CUBIC
AND ULTIMATE
GROUPS
COMPLETE
SYMMETRY
OF
BY 1. S. ZHELUDEV*
(Department of Physics, Indian Institute of Science, Bangalore-12, India)
Received Or
15, 1963
(Communicated by Dr. R. S. K¡
1.
r.A.Sr
[NTRODUCTION
THE idea of complete symmetry of tensors was introduced in our previous
papers. The groups of complete symmetry of polar and axial tensors of the
second rank have been described earlier (Zheludev, 1960 a). Ultimate groups
of complete symmetry for these tensors were also described (Zheludev, 1960 b).
All crystallograpkical groups of complete symmetry and the changes of
complete symmetry during ferroelectric and ferromagnetic transitions in
the crystals were presented in a subsequent paper (Zheludev, 1964). The
relations between antisymmetry, magnetic symmetry and complete symmetry
are given in the paper by Zheludev (1960 c). Cubic groups of crystal symmetry
will be determined in detail in this paper [in the paper (Zheludev, loc. cit.)
they were presented without a conclusion]. Ultimate groups of complete
symmetry of finite figures and the relation between complete symmetry and
physical properties of the crystals are also discussed here.
2. Cumc
GROUPS OF
COMPLETE SYMMETRY
As it was shown earlier (Zheludev, 1957), crystallographical groups of
usual symmetry can be obtained asa result of scalar (including pseudoscalar)
and vector (including pseudovector) actions on the elementary parallelepiped
of the cubic system. The groups of symmetry, in this case, are found a s a
result of superposition of the elements of the symmetry of the original elementary cubic parallelepiped (cube) and external action. Within the
limits of usual symmetry, it is possible to establish five above-mentioned sets
of values differing in symmetry and resulting in the groups: m3m (91
* Visiting Professor from the Imtitute of Crystallography, U.S.S.R. Ar
of Soience,
Mosr
U.S.S.R.
§ The symbols of the groups of usual and complete symmetry assumed in the international
convention ate given. The notations by Shubnikov (see Zheludev, 1964) are given in brackets.
Al
191
192
I . S . Z~LUDEV
scalar action 432 (3/4)--pseudoscalar action, 4"3m (3~)--vector
action,
m3 (~/2)--pseudovector action, as well as in the group 23 (3/2) presenting
combinations of the above actions. The original elementary parallelepiped
in this case should be considered a s a coloured cube. This parallelePil~d
is equal to the grey (not painted) parallelepiped subjected to the scalar action
[group m3m (~/4)].
From the point of view of complete symrnetry, it is possible to consider
cubic groups of crystal symmetry in a more detailed manner, since in lhis
case, one succeeds in ascribing different symmetry to the broader class of the
combinations of actions than it is possible in the case of usual symmetry.
In this case, the initial elementary parallelepiped is considered to be a " g r e y "
(not coloured) cubo. Sucia a cubo, by definition, possesses all elements of
symmetry of the figure belonging to any cubic group: three axes 4, 4", 4 and
m
4, four axes 6, 6, 6, six axes 2, 2, 2, 2, nine planes and nine antiplanes of
symmetry, the centre of symmetry and anticentre of symmetry. From all
external actions, it is (bese groups that have the symmetry of cubic groups
that will be considered here. Such groups of symmetry possess more than
one operation of rotation around the axis having the order higher than the
second one. It is easy to see that scalar effect (symmetry ~ / m . oo: m),
pseuodoscalar effect (symmetry c~/_ra,c<~:m.) a n d a combination of theso
effects (symmetry ~ / ~ : 2) satisfy this condition. Besides,,the set of four
polar vectors directed perpendicularly to the edges of a cubic tetrahedrona
the same set of axial vectors and any combination composed of the two latter
effects in a combination with scalar and pseudoscalar effects comply with the
above-said condition.
Cubic groups of completo symmetry will be found in this case, by considoring the superposition o f the olements of completo symmetry using
generalized Curie principle of symmetry (Zholudev, 1964). It is obvious
that cubic groups will be obtained in case of scalar and pseudoscalar action
during any mutual oriontation of a "grey " cubo and actions. The group
m3m. (~/4) will answer for scalar action, while the group m'3m' (6/4)
(a new group as compared with usual symmetry) will satisfy the pseudoscalar
action. The group 432 (3/4) will satisfy the combination of scalar and pseudoscalar actions. In the case of rector and pseudovector actions (above-mentioned sets of four vectors or pseudovectors), cubic groups can be obtained
only when the vectors aro oriented along the space diagonals of the cubo
Cubic and Ultimate Groups of Complete Symmetry
193
In lhe case of polar vectors, a superposition of the elements of symmetry
of the action with the eJements of symmetry of " a grey" cube results in a
group m'3m (~/4).
~
In case of axial vectors it results in the group m3m'
m
c~'~).
Cubic groups can also be obtained in the case of various (double, triple
and four-fold) combinations of scalar, pseudoscalar, vector and pseudovector
effects. It appears that there are 11 cubic groups which differ in complete
symmetry (Fig. 1). One of these groups 23 (3/2), the least symmetrical one,
(as is seen from the figure) can be presented by five different combinations
of the effects (four triple combinations of the type-scalar-vector, pseudovector; pseudo-scalar-vector, pseudo-vector, etc.) and one combination
including all types of the effects (scalar, pseudoscalar, vector and pseudovector).
The rest of the groups of complete symmetry (except the group 23) have only
one interpretation with the help of ~ctions.
m3m
m t 3 tot
h~2
m'3m
rn x m '
4'32'
~ ~m
q'3 m'
•5
mt3
25
2z
25
25
25
Irlo. 1. The interpretation of cubie groups of complete symmetry is attained by means of
sealar, pseudosealar, vector and pseuodovecor actions.
It is the cases of scalar and pseudoscalar effects of one sigla that are
considered in the figures presented in Fig. 1. At the same time, for instance,
another presentation is conceivable for the group m3m (~q
In this case,
the scalar effect will be a " n e g a t i v e " one (•ydrostatic compression in particular) in contrast to a "positive" effect which is shown conventionally by
194
I . S . ZHm_~UD~V
a black circle. In its turn, the figure presenting the group m'3m' (~/4)
can be considered as the " r i g h t " one, while the "left " figure belonging
to this group which can be conventionally designated by a circle with an
arrow directed to the left is also conceivable, etc. All sucia cases that do
not result in new groups of complete symmetry are not considered in Fig. 1.
We do not know of the other types of effects (except those shown in
Fig. 1) which would also result in cubic groups. It is therefore, possible
to assert that there are only 11 cubic crystallographical groups of complete
symmetry. All of them can be considered to l~e a result of external actions
(described by tensors of the second rank) on the " g r e y " cube. Practical
advisability of tensor interpretation of the groups of complete symmetry of
the crystals will l~e considered below.
The set of the elements of symmetry of the figures belonging to cubic
crystallographical groups is given in Table I. The axes of symmetry are
TABLE I
Cubic groups of complete symmetry
Notation
of the
classes by
Shubnikov
International
notations
of the
classes
1
3/;2
;23
3L~4L3
3/2
2
3/74
4'3m'
3L~~4Ls6P
3/2
3
4
5
6
3/74
91
91
3/4
43m
m'3
m3
4'32'
3L;24Ls6P
3L~24L~33PC
3L~24L~33PC
3L424L36L~
3/74
3/2
91
3/2
7
8
3/4
91
432
m'3m
3L44L86L~
3L%4~'4L~36L~
6P3PC
3/4
3/74
9
91
m~m'
3Lq~4L~36L2~6P3PC
6/2
10
g/4
m'3m'
3L;~4L;36L.~~9PC
3/4
11
91
m3m
3L;~4L~86L:9PC
91
No.
The formula
of complete
symmetry
m
m
Subclass
of usual
symmetry
Cubic and Ultimate Groups of Complete Symmetry
195
designated by the letters L indicating the order of cornrnon axes on the top
and the order of the antirotation, mirror and antimirror axes below. The
plane of symmetry is designated by the letter P, an antiplane is designated
by P, the centre of symmetry is designated by C and an anticentre by C.
3.
ULTIMATE G R O U P S OF COMPLETE SY~MMETRY
OF FINITE F I G U R E S
Ultimate groups are those that descril;e the s)mmetry of the figures that
have symmetry axes of infinite order (oo) (sphere, cylinder, cene, etc.).
As was shown earlier (Zheludev, 1960 b)there are 14 ultimate groups
of complete symmetry for polar and axial tensors of the second rank. Ultimate groups for finite figures can be obtained by arranging into rows crystallographical groups having identical type of symmetry. 79 erystallographic~l
groups of lower and medit.m system can be arranged into 24 such rows
(Zheludev, 1964). 11 out of these rows end in ultimate groups of complete
symmetry of tensors of the second rank. These rows are as follows:
m . l : m , m . 2 : m, m.3 :m,
..
m.oo:
1 .m, 2.m, 3 .m
..
oo.m
I:2,2:2,3:2
..
c~:2
m
13 other rows
2: m; 4 : m
2.m;
m
2:2;
4.m
u
4:2
9
~
B
do not end in ultimate groups of complete symmetry of tensors of the second
rank, and, generally speaking, they cannot end in ultimate groups in the
sense of this word, since they do not result in the groups that have simple
axes of symmetry of infinite order (c~). At the same time, it is possible to
introduce the idea about ultimate groups for sucia rows. In this case, ultimate
groups will be of conventional character, since they cannot be presented as
finite figures by means of spheres, cenes and cylinders. The notaª
of
tkr axes of symmetry ~ , ~ ,
oo in these conventional ultimate groups d9
gle
196
I . S . ZHELUDEV
not indicate that they are present in some ultimate figures but point to
the fact that when proper rows ate continued, the figures presenting ~hem can
have the axes N, ~'~, N (where N is the order of aja axis) of any hi~,h order.
Some of the latter rows have the forro
2.m; 4.m; 6.m...oo.m
m
m
1
2".m; ~.m; ~.m...~o.m
Three out of the ultimate groups of symmetry of tensors of the second
rank, namely, the groups oo/m.oo:m, oo/m.oo:m and oo/oo:2 can be
considered as the ultimate ones for the cubic groups 0/4, £
and 3/4 respec-
tively. However, this " l i m i t " is not so obvious as in the rows of ¡
figures, between above cubic and ultimate groups there is only one figure
possessing corresponding type of symmetry
~/4;
3/1"O; oo/m. oo: m
6/4;
3/10;
m
oo/m.oo: m
i.i.
3/4;
3/5;
oo/oo: 2.
The group 3/1-'6 can be obtained as a result of scalar action on "grey" icosahedron.
The group 3/1"-6 is obtained a s a result of pseudoscalar action, and
group 3/5 is a result of combined (scalar and pseudoscalar) action on the
icosahedron.
For the remaining eight cubic groups it is also possible to introduce
the idea about conventional ultimate groups. Note that in lhis case, there
are no groups at all having lhe same type of syrnmetry tetween cubic groups
and ultimate groups. When writing the symbols of conventional ultimate
groups, the symmetry of the diameters of "ultimate " spheres corresponding
to lhe symmetry of the directions in the cul~es representing one of another
group of symmetry (~/4 ~ c~/m. oo: m, 3/4---~ oo/oo: 2_ etc.,) will be shown.
Thus, together with the 21 conventional ultimate groups introduced,
35 ultimate groups of complete symmetry of finite figures are considered
here. Theseareasfollows: oo/m.oo:m; oo/m.oo:m; 00/00:2; m . o o : m ;
m.oo:m;
m.oo:m;
m.oo:m;
o0:2;
oo.m;
oo:m;
oo.m;
~:m;
Cubic and Ultimate Groups of Complete Symmetry
o0:2;
co;
co/m.oo:m;
o0/m.co:m;
00/00:2;
o0/oo.m; co/oo.m;
o0/oo; m . o o : m ;
m.~:m;
c o . m ; o0.m; ~ . m ;
oo.m; 00:2;
oo; o0; ~ .
,~/oo:m;
oo:m;
197
~/co:m;
oo:rn;
oo.m;
The systematization of
crystallographical groups according to their classification into one or
another ultimate group is presented in Table II. Since some of the first
terms in the rows represeat the identical groups but are written dowrt in
a different manner, let us show the connection between the different notations
1.m = ' i ' =
l:m=
m
m.l:m=m.l:m=
1:2:=2
2.m
2.m=2:m
m.2:m = m.2:ra
2 . m = 2: m
m . 2 : mu = m . 2 : m
1:2=2
m.l:m =2.m
1.m= l:m=m
2".m = 2 : m
m.l:m=2.m
2:2=2:2
~.m
~--- 2 : m .
Family tree of ultimate groups of complete symmetry is given in Fig. 2.
oo/o~.m
oo/oo.~m
ooloo
,,o/oo._~
ooloo~
ooloo=m
o0/oo:~_
0<~~
~oloo.m
r
n.
O0/O0
ooloo=~
1
O0/mL~:rrt
~T
~'
oo 2
co m
oo m
t
oo: 2
c~.rn
o~.rn
oo/m 10r en
-1
oo:2_
oo:m
o0.~
1-
oo:2
oo:,'n
oo.m
1
~:2
r
(~J~.m
l
ao:Z
oo~m
oo-m
Fto. 2. Family tree of ultima(e groups of complete symmetryof fmi(e figure:
i
6.m
~,m
~.m
"4.m
O0,m
4. m
3.m
6.m
2.m
I~
!
6:2
j
i
i
6:2
2:2
3.m
1~
3:2
1:2
i
i
OO~
~:2
,
4:2
6.m
4.m
2,11'1
6~2
6.m
2.m
3:m
4:2
~.m
oO
~.m
m
6:m
4:m
2:m
m . OO : m
m.O:m
6om
~.m
3:m
1 :m
U _
~o.m
4"oro
iI 6 . m -
i 4:m
2:m
m.6 : m
m.4 :m
m.3 : m
r..4 :m
m.3 :m
: m
m.2 : m
~/~
~/2
m.1 : m
s/4
m.2:m
3~
m.l:m
1 :m
m.4 : m
:m
~ 2:2
i
m.~:m
m.2 : m
~lm.~
I m.6:m
4:2
:m
3:2
m.4:m
~/m.~
~, I ~~:m!~Im.
~,I ~4
~
2:2
m
1:2
m.6:m
m.$:m
O 0 [ m , 00 t
6/4
~[~
6
2,1'11
2:m
_~q : rrf
_I
i '4.m
: HI
I
oO
OO.m
~
i4:m
i'
m.~o :m
i
~3
8.m
6:m
2:m
i4:m
I
:m
m,6:m
4
006
~.m
l
CO:m
ni.~
m.4:m
m.3:m
m.6:m
m.3:m
m.2:m
m.l:m
m.2:m
i
[ m.4:m
m.l:m i_
0 0 / ~ .TI1
: fil
i
s/2
,o/oo
s~
=/oo . ~
s/~
Systematization of crystallographical groups of complete symmetry
T,~LE II
O0
Cubic and Ultimate Groups of Complete Symmetry
199
4. COMPLETE SYMMETRY AND PHYSICAL PROPERTIES OF THE CRYSTALS
A geometrical interpretation of the cubic groups of complete symmetry
by means of scalar and vector actions gives an obvious opportunity for diseussing the physical properties of the crystals. For instance, it is seen from
Fig. 1 that an optical activity is possible in cubic crystals of the classes 6/4;
m
3/~; 3/4; 6/2; 3/2.
This is determined by the presence of pseudoscalar
m
actions in the figures representing these classes.
There are polar directions
in the crystals of the classes 6/4; 3/4; 3/4; 6/2 and 3/2.
m
m
This means that
m
piezoeffect is possible in them. The presence of polar directions corresponds
to the presence of polar rector components in the actions presenting the
symmetry of the listed classes, etc.
Scalar (pseudoscalar) and vector (pseudovector) components are shown
as main components in Fig. 2. The subjected groups in which there exists
one or another action ate also given. While using Table II and Fig. 2 ir
is possible to select the groups containing one or another main action without
using geometrical interpretation. So, it is possible to find out from Fig. 2
and Table II that there are axial directions in the crystals of the classes
6"/4; 3/4; 3/4"; 6"-/2; 3/2, etc.
Geometrical interpretation of all groups of complete symmetry appears
to be rather extensive. However, the most essential results can be obtained
considering only the most important groups. First of all these are polartensor groups, subjected to the group m. co: m. The most highly symmetrical
groups subjected to the above group are crystallographcial groups m . 6 : m
and m.4 :m. AII groups of usual symmetry (except cubic group) are subjected to these groups. Hexagonal and tetragonal prisms coloured into
certain colour give the symmetry of the crystals belonging to these groups.
One colour of the figures representing these groups in complete symmetry
allows us to consider the groups of usual symmetry as polar ones by the sign
(-k or - ) . It is easy to see by analogy that the groups m.6 : m and m.4" m
~
m
m
can be presented as prisms possessing a certain sign of the enantiomorphism
(the crystal group polar by the sign of enantiomorphism, i.e., either right of
left ones). The presence of axial-tensor components in the actions is
characteristic for all groups representing the latter groups.
The groups subjected to the group m. co:m, are of two kinds. In its
m
turn, each of them can be interpreted in two different ways.
For instanr
200
I . S . ZH~LUD]Yr
the group m . 4 : m can be interpreted by means of polar tensors (Fig. 3 a, b)
m
and by means of axial vectors (Fig. 3 c). The group m.6: m can be presented
by means of axial tensors (Fig. 3 d) and by means of polar vectors (Fig. 3 e, f).
The same is the situation during the interpretation of the groups subjected
to the group m. co: m (Fig. 4). In connection w;th the possible double
m
m
interpretation of the groups in question m. o o : m and m. co:m, crystallographical groups are divided into two parts (vchere ir is necessary) in Table II.
For instance, the groups including vector and axial-tensor components in the
row ~o.m have the axes ~ and those including pseudovector and polar-tensor
components llave the axes 2" and 6", etc.
"-x
c
Q.
T
j
1
l
91
Fio. 3. Geometrical interpretation of crystallographical groups subjected to the grou
na. oo:m. (a, b) Group m . 4 : m . The interpretation is attained by means of polar tensor
(space and plane variants).
vectors. (d)group
(c) Group m.4: m. The interpretation is attained by means of axia|
m.6:m.
The interpretation is attained by means of axial
tensors.
m
(e, f ) Group m.6: m. The interpretation is attainecl by means of polar vectors (spacv and plane
variants).
The use of Figs. 3, 4 and the Table allows to select the groups of complete
symmetry describing the crystals of intermediate and lower systems possessing any properties. It follows frcm Table II and Fig. 2 that axial (pyromagnetic groups of complete symmetry (subjected to the group m.oo: na)
Cubic and Ultimate Groups of Complete Symmetry
r
201
e
Q
d.
Flo. 4. Geometrical interpretation of the groups subjected to the group m. oo:m.
Group m.4: m.
(a, b)
The interpretation is attained by means of axial tensors (space and plane variants).
(c) Group m.4-- ra. The intcrpretation is attained by means of polar vectors. (d) Group m.~: m.
The/nterpretation is attained by means of polar tensors. (e, f ) Group m. 6: m.
The interpreta.
tion is attsJned by means of ax/al vectors (space and plane variants).
are 31 groups: 1; 2; 2; 2; m; m; 2 : m ; 2 : m ; 2 : 2 ;
2.m;
m
2.m;
m
m . 2 : m ; 4; 4 : 2 ;
4 : m ; 4.m; m . 4 : m ;
6-m; 3 : m ; m . 3 : m ;
6; 6 : 2 ;
6:m;
4; 4.m; 3; 3 : 2 ; 3.m; 6;
6.m;
m.6:m.
~Ihe crystals
belonging to the groups subjected to the group oo: 2 h~tve directions which
m
are both polar and axial ones. Henee, these crystals can have simultaneously
both pyroelectric and pyromagnetic properties, etc.
Concluding the discussion about the connectŸ between 1he complete
symmetry and the physical properties of the crystals let us stress the
remarkable fact that all cubic groups can 1;e interpreted a s a result of actions
deseribed only by polar and axial tensors of the second rank on the " g r e y "
cube. In this connection it is possible to assrme ~hat any action in some
point of spaee can be reduced to those mentioned here. ~Ihen, in the case
of empty isotropic space, the action (or the system of acting forces) should be
redueed only to those which are presented in Fig. 1. Apparently, when
202
I . S . 7_,mmtrDEv
considering this action, it is possible, in particular, to have an idea about the
symmetry of isolated fundamental particles.
5.
SUMMARY
It is shown that the systems of definite actions described by polar and
axial tensors of the second rank and their combinations during the superposition of their elements of complete symmetry with the elements of complete
symmetry of the " g r e y " cube, result in 11 cubic crystallographical groups
of complete symmetry. There are 35 ultimate groups (i.e., the groups having
the axes of symmetry of infinite order) in complete symmetry of finite figures.
14 out of these groups are ultimate groups of symmetry of polar and axial
tensors of the second rank and 24 are new groups. All these 24 ultimate
groups are conventional groups since they cannot be presented by certain
finite figures possessing the axes of symmetry oo, c~, c~. Geometrical interm
pretation for somo of the groups of complete symmetry is given. The connection between complete symmetry and physical properties of the crystals
(electrical, magnetic and optical) is shown.
6.
ACKNOWLEDGEMENT
The author of the paper is thankful to Professor C. V. Raman, Professor
R. S, Krishnan and Dr. P. S./qarayanan for the fruitful discussion of some
points in this work and their help.
7.
REFERENCES
1. Zheludev, l . S .
..
Kristallografiya, 1957, 2, 728:
PhyMcs-Crystallography.
2.
..
Kristallografiya, 1960a, 5, 346: English Translation.
Physics-Crystallography, 1960, 5, 328.
Soviet,
3.
..
Kristallogra]iya, 1960b, 5, 508: English Translation.
Physics-Crystallography, 1961, 5, 489.
Soviet
4.
..
Izvest. Akad. Nauk SSSR.,
5.
..
Proe. Ind. A›
English translation.
Soviet
Ser. Fiz. 1960c, 24, 1436.
Sci. 196~, 59, 168.