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THE STRUCTURE OF TRICALCIUM ALUMINATE

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The Pennsylvania State College
The Graduate School
Department of Chemistry
The Structure of Tricalcium Aluminate
A Thesis
by
Lynn Joseph Brady
Submitted in partial fulfillment
of the recuirements
for the degree of
Doctor of Philosophy
August 1940
Approved:
A CKr!OVVLPD GKMP N T
Thanks are yre.tef nlly extended to Dr.
Wheeler P. Da vey for Viis aid in t‘is v»ork.
The retro Erra.^.hic nicroscooe v.as kindly
loaned, by Dr. .Wary L. 'Villard.
TABLE OF CONTENTS
page
INTRODUCTION
...................................
1
PREPARATION OF SAMPLES .........................
2
APPARATUS
3
.....................................
THE STRUCTURE
..
...............
8
DISCUSSION OF THES T R U C T U R E ...................
14
S U M M A R Y ........................................ ■
17
R E F E R E N C E S .....................
19
APPENDIX I
Calculation of Lattice Intensities.
APPENDIX II
Calculations in Tabulated Form
Showing Structure Factors.
APPENDIX III
The Srace-Grouo O S .
h
TABLE OF CONTENTS
rage
INTRODUCTION ...................................
1
PREPARATION OF SAMPLES ........................
a
APPARATUS
3
.....................................
THE STRUCTURE
. . . . . . . . . . ; ...
DISCUSSION OF THE STRUCTURE
SUMMARY
. .
8
.................
14
.......................................
17
REFERENCES
.....................................
APPENDIX I
Calculation of Lattice Inter.siti
APPENDIX II
Calculations in Tabula ted Form
Showing Structure Factors.
APPENDIX III
The Srace-Grour O S .
h
19
TABLES
I Space Group
0^
Coordinates of Final Structure
Adopted for Tricalcium Aluminate..........
II Diffraction Date Tricalcivun Alu.ulnate. . . . .
SO
21
FIGURES
1 Recorder Circuit
.................. 22
S Power Circuits . . . . . . .
5 High Potential Circuits
...................
25
........................
4 Balanced Filter Diffraction —
MaCl Crystal
. .
24
25
5 A Short Portion of s Typical Diffraction Record..
26
6 The Diffraction Pattern of 5Ca0.Al20 3.... .......
8
7 Elevation and Planes Showing Structure Assigned
to Tric&lci.ur-i A l u w i n s t e ........... .
8 Intersections
27-51
-Ca-O-A1-0-Ca- Chains with Faces
of the Unit Cube
........................ 32
Repeated attempts have been nacle to determine the
crystal structure cf tricalc iu.,; a ruminate (oCaO ‘A1203)
■2-5-4-5).
(1-
It is known to be optically isotropic so its
crystals must have cubic symmetry.
No method, is known by
which crystals can be made large enough to permit goniometric measure sonts.
Because of this, and because of the
large number of atoms in the unit cell, it is necessary to
use the most general application of the theory cf space
groups in determining the crystal structure.
It Is the
purpose of this paper to propose a structure for tricalcium
aluminate which gives quantitative agreement between the
calculated, and the observed Intensities said positions of
the diffracted x-ray beams.
Although previous! workers have agreed on the position
of the strongest lines in the diffraction pattern, no two
workers have agreed on the Intensity, the position, or the
number of the weaker lines, nor have they agreed on the in­
tensities of the stronger lines.
Harrington ( l) found a number of lines in his powder
photographs which could not be accounted for on the basis
of a cubic crystal having an edge of 7.62 A.
lie assumed,
therefore, that this compound was only pseudo-cubic.
ination of his published data shows, however,
e
lines fit a cube v/ho.:.o edge is 15.24 A .
Exam­
that all his
Steele and. Davey (2) found three faint lines in their
photographs which w-oy were able to account for qualitative­
ly on the basis of a unit cube of edge 7,62
a
.
un this
basis they proposed a structure which obviously can net be
correct if the edge of the unit cube is 15.24 A°.
Lagerqvist, Wallmark, and Westgren (5) found so many
lines in their photographs that they hacl to assume the
length of the u >.- l.cell
for all of thou.
. be 15 .22 A in order to account
They found their lines to natch portions
of a simple cubic pattern, but they have not as yet p r o ­
posed a strue t u r e .
Hondo and Yamauchi (o) and Solacolu (4.) while propos­
ing no structure agreed that the length of the edge of the
unit crystal was approximately 15.2 A .
It is evident from the above tint, if we aro to arrive
at a tenable solution of the structure of oCaCuAlgOg,
specimens must be of high enough purity to justify precise
determination of the diffraction patterns.
froperation of 5ais.ples
Early in the course cf this wort: attempts wore made to
make tricalc inn aluminate using the technique of Lagorqvist,
Wallnark, and Westgren.
Those workers fused 75 mol % CaO
and 25 mol >6 AloO,- together us In m tho oxv-hvd.ro gen blovnioe.
.
'
O
'•
»•
cm
a
The fused mass was then ground and heated at 10 0° G. for a
period of ten days.
Although their [published procedure was
followed carefully the petrcgraphic microscope showed that
anisotropic material was present at the end of the twelfth
day of heating at 10u0 ° G.
Shepherd, K,unl:in, and Wright (b) showed that when
5CaO »Alo0o is formed fro.m the melt, crystals of the trical­
cium. aluminate frequently surround particles cf CaO, with
the consequent local formation cf 5GaO *5Al;:;>0g.
They fur-
ther showed that a period of 21 days of heating at 1400°
0.
is necessary to secure complete elimination of CaO and
9°5
is o .
In the present work, the starting materials for the
formation of all the calcium aluminates were prepared by
separately igniting precipitated chalk and 1Bakers1 analyzed
aluminum oxide in platinum dishes at 1000°
C. for six hours,
The resulting CaO and AloO,;,v/ero ground and thoroughly
mixed in the ratios of 5-1, 5-5, 1-1, and 5-5.
These m i x ­
tures were fusea at least once and were then ground to
fine powders.
Each of these specimens was subjected to
several cycles of sintering end grinding as in the work of
Steele and Envoy.
final purification was .made on the
basis of density using methylene iodide and nesitileno.
The resulting four purified products w e m
exa. dried for homo­
geneity under a petrographic microscope and thoir refractive
indices determined.
The identities of those specf.en sere
considered to be established when they showed b e
accented
densities and refractive indices.
Apparatus
The diffraction pattern is measured by ...eans of a
Geiger-Million counter.
The Geiger-Mill] or circuits sue?:, as
Gingrich, Evans, and Edgerton (7) have developed are well
adapted to x-ray measure’:ents where t?:o x-ray tube is ope­
rated under constant or near I;/- constant conditions,
liov;-
over, to make a circuit universally adaptable it should be
made independent of fluctuations in the source due to chang­
ing lino voltages on the x-ray filament transfer .ter.
t
ouch
a modified, circuit, completely a.c. operated, was construct­
ed which records the average intensities of the x-ray
beams with a recording galvanometer (8).
In this circuit pulses from the Geiger-i-filler counter
are amplified by two 6J7-G tubes in series to secure larger,
more nearly uniform pulses.
'these pulses are then c o n ­
verted to a uniform size by means of the relaxation oscil­
lator type circuit.
Pulses from the uniform pulse generator
are amplified and fed into a so-called "tank" condenser
having a fixed high resistance leak.
The potential
of the
condenser is thus a measure of the average rate of arrival
of x-ray quanta at the Geiger-m tiller count o r .
This poten­
tial is then recorded by a recording galvanometer in terms
of the current leaking off the condenser through the resis­
ts no e .
The complete circuit is given in Fig. 1 .
Fig. 1.
Kecording Circuit
It wile bo seen that the circuit of Gingrich, Evans,
and Edgerton has been altered to Include an adoitional
tube, (T-6) , in the vacuum tubo voltmeter circuit.
This
is adued to cake the average output current independent of
fluctuating line voltages on the x-ray filament transformer,
will be observed that as tire current through the x-ray
tube Increases, the grid bias on the tube (T-o)
increases
proportionally with a consequent decrease of current deliv­
ered per pulse by this tube to the electrical tank,
(C-20) •
Since the rate of the Geiger-hhller pulses increases nearly
linearly with Increasing current through the x-ray lube it
\
is apparent that by adjusting R-19, and by suitably choos­
ing the sizes of the
input condensers to the tube (T-6),
and (1-7), a sotting can be reached where fluctua.tions of
the x-ray filauent transformer line voltages v;il.l not cause
appreciable variations in the output current.
Two electric
"eyes" (T-4) , and (T-5) have been placed In the circuit in
the iiianner shown.
These tubes servo
as scales so that
settings of R-19 and the input condensers which have given
the desired results can be duplicated at will.
The power circuits and the Lifschutz high potential
circuit are shown in Rig. 2
Fig. 2
ir'ower 0 ircui
and Fig. b resuectively .
Fig. o
High tot cut:
no fact tha
Jircv. It
h e cutout curr
iris oeon made Inde-
pendent of the- current tlwough the
>ay tube removes the
6
last objection to a completely automatic x-ray diffraction,
apnarabus.
Toe Geiger -lVii\llor counter and the first ampli­
fier are mounted on tho movable arm of an x-ray spectro­
meter.
This arm is driven over the desired range by means
of a synchronous motor and a system of worm gears.
This
keeps the rotation of the arm rigorously in step with the
recording galvanometer.
Obviously, then, the galvanometer
gives a record of x-ra;y intensities vs. angle.
Molybdenum
radiation was used in conjunction with
balanced filters •-f
structed jointly
College.
1
Zrm.
... -,d S r O .
These filters were con­
* ,•->. Bell of The Jrennsylvunia State
The close balance obtained is shown in Tig. 4.
Fig. 4.
Balanced Filter Diffraction - ilaGl Crystal.
in which the shaded area represents the difference between
the transmission of the two filters.
Six diffraction pattern- of triealeium aluminate w*re
obtained, using filters of & /bo and SrO on alternate ru:is.
Because of toe ro.ndo;;incss of the x-ra". ouanta registered bp
t
O-
the counter and because the apparatus used has an exponen­
tially decaying memory the galvanometer record of x-ray
intensities shows considerable statistical deviations.
To
minimize this as far as is practical the width of the slit
system might be increased.
But, previous workers have
reported many lines relatively close together in the dif-
fraction pattern so it is obvious that the slit v/idth
should be kept as small as possible.
If the nuuber of
readings at a given angle is increased the actual in­
tensity of the diffraction r.iuy
method of averages.
be obtained by the
The sase results may
be obtained,
however, if the angular rate of rotation of the spectro­
meter arm Is made very slow and a smooth line drawn
through the mean positions ofAdiffraction pattern.
It is
this method which is used in this work.
A very short portion (angular length 2 G « 1 degree)
of a typical diffraction pattern Is shown in Fig. 5.
Fig. 5.
A short portion of a typical diffraction record
Reference to Fig. G shows that Fig. 5 actually represents
portions of too
diffraction peaks
H- lx 15 126 and
128.
corresponding to h* + k
The average intensities, as measured by the .heights
of the diffraction record above the cero of ordinates, are
determined for both sets of
o.inute of arc 2 G.
filters at intervals of one
The differences of the averages are
plotted point by point on a scale for 2 G much smaller thai
the original record and a smocto line is drawn through the
points.
On drawing this smooth line
.apparent 'peaks1
which obviously do not have thecharacteristics
by the slit v/idths are ignored.
demanded
8
8
The complete diffraction pattern is sliovm in Fig. 6 .
Fig. G.
The diffraction pattern of 3CaO ‘AlgOg
(The numbers directly above the peaks refer to h.2 + k ?'-+ 1*)
- 16
-14
-10
•o
»»■ ,<o
«J CM
-J't
5' .....
1
0'
10*
15'
20'
25'
§
s
A
;
2>0‘
35'
a a 23
i I!
AO'
26
The left end of the pattern up to about 2 <&= 4 degrees show
the shadow effect of the slits.
It is up srent that many
portions of too pattern contain unresolved lines, tho most
notable of these being in ti.e vicinity of 2 Q = 14 to 15
degrees.
similar studios of the diffraction patterns of the
three other calcium aluminates shows that no peak in the
diffraction pattern of Fig. 6 for tricalcium aruminate-can
possibly, belong to the diffraction pattern of one of the
other compounds.
The Structure
The x-ray diffraction pattern of powdered tricalcium
aluminate shows lines which are characteristic of a simple
cubo, of edge 15.255
a
.
A smaller unit of structure cannot
possibly account for a large number cf the lines in the
diffraction pattern, v/hile no lines present require a larger
9
unit cell.
On the basis of its density and the size of the
unit crystal the chemical formula of this compound is
(3CaO •AlrjO^) £>4 .
This means there are 72 Ca, 48 Al, and 144
0, a total of 264 atoms, in the Uuit cell.
Since no crystals can be made largo enough for symmetry
observations or Laue photographs, none of the usual methods
for establishing the space-group is available.
Moreover, it
tM\
is obviously unpractical to examine every possible arrange­
ment of the atoms consistent v/ith cubic symmetry because of
the large number of atoms present in the
unit cell.
This
means that more general .methods must be used to establish
the space group.
Fortunately, many space groups v/hich would
ordinarily have to be considered may be eliminated at onee ofi
the basis of ''forbidden lines" in the
( T ).
d i f f r a c t i o n pattern
This procedure leaves thirteen space groups v/hich
should be considered in order that no possible structure may
I
be overlooked.
V-
S'
The allowed space groups are, -- T ,-T , T ,
0' , 0 , o6 , 0 , T^ , T fc , 'l| , tX , (A
, and 0^ .
Even this
number is too large to' permit a consideration of every pos­
sible arrangement of the atoms in ore or to establish the
correct order cf symmetry.
Of the various possible procedures, the one v/hich seems
to inve lvo the least labor is or; follows:
(0.) only those
arrangements of atoms are considered .filch arc consistent
with known pnysica 1 and chemical data;
a >correct*
(’
0 ) the search for
structure is started with the space grouo havin'"”
10
the highest order of synjet**;/ and, if necessary, continued
in turn among those of progressively lower order until a
reasonable structure is found v/hich fits the
tively.
data quantita­
Having found a suitable structure no further inves­
tigations need be nads- on space groups having a lower order
of symmetry.
This is justified, of course,
solely in terms
of the universally accepted theory that crystals usually
grow with the highest symmetry permitted by the atoms and
radicals v/hich must be packed together.
Since the above
procedure gives the most symmetrical structure v/hich fits
the diffraction data quantitatively, it necessarily gives the
most probable structure of all those which a more extended
investigation might show would fit the data equally w e l l .
Of the space groups listed above as being possible, these
in ti.o 0^ family have
the highest sum ,etry.
!
berv .f this family,' 0,
3
and Of,
Only two nem-
, need to he considered
since the other .^embers are eliminated on the basis of for3
bidden linos.
Of the two, 0^
r,' r r t
is the most promising since
it is inherent that this space group gives no diffraction
fro..:) the planes 100 (1) , 100 (5) and 111.
experiment (sec* Pig. 5) .
of symmetry than 0^
.
This agrees with
It has, moreover, a higher degree
It will, therefore, be considered jn
detail.
The diffraction pattern has strong lines corresponding
to a body centered cube asd shows no peaks having measurable
intensities where h 4 -+• k ’*•+ 1 “
is less than 10.
The body
11
centered pattern may be partially accounted for by placing
24- Ga in the equivalent positions (k) v/ith coordinates ( 0
-j f) , and 8 Ca in the equivalent position (e) (9).
Space
considerations makes it necessary to place a large number of
oxygen atoms on the faces of the cube and it is safe to
assume that these atoms are near the calcium atoms previous­
ly considered.
This is as far as we may proceed without cal­
culating intensities and considering packing sizes of the
atoms.
It is assumed that Goldschmidt*a ionic radii are reason­
ably accurate but are not to be regarded as strictly inflex­
ible.
No structure has been eliminated on the basis of pack­
ing dimensions unless the available space is exceeded by at
least bO per cent.
The theoretical intensity of each lattice reflection is
calculated with the use of the v/e11-known equation
i = k
U_..mmlULa—
sin' U • cos 0
P •f
(i)
where p is the number of planes in a given form v/hich contri­
bute to the diffracted beam, and
k is a proportionality
constant involving among other things the geometry of the
slit system.
P is obtained as follows;
tors are calculated separately for eac:
The structure fac­
point position, and
are then multiplied by the corresponding atomic structure
factors.
The summation of these products is P.
No attempt
need be made to evaluate k since only a ratio of intensities
12
is desired.
The atomic factors of liartree are used in the
present work, and the structure factors for particular point
positions are derived from the stru.cture factor for the
general point positions (10) in the usual manner (ll).
In
calculating intensities in the present work no corrections
are made for extinction or temperature effects.
The diffraction pattern (Pig.- 6 .) shows there are no
measurable peaks due to the planes 100 (2), 110 (1), 111
(2) and 210.
This means that P of equation (l) must be neg­
ligibly small for each of these planes.
Because of this,
the calculated intensities of d. ffracked beams fro.; these
planes are used, as the initial criteria of the "correctness'''
of assumed structures.
In arriving at a solution, the
method of trial and error is used, varying the parameters
of the equivalent point positions, until it is shown either
that the assumed structure shows oromise or that it
must
be rejected on the basis of the Intensities calculated for
the above planes.
Several thousand combinations of equivalent point
positions with varying parameters must be tried, the- best
ones being retained.
nated as far as
Those which still remain are elimi­
possible bp comparing the calculated
lattice intensities for still other planes wit I-' the x-ray
diffraction pattern.
Of too coordinates of space-group Qj?
vestigated,
v/hich were in­
those shown in Table I correspond to
13
Table I
bpace Croup ulf_
Coordinates of final structure
adopted for Tricalcium Aluminate
calculated lattice intensities which agree with the ob­
served x-ray intensities within the limits of experimental
error.
These calculated and observed diffraction data are
tabulated in Table II.
Table II
Diffraction Data
Tricalc iuoi A ruminate
This structure nay be visualized b; the aid of Dig. 7.
*
Fig. 7.
Elevation and planes showing
structure assigned to TricalcI urn. Alur.iinate.
(Dimensions are expressed In 64ths of cube edge.)
The calculated intensities are closely der-endent upon
the exact values assigned to the various atonic coordinates.
For example,, If the parameters for the equivalent ps int
, tuition ( i) are changed from (5/32 , 5/o2 , 5/32)
to (l/8 ,
-/':> J-/'h) the calculated and observed intonsiti.es si..own
In
14
Table II for tho (400) line become identical, at the expense
however, of a poorer fit for so s of the other lines.
Lore-
over, in view of the large auiber of atone, to bo considered,
the effect on the calculated lattice intensities of even one
percent error in atomic structure factor is very largo.
The
close agreement shown in Table II between calculated and ob­
served intensities for o4 lattice reflections is therefore
highly satisfactory,
further adjustment of coordinates with
still closer limits is obviously hardly justified.
Discussion of the Structure
Fig. 7 shows that there are layers cf single planes
cutting the Z-axis at 0 , l/l, l/2, and o/4.
bandv/icbed In
between each pair of these is a set of three planes v/hich
are very close together sc that they may be visualised as
a single puckered plane.
These puckered planes cay be
thought of as cutting tho Z-axis at l/h, b/8 , 5/8, and 7/8.
The following discussion has beer, worded as though these
puckered planes were true planes.
This simplifies the
*
structure without causing it to lone Its chemical signlfic anc e .
Each true plane at 0, 1/4, 1/2, and b/4 is seen to
contain continuous, intersocting, chains of
- Ga - 0 - A1 - 0 - Ga -
parallel to tte X a ai Y axes,
and analysis of the various layers of Fig. 7 shows that,
as Is demanded by cubic syv, etry, there are similar chains
parallel to the Z-axis, which Intersect with those chains
%
.15
which are parallel with the X and Y axes.
The ends of
these chains cat the faces of the unit-cube as shovm in
Pig. 8.
Pip. 8.
Intersections of •— Ga - 0 - ALL - 0 - Ca Chains with
Paces of tho Unit Cube
These chains obviously fora a t’
nr ee ~d ia en a io nal net-work
of coarse aesh and. it works cut that there Is
rooa between chains for ti
atons.
X:i t o
true plane:
' f*
'a .
dr
'' r\
tv,
( Jo
KJ
A
j
d.A
ifficiont
1
4
.a •.
A
(')
.. V : t i e .
.c-‘ cat the Z-axis at
W
-
/
.i
/ ->
and d/4 are elyht Ga atc.is, (wyckoff (if) coordinates hie),
at tho corners of sawares sec1': that If o eiyht atons lie at
iO corners of a cube whoso eclso is half hi
cube a.,.d. whoso cent;
coincides wit'h tho center of the
nnit-c ubo .
os 1 io furthor awa'
Theso a
est noifibbers on al 1 sides than
whole structure.
itons an a
iron if'olr
jo'l.r —
a to:.,s In tho
If wo arc to rrrnUiO any really ionised
in trie ale iuu a.laminate, they :..ust be these
s.food. atoms
oiyht Ca at wye
out of a total of SG4.
(if.;]it-Ca is a. strucoar'.il cube hayiuy tw: 101 Ga at diayonally opposln
corners, and six 0 at the re .alni:
at the cen
C1P'”■
i, u .
%
16.
cubes are at tee 1/8 £'8 5/8 levels.
the 5/8 and 5/8 levels,
Their tors are at
ill tee C a , except those at
Wyckoff1s 8e and 16i and those in the
chains, ere
-C e.-Q-A1-0 -C a -
-ttsched to oxveen in clusters mare un of
four Ca and eirht 0 atoms.
each of toe four Ca atoms
in a .riven cluster lies at s corner of a senrare cross in
one of the faces of the u'it-enbe, and toe emr-ty center
of the sous re cross c'U'U at the center of the cube face
(V.y ck o f f 1s 1P i ano 1P i) .
Th e eivht 0 atoms in the
cluster are sraced around the four Ca atoms in such a
way as to form a closed network in wnich adjacent Ca-0
bonds form right angles, and the 0 atons form a layer
around the cluster of four Ca atoms.
serve to tie narallel
Tnese clusters
-Ca-O-Al-Q-Cs- chains together.
For all four of the Ca atoms in a given cluster, the
O-Ca-O
grouping which is characteristic of CaO can be
seen, but the coordination is different.
There remains a group of twenty-four A1 atoms at
WyckoffTs £4r.
Facb of these is found tucked away in one
co rner o f e ach in t er se ction o f tw o
-C a -0 -11 •-0 -C a -
c:i a in s .
Our structure is seen to be notably free from Al 0 2,
Al 0 5 , and AloCh groups.
both
0 and
Ca
fverv Al atom is associated with
j
eto^s, jixcemt far ei rht isol. ted Pe Ca
atom' , the Cs. atons see in to 00 assoc ir bed in one ssy or
17.
another directly with 0 atoms.
A few of these seem to
be associated directly with. both. 0 end Al atoms.
The
structure definitely does not show discrete molecules
of tricalcium alumina te, nor does it show more than eight
isolated Ca ions,
ji.lthough the structure is obviously
different from thmt re"orted by bteele and. fevey^’ \
nevertheless it shows rather simils r tyres of chemical
combination in that C a , il, an.? 0 are a11 ci roctlv and
intims iely associated with each other.
Tne lacw of any
definite organized 3rvr-v o;? positive and nefsr ivs? ions
orobs blv accounts T ?r so"!? oj’ i ts "'t'>'ir’:'r’ti-s.
vag
existence o~ clusters vw’ioh s-'ov/ a stoichiomer.ric
comnostion of Ca 0 croba blv accounts for the
artial
deco‘"uosi tion of tricslciai.? slurninate to Ca 0 and
A1 P0 3 at temrurstures below the melting noint (6) .
The
structure contains ewntv s'uccs of aa'fu icient size to
account for tue tendencv to te ■<e un ’water of crystallisa­
tion.
Cirmrs ry
Trice.lcium a 1-amir;ate of biro s unity v;& s c.-^e ■-area
hv t:,e metnod of a:to-elo a>■r fever,
and its :..-rav a iffrsc-
1.1on us ••tern was taven a s m s "fo K
1 ;•ha rw-s isol -fed by
ba.Lanced filters and re core c-:d b” an autsum 11 c aewaer- oilier
18.
counter
i t
a ’^ - r a t n s .
con tains
tie
ver'v
c ro s s e d
7.°
b ig b
s.s
C f,
The
I f
gave
edge
A l,
s y r- r e t r y
calcu ls tee
tfiese,
i f
fs v in s
c o rre c tn e s s .
vvere
The
of
f is h
e
a rf
degree
•? t o t a l
ca.icu jyfo d
141
cube
0 .
1
sopce-grour
tf.oore1n.cFl
fo r
the
o f
o' r
of
is
s tru c tu re
0^
,
assumed
in te n s itie s
of
A,
of
is
of
c iffr & c tio n
;'lsn es.
scro
and
as. v i n g
(Pm5n) ,
n ro b a b ility
in te n s itie s
14
1 5 .i f 5
Jf
or
extremely Ion values.
fone o f ±bose If1 lines could be
found e:ynT'imer>te lly.
Tlie
reayiyLnp
7.4
lines
uere
1o u n d
exoeri/uental.ly and tneir calculated and observed
intensities screed
n it f in
foe
error of oxoeriner.t.
7be ■fy"sica 1 and cbesLCr 1 sienificsoce of the
oro’'-osed strocinre have been dir;cussed.
It is found
fast tbere rye no nolecoles of tricslcium cliini-sate,
no
11 Cf, Al 0 3 nor flo03 groupings, fit tbst tsere are
clusters of CaO.
Tnese f u r y
ulus tie existence of
"holes'’ in f';e structure an'count sell for tfe lack o
definite "siting noint, inc. nument selling, srf tend
to tote iic 'rater of crystallisation.
■rcy
%
References
(1)
Harrington, E. A., An. J. 3ci. 13, 467 (1027)
(2)
Steele, P. A., and Davey, Wheeler P., J . An. Chern.
Soc. 51, 2283 ( 1929)
(o)
Kondo, S,, and Yanauchi,. T . , J . Japan Ceram. Assoc.
4 2 . 813 (1934)
(4)
Solacolu, S.,
Ibid., 319
(5)
Lagerqvist, K . , Wallrviark, 8 ., and V/estgren, A.
anorg. und allgeH. chein. 254, 1 (1937)
(6)
Shepherd, Rankin and Wright
Ah. J. Sci. 28, 293
(1909)
(7)
Gingrich, Evans, and Ed.gerton
Rev. Sci. Inst. 7,
451 (1936)
(8)
Brady, L. J., P h y s . Rev. 55, 1155 (1939)
(9)
Internationale Tabellon
zur Beetinnung von Jhristall-
strukbur en (Gebriider Borntraeger Berlin 1935)
(10) Ibid. , 1, 860
(11) Ibid., 1, 87
^
(12) Wyckoff, R. W. G . , The Analytical Expression of the
'Results of the Theory of
Institution of Washington
Space-Group3 (Carnegie
- 1922)
Zo
T b ble I
Srsce Grout)
Oj^
Coordinates of fincl structure
pfo'otec’ for Tricolciiirn iluninate
:Xement
Eoniva lent
point
position
■
Ca
'
l.yckoff1s
eapivalent
roint
oosition
k
(Pi r)
1/4 1/4
e
(8 e)
1/4 1/i 1/4
(IB P.)
5/ P S
(If. i)
1/8 0 1/1
(1: j)
1/8 I/: 0
h
-
_
Parameters
of
ecori.vol.ent
mint
m a i t ion
_
k
_
.
.
.4/ PS B /PS
. . . . . .
(:4 r)
0 1/8 1/8
(€■ f)
1 /i 0 1/i
a
(8 r)
1/4 1/i 0
f
(IS a)
1/A 0 0
(PA r)
0 :/5 /G4 1/4
(:;4. r)
0 1/4
■.I
v p 'pa
2 Q - cl
k3
(24 r)
0 7/64 1/4
k4
(kA r)
0 1/4 7/64
1
0-3-1
1/8 5/S? 11/52
21 .
Table II
Diffraction Data
Tricalcium 5 1urninste
it? 5t?3
i.H-‘j-
"s" r *■
Q
/•%
4 1 ? ’
1
O
49
100
0
110
0
1 8.2
Wo
« va
q HIM*
0
3.0 . >:
0
100
5
2910
6.6
6
211
8
9
110
(9)
0..16
13.8
88.9
0
100
310
111
521
(8)
0
67
0
70
5.6
1 7.6
0
80
100
941
(4)
5° 2 5 ’
16
6°o2 ’
6° 4 2 ’
21
84f 25
L
pOAO t
!r.J6
*..)6' •
9°19 ’
3 v
l,
<
111
5 5 ’
7 °o 4 ’
8 ° 30 ’
8 ° 5 8 ’
^
•?: is
?8
10
n
»2
x
ts« "
2r-SL
4
12
14
4
jb/4ties
a
211
100
0
40
'45
810
910
ft /}<T'
• >*t<9
48
rc. - z ,
111
720
4.817
9.14 2
4.167
5 .4
1 0.6
8 .78 4
3 .808
188
8 .25
0 .5
11.0
0 .8 8
818
3 .7 4
3.12 0
8 .04 3
8 .110
8 ,0 5 6
2.991
8 .988
8O
*3
1000
N
(*)
;■ . 6 9 2 '
8.40 8
2 .871
602.
? . 40 P
c
i- * ■r■ o p
9 .5
58
1 5.8
65
(4)
2 .128
8 .19 8
20
c » a;
54
(5)
a. 3 0
510
110
4.847
0
40.1
(■)
0
0
0 .5
(8)
(4)
1 1 .1
111
18.3
185
o
y
0
0
0
100
0
20
150
55
1000
40
55
%
9 .8
50
0
641
1 0 5*25 ’
56
r6 2
iq °4 4 I
64
9- Oi
100
1 2 °1 9 ’
1 S °1 2 ’
84
96
4 91
211
190
100
321
789
'■■■ • 8-^
J*
r.
**■ • r'
•J >* ‘ 3
6.0
cr o /•
0
60
20 .9
105
550
‘. ''I • Q,’
3
17 5
565
59
‘ 30
J_ ■ .’> ♦1-
5 0 .5
5 3 .1
85
1.357
5-5.0
57 .0
50
(9)
1.93 8
I. . 9 3 5
1 .906
1. » P-. 3 Aa
.a
1.904
o ppr
.L a
...
1i. * n
cr c
■ • ■*
52 cr:
f' - ~\
T<Of
~1U?
.■ *'-.O
»■
t
4 80
(4)
(T:)
(4)
fib)
(O’
1.555
i ~,c
911
112 1
1 5 °1 0 ’
.198
1051
...
1. 856
(5)
1
19°49' fpl2
20°24!
214
(8)
( ~)
(8)
(?■)
1.547
1.206
1.100
1.067
1 547
1.5 05
1.100
1.067
188
9.7
57.4
6.1
1.047
1.046
11.7
1.016
1.018
17.9
19.7
.
oil
16.1
9 5 .y
10.1
10
57
97
10
(?)
bj
(4)
1-- 1
..
110
■310
111
711
551
720
641
521
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12‘8
15 17'
160
17°07»
195
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T- I
83
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T-2
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X
A W. NEON'S,RESISTORS
REMOVED FROM BASES
■■
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REMOVED FROM BASE
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ZrO* _
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4DIFFRACTION -N a C I
CRYSTAL
—*— *— *—
Mo X-RAY T U B E - 4 3 K.V. P E A K , 10 MILLIAMPS.
Z(o
.A short portion of a typical diffraction record.
30 ' SO
30
.
F;3- S
Fig. 7
27
64
S4
5G-AL
Elevation and Planes
Showing Structure
46
S3 40
Assigned to Tricalcium
42
Aluminate.
32
24
(Dimensions
are in 64ths.)
ZS
Z2
16
8
10
Ca
Cct
Fig. 7
Fig. 7
*3
Y
Ca
4
Fig. 7
JO
Y
Cu.
it 4 4L
32
3Z
3 i£
a
♦X
a
Fig. 7
Ca
at;
C q.
4
Fiy. 8
Intersections
-Ca-Q-Al-O-Ca-
Chains with Paces of the Unit Cube
Appendix I
The theoretical intensity of each lattice reflection
is calculated by the well knov/n equation
I = k ^J^cos^gQ- p F 2
sin"© cos ©
where
p
is the number of Dianes in a given form which
contribute to the diffracted been, and
k
is a propor­
tionality constant which need not be determined since
only a ratio of intensities is used to compare
theoretical and experimental intensities.
F
is obtained as follows: The structure factors are
calculated separately for each mint, position and are
then .multidied by the corresuoncling atomic structure
factor.
The summation of the products is
F.
As an il­
lustration the theoretical lattice reflection for the
440 rlanes is given.
The structure factor which has been calculated, for
the general uoint position of the snace group
A = 8 ■ cosfwhx cosiwky cosh vis
t cosSttIx cosfirhy cosimks
+ cos-brkx cosfvly coslbrhs
+ costt ( h+k-fl )
:cosivkx cosivhy cosivls
+ cosSwhx cosSrrly costvks
Oj
is
2,
+
c o s SttIx
cosgvky cosSrrha
B = 0
For the calculation of the structure factors for the 440
planes the eouations become
A = 16
COS 07 tx c o s S t t j
+ cosS7ry cos8Tra
+
cosQttx
coS'Sirs
The specialization of the structure factor for particular
uoint positions can he easily derived from this equation
by replacing
x, y, and s
by the corresponding coordinates
and dividing by a suitable number to allow for the
reduced multinlicity.
The general coordinates for the enuivalent point
position
(k)
are o,y,s.
If we assume
y = e = 1/4,
the struct/are factor becomes
A = if
The factor
1/2
1 + 1 + 1
=14
represents the reduced multiplicity.
It is arrived at as follows:
A
is calculated for the general point oosition which
contains 48 terms.
position
for
A
(k)
Since
A
for the equivalent point
can have but 24 times the general equation
is divided by S.
s.
The structure factors are calculated for each ecuivalent
ooint nosition in exactly the same manner and the struc­
ture factors for the three elements, Ca. Al, end 0, are
summed independently.
In the case being discussed the
sums for Qa, Al, and 0 are 32, 16, and 54.7 respectively.
The details of the calculation are given in .Appendix II.
These structure factors are multinlied by the corresponding
atomic structure factors and the results summed and
sous red to give
and
p
8.8 x 10' .
The values of
are determined in the usual manner.
kl440 = 602
and 7^
x 1000 = 1000
I4 4 0
--—
sin J9 cos 9 '
This gives
II-l.
Appendix II
Calculations in tabulated form
showing structure factors and
calculated x-ray intensities
planes
k
e
i
g
h
Ca
100
0.00
0.00
0.00
0.00
0.00
0.00
110
0.00
0.00
6.94
-4.00
-4.00
-5.08
111
0.00
0.00
0.00
0.00
0.00
0.00
000
-8.00
-8.00
-6.15
6.00
8.00
-6.15
810
0.00
0.00
0.00
-4.00
4.00
0.00
811
0.00
0.00
-1.89
0.00
0.00
-1.89
820
-8.00
8.00
3.35
4.00
4.00
11.35
300
0.00
0.00
0.00
0.00
0.00
0.00
310
0.00
0.00
-8.73
-4.00
-4.00
-16.73
888
24.00
-S.00
-3.90
0.00
0.00
15.10
381
0.00
0.00
5.34
0.00
0.00
6.34
II-2.
400
£4.00
4P1
0.0
422
-8.0
500
3.00 -11.31
4.00
4.00
0.0
0.0
0.0
0.0
8.0
-1.7
0.0
0.0
-1.7
0.0
0.0
0.0
0.0
0.0
0.0
430
0.0
0.0
0.0
4.0
4.0
8.0
510
0.00
0.00
431
0
0
440
-■4
8
620
630
-4
8
0.0
-9.7
1 1 .5
-4
8.2
4
16.2
A
0
0
0
0
0
0
0
0
-4
A
0
-6
94
790
-1.7
28.70
0
0
-1
0
0
*
^.
641
642
732
0
0
0
-8
8
/!.
0
0
0
*ryt•'
t'.
0
0
11-s.
651
0
0
1.6
800
f.d
8
0
R/1C'
0
1.6
40
-8
P A.A
0
-16
p
-i-'.e
0
i 10 00
860
0
-8
Q
-14.8
8
0
D
0
Aw
ill 11
0
1.44
0
0
1.44
AI D 51
0
-1.60
0
0
-1.60
0
12.1
880
11 40
n
0
12.1
0
24
8
0
12
■4
8
-8
-4
-4
16
0
IP
IP:
56
888
:14 ??_
-0 10 2
14 40
-0.9
24
P'4
0
-8
_n
1
56
15.].
10.78
A
A
0
p ‘
.'.A
*
\
II-5.
F :Ce.
o.
f
Z.AI
F Al
0.00
0.00
0.00
0.00
0.00
0.00
0.00
-56.00
15.00
-0.00
-5.00
4.00
15.00
145.00
0.50
0 .50
0.50
0.00
0.00
0.50
0.00
-107.00
8.50
5.50
5.00
4r.00
16.00
157.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
: .10
4.00
5.80
f .00
-0.00
180.00
0.00
-5.00
-8.00
-5.00
-8.00
-75.00
0.50
o.oo
0.50
0.00
0.00
0.00
0.00
-876.00
-4.00
-5.00
-8.00
00
-4.50
-58.70
4 .no
5P.5Q
4
8 54.00
0.00
00
55.80
0.50
o.oo
.00
-If .00
- 5 .. •
no
.
J
-155.. on
-1
0 00
J
m
‘ /
3.00
II—6.
458.00
-16.00
6.00
6.00
16.00
0.0
0.0
-55.1
0.0
-4.0
4 ,0
0.0
0.0
0.0
6.0
.0
4.0
-8.0
-75.6
0.0
0.0
0.0
0.0
0.0
0.0
0.0
119.0
0.0
0.0
0.0
0.9
0.0
0#Q
-145.7
-4.0
-6.0
-6.0
4.0
-4.0
“O 6.8
168.0
-4.0
-6.0
-8.0
4.0
4.0
-86.8
469.0
o
•(o.>•‘
J
6.0
16.0
16.0
148.0
661.0
0.0
-6.0
-6.0
-4.0
-8.0
oo
6.1
4.0
4.0
0.0
-6 .].
-52.7
0.0
0.0
-4.0
4.0
0.0
0.0
0.0
27.0
84.0
6.0
6.0
16.0
48.0
410.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
4.0
-4.0
0.0
0.0
0.0
55.6
-
-
6.00
75.00
*
-51.0
0.0
-6.0
-6.0
-91.8
4.0
8 .0
v .C
-4.0
-8.0
4.0
-66.0
II-7.
8.0
c
.
-4.0
-4.0
6.0
6.0
11-.0
32.0
64.0
-0.0
.0
7.0
4.0
0.0
0.0
764.0
8.0
6.0
6.0
17.0
,0
242.0
-158.3
8.0
2.0
8.0
4.0
16.0
120.0
0.0
8.0
2.0
8.0
4.0
16.0
180.0
14.4
0.0
2.0
:.0
.0
0.0
0.0
-16.0
0.0
8.0
8.0
-4.0
0.0
0.0
181.0
-4.0
2.0
8.0
-4.0
-4.0
28.2
532.0
24.0
6.0
6.0
12.0
48.0
316.0
146.0
-8.0
6.0
6.0
12.0
16.0
104.0
486.0
84.0
6.0
6.0
12.0
48.0
298.0
126.0
0.0
-6.0
-6.0
-18.0
-184.0
-140.5
90.0
0.0
-6.0
-6.0
-18.0
0
-140.5
-140.8
-8.0
2.0
2.0
4.0
0.0
0 0
>0.3
-4.0
8.0
479.0
8.0
-184.0
•
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w
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-8.0
£.0
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4.0
0.0
0.0
197.4
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6.0
6.0
lfc.O
16.0
91.4
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o
CD
CC;
I
CO
o
CO
CO
i
•
I
CD
O
o
o
o
o•
o
'CO
1
CO
oo
cc
I—I
*
6
O
CD
I
CO |
11-10.
8.61
8.61
8.61
-58.62
-
+
-
-5.4
7.7
7.7
7.7
7.7
0 . 0
0 . 0
0 . 0
0 . 0
0 . 0
7.7
-7.7
7.7
-7.7
20.7
0.2
0.2
6.4
6.4
-1.7
-1.2
-1.2
1.2
1.2
-15.15
14.1
14.1
-6.8
-6.8
40.0
-8.0
- 8 . 0
-8.0
-8.0
-11.5
8.61
-5.55
8.2
—CC Oi
o
.o
5.55
-8.2
C O
O
-5.55
5.55
-29.0
29.5
9.7
-9.7
1.2
-22.2
-22.2
-24.0
+ *
-
+
-
+*
-
+
-
7.4
7.4
7.4
5.65
-1.5
1.5
1.5
0.0
7.4
-1.5
__________
-5.1
- 8 . 2
11-11.
0.7
0.7
6.9
6.9
0.0
19.5
19.5
19.5
19.5
16.0
1.6
1.6
1.6
1.6
4.5
11.8
11.8
-9.1
-9.1
6.65
6.65
-4.45
-4.45
-4.45
-4.45
4.2
-4.2
-4.2
0.0
1.2
0.8
0.8
0,0
2.1
8.1
-5.9
-5.9
0.0
17.0
17.0
17.0
17.0
0.0
5.0
-5.0
-5.0
5.0
-8.0
17.0
17.0
17.0
17.0
0.0
4.7
4.7
4.7
4.7
0.0
-14.9
-14.9
-14.9
-14.9
0.0
-7.8
-7.8
-7.8
-7.8
-10.5
4.9
-
1.2
-
6.65
6.65
-8.0:
‘
-
-99.6
-4.48
-
-
-
-
-
%
11-12.
1.6
1.6
1.6
1.6
-4.5
-4.8
-4.8
-4.8
-4.8
-8.0
The values .for
, k 2, k 3 , k4 for this plane are
inherently su.ch as to conceal each other and are
merely recorded as to the signs of the c;riant itie s.
11-13.
F 0
£F
(e.F)
0.00
0.00
0.00
-4.34
-45.00
43.50
0.00
0.00
0.00
0.00
-6 .IB
-53.40
-3.40
11.56
5.56
88.80
S3.30
0.00
0.00
6.50
-98.00
-180.00
-75.30
0 •00
0.00
0.00
8.36
67.00
348.00
18.79
-14.70
-14.70
0.63
5.10
58.80
t
0.00
1,390
790.
09.5
4875
1510
870
700
68 £
410
0.00
68,500
*92
8,209,
3, 400
330
11-14.
-£.1?
-16.70
510.00
-3.4
-55.8
8.0
;60,000
500
-55.8
580
178
179.0
78.3
6,130
150
0.0
0.0
0.0
50.7
135.0
854.0
64,500
14 4
11.6
74.5
106.0
11,6:00
184
0
,116
1£4
038.0
■-"60,000
114
-15.15
-85.8
54.7
8.0
» ■o. < > n
•
- 94 .
9,000
150.4
165.0
■5,700
J
~~ *
6.3
1.8
■no
-
5.1
-
8.8
55.
0.0
.
0.0
58.4
144
76
75.8
4 ,500
76.0
600
66.0
-59.8
1,540
66.0
16 6 .Q
65.0
3,035
63.0
0.0
155.0
15,600
56.8
-564.0
-67.0
4.5
—
4Q
11-15,
15.2
67.0
54.0
9 ,916
56.8
93.2
400.0
1134.0
1,500 ,000
54.0
10.6
40.0
-144.0
20 ,700
48.0
-2.7
-9.3
497.0
247 ,000
55.4
-5.0
-10.0
-48.5
.
c..,550
55.7
-5.0
-10.0
110.0
18 ,100 .
55.7
0.0
0.0
14.4
807
8 6 .8
-0.8
-2.5
-13.5
528
'6 .5
-7.5
-8 5. 6
70.0
A ,900
8 6.8
6'v.9
258.0
1106.0
1,210 ,000
86.0
-10.8
-51.5
190.9
:'2 ,600
;0 .2
6Q .n
125.5
•-7.9
Qn ri,
y 900
16.4
53 .0
881
14.6
-155.6
1°4.0
4 :■ ,900
16 .6
-05.0
-5 52.0
MO ,300
14.6
18,0
■* *•
-41.*
Ac *n
-r1
*
*
11-17.
p
kI
0
—
6
*
x 1000
I440
Calculated
0
1
x 1000
-£440
Observed
0
18.2
50.2
0
0
0
0
5
0.3
0.5
0
12
6.6
11.0
0
6
15.8
22.9
0
0.0
0.0
0
--------
--------
PR.4
49
70
4
3.4
5.6
0
24
18.8
51.2
20
41
II-IS,
132
54
2.5
12
11.1
12
111
218
3.74
18.3
185
100
0
20
150
12
.2
23
38.2
55
24
1000
12
9.5
12
84.7
15.8
1000
40
1-1
84
50
;0
2.9
4.8
0
24
G .6
6.0
0
84
35.16
58.4
60
12
84
11-19.
2ft
5
24
12
3
211
20.9
105
350
54.8
175
365
59
240
20.5
34.]
25
54.0
57.0
50
6
188
,312
12
9.7
16.1
27
57.4
95.2 ,
97
6.1
10.1
20
12
24
24
24
4
12
200
z:::z:zz::::::z+
12
12.7
21.1
35
11-20
24
24
t
a =
17.9
1 + cos "20
?5
sin'"9 cos 0
29.7
80
Diagram
IT-I
of
to
Table Shown Oh Paaes
n-ZI
Inclusive.
1
S
$
I3
17
z
6
/0
I4
!6
3
7
//
/S
/3
4
Q
IZ
!G>
20
Appendix
III
The Sriace-Group of
h
35
s
^ -
U < i u in '.p 'iip l» « - n
Of ,'-1— P m i n .
11:
i ’ ll i l l t l:i
■>: («) (H ill: ,i 11 _
(i: (/,) (111; 2O p
(c)
('0 ] 2 0 ;
,S: («) •I1 1111.•
■IS: (/) ;»•()(>:
xOO;
2 20; p to
! J ll:
>PP 201
u > p ;J20
l :i .
•I 1 1 !
0x0;
0x0;
11 : i.
illOO.e:
00x :
0 20 ; 11(i A
:i :t I
•1 1 I ‘
i .: X.
l __ X,
i 0: o p
i .
PP
:$. n i l . 1 :i 1 . 1 1 :$
4 I j ■i I -i ■ 1 1 1 - 1 4 1
l i -• .1'. \ • ! 1
A
i l — X, 1 . 1 1
-A
l
l 1- x, 0 ; 0. P
1
0
T
T
T
): 0, p
0
('/) .!•() 2 : p rO ; 0 2 , r ; 2 - ; x ,
x O p 2 x 0 ; o p u ; T__ x t
O .rp 2 0 x i . .»■, .1 0
(/') ,r
x 2 0 ; O.i- p 20 x ; I __ x t 1 0
1«>: ( ')
.r.r.r .)• x x ;
X X X x x .i':
X X X ,f x x ;
XX X x x x :
2-1: (/) X
•I J T -P !-.r:
:], x, 2— x ;
V
4 >X
*’ 5 ■> j .»■:
2 -x ;
4.
(/,') <>.'/•- zOy;
0//2J 50 ,y;
0.1/5 5 0 //;
O j/s sO y;
4S: (/)
x.i/5
X//Z
xyz
xyz
Xl/Z
xyz
xyz
zxy;
zxjr.
zxy= •'•//;
s ■'■//;
5 .»•//;
2 •'•//;
5 .r//:
in (n)
in (/>)
in (<•) •'
in (e)
in (!)
in (#')
/'
in (?)
in (!■).
in (/)
•>
P
in
i
- ■ X, 2: 2 ,0 , 2 l- x
— x, p 2 ,0 , 2 — x
x:
x, A x : 2 ■ x , ; — ,i',
.<■ i
x i -!. :>*, A x ; 2-— X, — x. 2 : x :
.t T_ ■r, A— x : 2 — x , ; 1 x, 2 | - x ;
I __
A x : 2 | X,
.)
i
x 1 | x , 11.• 1
PI
1> r, 2 — .u; P - x , . , X ; Xy 2— X.
> 4 >'
2 — .1
' I •*'
— ■>', i : -ii i', 2 ■- .c; 2 I
; , X ; .V’, ' 1“ .*',
I i - x, : i. :i r, i _ x : 2 — Xy , X ; Xy \ — X,
I ! •( , 1, *: x
5 •'
: i. ii
:t •; x \
2— .1 i 4
— x, -I • 4J v, 2 | x ; 2 | Xy . ,.-c ; x, \ \ - X,
>•
1 1
~
1 . >/; 1 1 .</,
?/=(>;
//, P
P P l- =: I !
1_
i j __- • I __
— V, P
// 5 0 ; ■V I
- / / ; J -.V ,
//5 0 ; jj I — -> 1 ■//: p i'? / . T T__i _____ r
t- y, 2;
— ii, P
i/5 0 ; jj T ; -• T_ ” //■■ 1 - / / . ! ’ ! <■=! 2 1 y z x : M - ! t , l i. r, !, - ~ • 1 : =, P i //, P {- X . 1 ■■1- X.
y z x : i |. //. 1 — x, \ - —~ • }
2 - / / , .2-— X . T -l-x', .1__• 1
y z x : T__ '/. I
.r, 2 2 -I- y, p — x . T— x, U
. i
.1__z
y z x : I __ //. a — x , 2 -5- 2 __- j 1__ y 1 }■• x 5
2— X,
. 1
5 __~
y z x : i _ !f> I — .r, 2 - —r*
—
x
—
x,
-= .!- !/!!y z x : I _ ! / A -1- x, 2 - z : \ — =, i t - v , \ I- .t; . .1— x, l + z !
y z x I _j_ y> I — a-, 2 Z ' } + =. i — ?/, i r x . 1. -1- X, .1__ 2
y z x 1 -2- II, A 4- ■x, 2 - —Z'
I- 5, P |--//, .2-— X . J -|- X, A 1• -
/// 3
- /// ill ill
!h,r- 4 ’2 in
», ' 3 ’2
r„,
in in
3
P,
-
/p ,
2 '■ X
2— x
2 i
2—
I—
i i
Piiii k t sv m m c t l'ie :
'A ,
0,
0,
\ - X
1__ j-
( i t t e r k o m p le x e :
o»° — I in 3 in. (a)
( '/ ) •
( !/),
I*).
— I vi S in (b)
0 , :I— P m i‘ n (c)
0„' — J ^ i i i i n i (a)
<>!■' — / m i in (e )
— J’ in i n ( ;/)
— I m i n i (/)
— P i u i n (j)
1‘ i n i n (/,■)
- - I ’ m i i i (I)
in
in
in
in
in
in
in
in
in
in
(a)
(P
(c ),
(d).
(«)
(/)•
tr/)
(»')
(?)
(i-)
(i)
{>•)
i-\~v\
A— //;
■l-y;
PI-?/;
1- 0;
P i - //;
h+ y,
•>—
?/•
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