above and just
below 227°K
1914
COMMUNICATIONS
Y IG , we can rule out the suggestion by Taborov® th a t the attenua­
tio n is caused by ferromagnetic resonance, because we can resolve
the la tte r attenuation peak at a ll temperatures and ve rify th a t it
occurs at the magnetic field Values and orientations computed
from the work of Schlom ann." The results on one specimen are
summarized in Fig. 2. The curve A marks the onset of the attenua­
tio n by ferromagnetic resonance, as verified by its vanishing when
6 = 0° or 9 0 °," and by the fact th a t the magnetic field value fo r ||
rod is ve ry close to the value fo r magnetostatic pulse propagation
when the sample is in the r f magnetic fie ld ." Curves B and C
represent approximate positions of the extra peaks. B is dom inant
at temperatures above 227°K, w hile C is dom inant below this
temperature. The positions of the curves B and C seem to be
specimen dependent, and could reflect interactions w ith individual
domain walls, or groups of domain walls.
We have considered the possibility th a t the attenuation peak
is caused by a relaxation mechanism w ith a varying as
w V /(l-f-c d V ), where t= tc o exp ( A /k T ) is some relaxation tim e,
fo r example, th a t fo r electrons to hop between Fe®+ and im p u rity
Fe®+ ions.'"® However, to account fo r the shape of the peak in
Fig. 1, i t would be necessary fo r A to assume the unusually large
value of ~ 2 eV. I n addition one would expect a direct interaction
between the phonons and the Fe®VFe®'*’ io n s." I n conclusion, it
seems probable th a t the attenuation peak is caused b y in te r­
actions w ith domain walls, but the mechanism remains to ta lly
obscure.
* Perm anent address: C helsea C ollege of Science and T echnology, U ni­
v ersity of London, L ondon, E ngland.
‘ M . F. L ew is and E . P atterson, J. A ppl, P h ys. 3 9 , 1932 (1 9 6 8 ),
* E . G. Spencer, R . T . D en to n , and R . P . Cham bers, P h ys. R ev . 125,
1950 (1962).
* R . C. L e Craw and R . L . C om stock, In Physical Acoustics, W . P. M ason,
E d. (A cadem ic Press Inc., N ew York, 1965), V ol. I llb .
* J. K . G alt, B ell S ystem T ech. J. 3 3 , 1023 (1 9 5 4 ).
* C. K ittel and J. K . G alt, in Solid State Physics 3 , F , S eitz and D . Turn­
bull, E ds. (A cadem ic Press Inc,, N ew York, 1956).
“ M . A. W anas, J. A ppl. P h y s. 3 8 , 1019 (19 6 7 ).
’ F . G. W est, J. A ppl. P h ys. 2 9 , 480 (19 5 8 ).
» S . L ev y and R . Truell, R ev. M od. P hys. 2 5 , 140 (1 9 5 3 ).
• V . F . T aborov, S oviet P hys,— A coustics 10, 209 (1 9 6 4 ).
" B. K . Basu and P. P . Sethna, A ppl. P hys. Letters 9 , 341 (1 9 6 6 ).
a E . Schlom ann, J, A ppl. P h ys. 3 1 , 1647 (1 9 6 0 ).
a R . W . D am on and H . V an de V aart, J. A ppl. P h y s. 3 7 , 2445 (1966).
**M. E . F ine and N . T . K enney, P hys. R ev. 9 4 , 1573 (1 9 5 4 ).
R .H .B .N .e .
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