T H E C H A N G E O F C O M P L E T E S Y M M E T R Y O F
T H E C R Y S T A L S D U R I N G T H E P H A S E
T R A N S I T I O N S I N F E R R O E L E C T R I C S
A N D F E R R O M A G N E T I C S
(Department of Physics, Indian lnstitute of Science, Bangalorr
Received Sptember 23, 1963
(Communicated by Dr. R. S. Krislman, v.A.qc.) l. INTRODUCTION
COMPLEMENTARY and complete symmetries of scalars, vectors and tensors of the second rank and the derivation of complete symmetry groups for these values were given in the paper [Zheludev, 1960 (a)]. By definition, the elements of complementary symmetry are antirotation axes (antiaxes) and antireflection-rotation axes (antimirror axes). These ate denoted by
1, 2, 3, 4, 5, 6 . . . co and 1, 2, 3, 4, 5, 6 . . . co respectively (the antireflection m axisT, being ah antiplane, is denoted by m) and satisfy the following condi- m tions :
(a) a tensor has, a s a complementary symmetry elemont, an antiaxis of any order, if a simple rotation of the system through an angle eorrespond- ing to the order of this axis, causes all components of the tensor to change the signs without changing their absolute value;
(b) a tensor has, a s a complementary symmetry element, an antimirror axis of any order, ir all components of the tensor change in sign without change in value, ~hen the co-ordinate system is transformed in accordance with the corresponding operation of the mirror axis.
The groups of complete symmetry of the tensors are the groups which include both the ordinary symmetry operations (rotations and mirror rota- tions) and fhe operations of complementary symmetry (antirotations and antimirror rotations). "[he comparison of antisymmetry (Shubnikov, 1951), magnetic symmetry (Tavger and Zaitsev, 1956) and complete symmetry is given in the paper [Zheludev (1960 c)].
* Visiting Professor from the Imtitute of Crystallography, U.S.S.R. Academy of Seir162
Moscow, U.S.S.R.
168

169
As was s•own in the paper referred to earlier [Zheludev, 1960 (a)] there aro only ten groups of complete symmetry of a polar tensor of the second rank (determining, in particular, a scalar and an axial rector) narnely, oo/m. oo: m (scalar group), m. o o : m (axial vector group), m. oo:m, oo:m, m - 2 : m , m m . 4 : m , 2 : m , m . 2 : m , 2 : m , 2.
In a similar manner, it can be slaown that ten groups of complete symmetry for ah axial tensor of the second rank (which, in particular case, determines pseudoscalar and polar vector) aro : c~/m. oo: m (pseudoscalar group) ; m . o o : m (polar vector group), m. oo:m, c~:m, m . 2 : m , m . 4 : m , 2 : m , m . 2 : m, 2: m, 5.
2. PRINCIPLE OF SYMMETRY SUPERPOSlTION IN COMPLETE SYMMETRY.
ULTIMATE AND CRYSTALLOGRAPHICAL GROUPS
OF GOMPLETE SYMME'IRY
The interpretation of complete symmetry can ize referred not only to teI,sors but also to the finite figures. "Ihis can l:e easily done by considering the superposition of elements of complete symmetry of the figures present!ng lhe tensor symmetry. For this purpose, first of all, it is necessary to generalize the superposition principle (Curie principle) for lhe case of ccmplete symmetry.
The simplest formulation of the principle of symmetry superposition (Curie principle) is as follows:
"Ihe symmetly group of the composed figure will be the highest ccmmon suizgroup of the symmetry groups of the components, corresponding to the mutual orientation con- sidered of the symmetry elements of tIle parts.*
The natural extension of the principle of the symmetry superposition for the case of complete symmetry is a generalizati•n, stating that
"lhis means, for instance, •hat if lhe anti- planes m of the two parts do coincide, then composed figure will have the antiplane m ; if the plane m and the antiplane m of the components coincide, m then there will Ize no similarly oriented plane and antiplane in llae composed figure. The same is also true for the rotation axes, mirror rotation axes, antiaxes and antimirror axes. For instance, two mirror rotation axes coin- ciding in direction can forro mirror rotation axis in a eomposed figure.
* The cases of superposition resulting in enhancing the symmetry are not considered here.

170 I . S . ZHELUDEV
However, a r in direction of a mirror axis and ah antimirror axis of the components will not give any axis for the composed figure, etc. Another statement that can be made in case of complete symmetry is as follows :
Ultimate and crystallographical groups of complete symmetry will be obtainr below by considering the superpositions of complete symmetries of scalars, vectors and tensors.
It was shown in the paper [Zheludev, 1960 (b)] that there ate 14 ultimate groups of complete symmetries of scalars, vectors and tensors of the second rank. Eight of them are the groups of complete symmetry (mentioned in the introduction) of polar and axial tensors of the second rank having axes of symmetry of the infinite order (oo). Six new groups 00/0% oo.m, oo.m, oo: 2 co: 2, and oo aro obtained a s a result of various superpositions of the elements of complete symmetry of the figures which represent the above said eight groups. It is to be noted, however, that new ultimate groups can be obtained only in the superpositions in which the symmetry axes (oo) coincide with each other.
All the above-mentioned 14 ultimate groups of complete symmetry include all the seven ultimate groups of usual symmetry introduced by P. Curie, namely, oofm.oo:m, c~loo, m.oo':m, oo:2, oo:m, c~.m, oo.
There are twelve crystallographical groups among the groups of complete symmetry of the polar and axial tensors of the second rank. Various combi- nations of the figures having the symmetry of these groups result in 21 new crystallographical groups, namely: 4 : m ; 4 : m ; 4; 4; 4.m; "4.m; ~.m;
4".m; 4.m; 4:2; 4; 2.m; 2.m; 2.m; 2 : 2 ; 2 : 2 ; m; m; 2; 2; 1.
From the total number of 33 (12-b 21) crystallographical groups of complete symmetry, 20 groups represent all the symmetry groups of low systems (triclinic, monoclinic, rhombic) and 13 groups representa part of the tetragonal system groups.
These above-considered 33 crystallographical groups and 14 ultimate groups can be arranged in 11 rows, their type being as follows : m . l : m (2.m); m . 2 : m . . . m . o o : m

171
It is obvious that 33 new crystallographical groups m . 3 : m ; m . 4 : m ; m .6: m, etc., can be obtained by changing the axis oo for tlae crystallographical axis of the order 3, 4 and 6 (medium systems) in eacla row.
Having arranged some groups out of 33 in 11 rows, another 13 rows can be formed. They a r e a s follows:
2 " m ; 4 : m . . .
2 . m ; 4 . m . . . m and ate not completed by the ultimate groups but they can produce additional u
13 crystallographical groups possessing the axes 6, 6; 6. Thus, the total number of groups of low and medium systems found appears to be equal to
79 (33 + 33 q- 13). The number of crystallographical groups of the cubir systcm is equal to 11. They can te obtained by considering the symmetry of the cube distorted by scalar, rector and tensor influenr as well as by combination of these influences,
influenr exerted by polar, sr and axial tensors together.
Thus, the total number of crystallographical groups of complete symmetry is equal to 90. Thoy inr the 32 groups of usual symmetry. In complete symmetry, these groups are r as the groups which do not include the polar rector r (these ate polar, one-colour groups in the anti- symmetry). Table I shows complete symmetry for all the crystal r162 by enumerating all elements of symmeti-y belonging to the given class. The rotations are designated by the letter
the order of the axis being given in the indices. For simple rotations, they are given above, for antirotations, mirror-rotations and antimirror-rotations--below. The symmctry plane
(operation 1-) is designated by P, an antiplane (operation T) by P. The centre of symmetry (operation 2-) is designated by C, the anticentre (operation 7) by C. The symbols of the complete symmetry classes acr to the inter- national convention (international symbols) and the subclasses of usual symmetry for all r of complete symmetry are also given in Table 1.
Geometrir interpretation for some groups of complete symmetry is given in Fig. 1.
3. FERROELECTRIC PHA$E TRANSITIONS
By ferroelectric phase transitions are meant such transitions in which
- - I . there apl~ars an electrical polarization Ps in the initially non-polar para-electric

172 I . S . ZrmLtrDEV
9
J ts 9 I q s ~
,-'2__ a ,
-) s t z~.. / /
] s I*
9
~
- - . f S~
4.~
- - ~
$ 2
* 9 s
9
, , a
,
1 i i i ls
/ q e
I i i
I t I t "7
Fio. 1. Some of the groups of complete symmetry in ver and tensor" interpretation: a,
- - 2 : 2 ; b,--~~.m; c , - - 6 : 2 , d , m . 2 : m ; e,--4.m, f,--2.m_ g,--m.3:m_; h , - - 6 : m ; i,--4:2, unit cells of the crystals. T•e directions of polarization in adjacent cells of the ferroelectric crystals ate parallel and in the antiferroelectric crystals tlaey are antiparallel. As a result of the polarization, lhe cells become
and acquire the complete symmetry of one of the polar crystal classes.
Complete symmetry of the polar crystals is described by one of the groups of complete symmetry, these being the subgroups of complete symmetry of a polar vector m . ~ " m. There are 31 subgroups of the polar groups of m complete symmetry of the crystals
Table I):1; 2; 2; 2; m; m;
2 : m ; 2 : m ; 2 : 2 ; 2.m; 2.m; m . 2 : m ; 4 ; 4 : 2 ; 4 : m ; 4.m; r n . 4 : m ;
4; 4. m; 3; 3 : 2 ; 3.m; 6; 6.m; 3 : m ; m . 3 : m ; 6; 6 : 2 ; 6 : m ; 6.m; m . 6 : m . The same groups can be named the groups of complete symmetry of the
In ferroelectric cyrstals, one of lhese groups of complete symmetry has the domain--a region of spontaneous polarization with the same direction of Ps. In t•e antiferroelectric crystals, these groups of complete symmetry have the subcells,
the former unit cells of the para-electric modification.

173
TABLE I
Systems
N o t a t i o n o f
the class by national"
The formula of complete
Shubnikov symbols symmetry
Subelass of usual symmetry
1 2 3 4 5
2
1 Triclinic ..
~ , t 9 ,
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
. ~
~ ~
. ~
~ .
Rhombic ..
~ ~
~ ~
~ ~
~ ~
~ 9
Tetragonal
~ ~
~ I
~ ~
~ ~
~ ~
~ ~
~ ~
~ ~
~ ~
~ ~
9 ~
~ ~
~ ~
~ ~
~ ~
6 o
4 Monoclinie' ..
2.
2
2 m m
2 : m
2: m
2: m
2 : m
2: 2 m
2 : 2
2. m
2 . m
2. m m . 2: m m .2: m m . 2 : m m . 2 : m
4
4 t
4: 2
4: 2
4 : m
4 : m
I ' i
2'
2 m'
La~ = C
La~ ----- C
Llz
L ~
L'~ = P m
2'/m'
2/m'
2'/m
L l t = P
L'=~PC
L_2,P_. C._
L , W C
2/m
2'2'2
L ~=PC
L=2L2
.222 m ' m ' 2 m'm2'
3L 2
L~2P m
L1,PP mm2 L22P m ' m ' m L2~2L~~P2PC m'm'm' 3L~.~3PC ro'mm mmm
4'
4
4'22'
42'2'
422
4'/m
3 L ~ 3 P C
L~~Ÿ
L 4
L~42L~2L~.
L44L2
L44L 2
L2~.~jPC
4/m' L4~PC
6
1
1
2
1
2
1 m
2 m
2 : m
2
2 : 2
2 m
2 . m
2 : m
2 : 2
2 . m m . 2 : m
2
4
2 : 2
4
4 : 2
2 : m
4

174
Systems
I. S. ZI-IELUDEV
TABLE I
Notation of " I n t e r - the elass by n a t i o n a l "
Shubnikov s y m b o l s
The formula of complete s y m m e t r y
Subclass o f usual symmetry
1 2 3 4 5
28 T e t r a g o n a l .. 4: m
29
30
31
32
33
34
35
36
37
38
39
40
41
42
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
4: m
4 . m
4. m
4. m m . 4 : m m . 4 : m m . 4 : m m . 4 : m m . 4 : m m . 4 : m
71
"~
~ . m
_~.m
43
44
. . . .
. . . .
~ . m
7~.m
49
50
51
52
45 R h o m b o h e d r i e . . 3
46 . . . . 3 : 2
47
48
. . . .
,,
3: 2
.. 3. na
. . . .
. . . .
3 .m
~
,,
. . . .
.. ~
~ . m
53
54
55
,,
,,
,,
.. CŸ m
.. ~ . m
.. 6. m
~ ' 2 m '
~12.rn
3
3/2'
3/2
3 m '
3m
~'
~
~ ' / m '
] ' / m
I / r o '
3/m
4'/m' L~~~ ~PC
4 / m
4 ' m ' m
L4~PC
L~,2P2P
4 m ' m ' L44P
4 m m L~4P
4 ' / m m m ' L~4~~2L~~2L2~3P2PC
4 / m ' m ' m ' L4;4L~~SPC
4 / m m ' r o ' L4#L~~P4PC
4 / r o ' m m L4;4L~~PPC
4 ' / r o ' m m ' Lz4~i2L~.~2L~~2P3PC
4 / m m m L4;4LI~5PC
7[' L ~
L~; ~
~ 2 ' m '
71'2'm L2;2L22P
L~;2L~2P
L2~2L22P
L 8
L83L_2
LS3L s
Ls3P
D
La3P
L3e3C
LsaC
L8~3L2~3PC
La~3L~3L2~3PC
L~~3L~3L2~3PC
LS~3Ls~3PC
6
3
3
3 : 2
3
3 . m
3
3 : 2
3 . m
4 : m
2.111
4
4.111 m . 2 : m
4 : 2
4 : m
4 . m
7Lm m . 4 : m
2
2
2 . m
2 : 2
~4m

175
T ~ L E I
Systems
N o t a t i o n o f " I n t e r - the class b y n a t i o n a l
S h u b n i k o v symbols
T h e f o r m u l a complete o f s y m m e t r y
Subclass usual o f s y m m e t r y
1 2 3 4 5 6
56
57
58
H e x a g o n a l ..
. . . .
. . . .
3 : ra
3 : m m . 3 : m
65
66
67
68
69
7O
71
72
73
74
75
76
77
78
79
8O
81
82
83
59
6O
61
62
63
64
. . . .
. . . . m . 3 : _m
_m.3 : m
,,
. . . .
.. m . 3 : m
6
. . . .
. . . .
6 n
6 : 2
6: 2 . . . .
. . . .
,,
6: 2
.. 6: m m
. . . . 6 : m
. . . . 6 : m
,,
. . . .
.. 6 : m m
6 . m
. . . . 6 . m
. . . .
,, ..
. . . .
. . . .
. . . .
6 . m m . 6 : m m 6 : m m . 6 : m m . 6 : m
Cubic
,, i~
. . . . m . 6 : m
.. m . 6: m
.. 3/2
.. 3/_4
.. 3/~
.. 6/2
~'
~
~ ' m ' 2
~'m2'
~m'2'
~m2
6'
6
6'2'2
La~P
La~P
LS~3Ls4P
LS~3L23PP
L~a3L~4P
L ~
L e
L~3L23L2
62'2'
622
6 ' / m '
6 ' / m
6/m'
6 / m
6 ' m ' m
6 m ' m '
L%L~
L % L ~
Ls~e~PC
i
Ls~~PC
Le~PC
L % P C
L863P3P
- i
L % P
6 m m L % P
La~ ~~3L~~3L2~3P4PC
6 ' / m m m ' La6~g3LS.~3L2~P3PC
6 / m ' m ' m ' L6~6Lz~7PC
6 / m m ' r o ' Le~6L2~6PPC
6 / m ' m m LS~6L2.~6PPC
6 / m m m Le,6Ls~7PC
23
T 3 m '
3L24L 3
3LSa4La6P
~3m m ' 3
3LS~4Lq6P
3L~~4L3~3PC
^4
3
3 : m
3 : 2
3 . m
3 : m m . 3 : m
3
6
3 : 2
6
6 : 2
3: m
6
6: m
3 . m
6
6 . m
~ . m m . 3 : m
6 : 2
6: m
6 . m m . 6: m

176
1
84
85
86
87
88
89
90
Systems
2
. . . .
. . . .
.
.
.
.
. . . .
. . . .
. . . .
8/2
3/4
3/4
6/4
~/4
6/4
8/4
3
I. S. ZHELUDEV
TABLE I ( C o n t d . )
Notation of "Inter- the class by national
Shubnikov symbols
The formula of complete symmetry
4 5 m3
4'32'
432 m'3m
3L2~4La~3PC m3m'
3LZa~~4L3~6L2~6P3PC
3LL_~_~4L3~6L_~3P6PC m'3m' 3L4;4LS~6L~.,9PC m3m 3L4~4L3~6L~~9PC
Subclass usual of symmetry
6
Under ferroelectric phase transitions, spontaneous polarization can appear in the crystal along different directions. A new crystal symmetry
(more exactly, its domain or only a unit cell) will correspond to these different directions. In complete symmetry, the change of the symmetry during a ferroelectric phase transition can be found by the analogy, as it was made in case of usual symmetry (Zheludev and Shuvalov, 1956, 1957). Using the generalization of the Curie principle of symmetry for the case of complete symmetry, one can get the crystal symmetry (a new symmetry) after the ferro- electric phase transition by considering the superpositions of complete symmetry of the crystal in the para-electric (non-polar) phase and the vector
- ) , of spontaneous polarization Ps. For instance, ir is not difficult to see from
Table I that a superposition of the complete symmetry of a crystal belonging to the elass ~ 4 and the complete symmetry of the vector Ps (na. ve: na) results m m ='ir' ~ ' ) in a group m. 2, m (in the case of a coincidence of Ps with the axis' . When m N
Ps coincides with the axis ~, it results in the g r o u p 6 . m . In case of a coinci- w m dence of the rector Ps with the axis 2, it results in the group ~, etc.
The results of the change of complete symmetry of the crystal for cubic, rhombic, monoclinic and triclinic classes are given in Tables II, III*. The
* The tables of the changes of complete symmetry under the fcrroelectric transitions in the crystal r of other systems aro not givvn h9 since they ate too extensive.

177 number of crystallographical directions equal to the given direction in the class considered is given in brackets after the polar groups. This number is equal to the number of the unit cells (they have already a polar symmetry) required for obtaining the crystal symmetry in the high-temperature phase.
This means that when the crystal is broken up into domains having equal number of cells oriented along the crystallographical equal directions, it acquires complete symmetry of the para-electric phase.
Tables II and III show the possible polar classes of complete symmetry of the crystal after ferroelectric phase transitions. But they do not indicate as to whether a crystal of the given class can undergo any transition, ferro- electric or otherwise. They do not give also any idea about the sequence of these ferroelectric transitions. It is assumed in the Table that electric polarization appears in the crystal having a para-electric modification, the complete symmetry of which is shown in the upper line. This means, in particular, that if the crystal undergoes some ferroelectric transitions from one polar class into another one, then, every time, the transitions are through
"returning " to the para-electric modification. It is impossible to say
if the crystals undergo the change of the direction of spontaneous polarization exisfing before the transition wilhout " r e t u r n i n g " to the para- electric modification. These transitions will not be considered here. At present, the practical signi¡ of Tables 11 and Ili is not clear. But in spite of it, one can draw some useful conclusions. Thus, in most wqll-known cases, low-temperature modi¡ of the crystals are the characteristic ferroelectric modifications. These modifications are of lower symmetry than the para-electric ones. In some cases, however, in the crystals after the ferroelectric region, on funher cooling, there appears again a para-electric modification, its symmetry being higher. Typical example is Rochelle salt crystals. Above 24 o, these crystals have a para-electric modification, the symmetry being 2: 2. In the temperature range q - 2 4 ~ - - - 1 8 o, they have a ferroelectric modification, the symmetry of which is 2. Below -- 18 ~ they have again a para-electric modification, the symmetry being 2 : 2 . It can be seen that the cooling results in ah increase of the crystal symmetry.
For complete symmetry, this transition can however be written in the forro: m . 2 : m (2:2)--above 24~ 2._m (2) for-t-24 + - - 18~ 2 : 2 (2:2) m m b e l o w - 18 ~ C. (The subgroups of common symmetry of the groups of complete symmetry are given in brackets). In this case, the transition to the low-temperature para-electric modification goes without increasing the symmetry, since three groups have the order 8, 4, 4 respectively.

1 I

180 I . S . ZrmLtlDEV
4. FERROMAGNETIC PI-IASE TRANSITIONS
Ferromagnetic phase transitions here are such, under which there appears magnetic moment Ms in the non-magnetic cells of the crystals.* The direc- tions of Ms in the neighbouring unit cells in the ferromagnetic crystals ate parallel. In the antiferromagnetic crystals, these directions ate antiparallel.
As a result of the appearance of magnetic moment, the cells acquire complete symmetry of one of the axial classes,
such classes, the symmetry of which is one of the subgroups of the axial vector group (magnetic moment) Ms--the group m. ~ : m. The total number of such axial groups of complete crystal m symmetry is 31
Table I): 1, 2, 2, 2, m, m, 2: m, 2: m, 2: 2, 2.m, 2.m, m . 2 : m , 4, 4 : 2 , 4 : m , 4.m; m . 4 : m , 4, 4.m, 3, 3"2, 3.m, 6, 6.m, 3 : m , m . 3 : m, 6 , 6 : 2 , 6 : m , 6 . m , m . 6 " m.
These groups can be named the groups of complete symmetry of the
In the ferromagnetic crystals, the domain has a complete syrnmetry of one of these groups. In the antiferromagnetic crystals, a former unit cell of non-magnetic modificaion •as a complete symmetry of one of these groups.
The modifications of the complete symmetry, when magnetic moments
Ms appear in them along different directions, can be found in the same way as was done under the ferroelectric phase transitions.+ The superpositions of complete symmetries of the crystals in non-magnetic modifications and the symmetries of the vector of the magnetic moment Ms should be considered for this purpose. For instance, it is not difficult to see from Table I that superposition of complete symmetries of the crystal, belonging to the class
- 1 , .
~q and that of the vector Ms (m. c~: m) results in the group m. 2: m (when
Ms coincides with the axis 4). When Ms coincides with the axis 6, it results
..1, in the group 6".m. A coincidence of Ms with the axis 2 results in the group
2: m, etc.
The results of considering the modifications of complete symmetry of the crystals under the phase transitions caused by the appearance of magnetic
* t.e., those which have no spontaneous magnetic moment. t The modification of the magnetic symmetry of the crystal during the ferromagnetic, phasr transitions was performed for the first time by Shuvalov, 1959,

Change of Complete Symmetry of Crystals during Phase Transitions 183 moment Ms are given in Tables IV and V. In these tables, along with new groups of complete symmetry, is indicaŸ the number of crystallographical directions in the crystal that are equal to the direction in ,91 the magnetic moment appears. If after the phase transition, ferromagnetic has this number of the domains (in the antiferromagnetics--the number of unit cells), which are equally oriented along the equal directions considered, then the whole set of domains (cells) will have the symmetry of the crystal which it had before the phase transition.
The remarks similar to those which were made for Tables II and III are true for Tables IV and V. It is assumed here that all transitions arise from the non-magnetic modification. The transitions between two magnetic modifications are performed by " r e t u r n i n g " to the nonmagnetic modifica- tion, etc.
The tables similar to Tables IV and V can be used directly for investi- gating the physical properties and the structure of magnetic crystals.
The author of this paper expresses bis gratitude to Prof. R. S. Krishnan and Dr. P. S. l'4arayanan for their help and attention.
5. SUMMARY
The extension of the superposition principle of the symmetries (P. Curie principle of symmetry) for the case of ccmplete symmetry is given. The enumeration of all crystallographical groups of ccmplete symmetry is pre- sented, the number of elements having complete syrnmetry for each class of the crystals being indicated. "Ihe change of complete symmetry of the crystals under the'p¡ transitions is obtained by superimposing the elements of complete symmetry of polar or axial vectors on the one hand, and the elements of complete syrnmetry of the crystals on the other.
The tables of complete symmetry changes for the cubic~ rhombie, mono- clinic and triclinic crystals during the ferroelectric and ferromagnetie phase transitions are given.
REFERENCES
Tavger, B. A. mad Zaitsev, V. M.
Shubnikov, A.V. ..
Shuvalov, L. A.
JF.TP, 1956, 30, 564. (Eng. Ÿ 1956, 3, 430.)
"Symmetry and antisyrnmetry of the finite figures," Izd. Akad.
Nauk, SSSR, Moseow) 1951. ffristallografiya, 1959, 4, 399.

184 I. S. ZHELUVeV
Zheludev, I. S. . . Krystallografiya, .1960(a), 5, 346, Engli.qh Translation; Soviet
Physics--Crystallography, 1960, 5, 328.
. . lbid., 1960 (b), 5, 508; English Translation; Soviet Physics
--Crystallography, 1961, 5, 489.
Zheludev, 1. S. and Shuvalov,
L.A.
. . Izvest. Akad. N a u k S S S R , Ser. Fiz., 1960 (c), 24, 1436.
Krtstallografiya, 1956, 1, 681.
. . lzvest. Akad. N a u k S S S R , Ser. Fiz., 1957, 21, 264.