Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
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The following general rule exists: only those matrices O
i
are
permissible for which, if the non-zero coefficients are replaced
by 1, the corresponding transformation of the axes conserves the
Hermann–Mauguin symbol.
Examples
(1) When the three letters of the Hermann–Mauguin symbol are the
same, as in P 222, Pmmm, Pnnn etc., the Hermann–Mauguin
symbol does not change and all six matrices are valid.
(2) When the z axis plays a privileged role and when the x and y
axes are equivalent, only O
1
and O
6
apply. Examples are P222
1
,
Pbam and Ccca. In Pmma,thex and y axes are not equivalent
because the interchange leads to Pmmb (the non-equivalence of
the x and y axes can also be recognized by inspection of the full
symbol P2
1
=m 2=m 2=a).
(3) Matrix O
1
always applies.
13.1.2.3. Triclinic system
As stated above, P1 has only isomorphic subgroups and the
general nature of the matrix S [equation (13.1.1.3)] requires the use
of special techniques (cf. Chapter 13.2, Derivative lattices); they
apply also to P
1.
13.1.2.4. Parity conditions
In equation (13.1.1.2b), there occurs the choice of the origin by
means of s, the nature of the lattice by means of t
G
and the nature of
the symmetry operations by means of the column matrices w and w
0
.
The three factors, origin, lattice type, screw and glide components,
impose parity conditions on the coefficients of the matrix S. Only a
few examples are given here.
When (W, w) and W
0
, w
0
are operators of lattice translations of
H, say I, t
G
and I, t
H
, equation (13.1.1.2b) reduces to
S t
H
t
G
: 13:1:1:4
Example
In a tetragonal I lattice, t
G
is either an integral or a fractional
translation. If t
H
is
1
2
,
1
2
,
1
2
and the matrix S is replaced by T
1
,
one obtains
ORTHORHOMBIC SYSTEM
O
1
S
11
00
0 S
22
0
00S
33
0
@
1
A
Conditions: S
11
> 0, S
22
> 0, S
33
> 0, S
11
S
22
S
33
> 1
S
11
S
22
S
33
O
a
1
n
1
n
2
n
3
O
b
1
n
1
n
2
2n
3
1
O
c
1
2n
1
12n
2
1 n
3
O
d
1
2n
1
12n
2
12n
3
1
O
e
1
2n
1
2n
2
2n
3
1
O
f
1
2n
1
2n
2
n
3
O
g
1
2n
1
2n
2
2n
3
O
h
1
2n
1
1 n
2
n
3
O
i
1
2n
1
1 n
2
2n
3
1
O
j
1
n
1
2n
2
12n
3
1
O
k
1
n
1
2n
2
2n
3
O
l
1
2n
1
12n
2
2n
3
O
4
S
11
00
00S
23
0 S
32
0
0
@
1
A
Conditions: S
11
2n
1
> 0, S
23
n
2
< 0, S
32
2n
3
> 0
O
5
00S
13
0 S
22
0
S
31
00
0
@
1
A
Conditions: S
13
n
1
< 0, S
22
2n
2
> 0, S
31
2n
3
> 0
O
6
0 S
12
0
S
21
00
00S
33
0
@
1
A
Conditions: S
12
2n
1
> 0, S
21
n
2
< 0, S
33
2n
3
> 0
TETRAGONAL SYSTEM
T
1
S
11
S
21
0
S
21
S
11
0
00S
33
0
@
1
A
Conditions: S
11
> 0, S
21
0, S
33
> 0, S
2
11
S
2
21
S
33
> 1
S
11
S
21
S
33
T
a
1
n
1
n
2
n
3
T
b
1
n
1
n
2
4n
3
1
T
c
1
n
1
n
2
4n
3
3
T
d
1
n
1
n
2
2n
3
1
T
e
1
2n
1
12n
2
2n
3
1
T
f
1
2n
1
2n
2
12n
3
1
T
g
1
2n
1
12n
2
12n
3
T
h
1
2n
1
2n
2
2n
3
T
i
1
2n
1
12n
2
n
3
T
j
1
2n
1
2n
2
1 n
3
T
2
S
11
00
0 S
11
0
00S
33
0
@
1
A
Conditions: S
11
> 0, S
33
> 0, S
11
S
33
> 1
S
11
S
33
T
a
2
n
1
n
2
T
b
2
2n
1
1 n
2
T
c
2
n
1
4n
2
1
T
d
2
n
1
4n
2
3
T
e
2
2n
1
14n
2
1
T
f
2
2n
1
14n
2
3
T
g
2
n
1
2n
2
1
T
h
2
2n
1
12n
2
1
T
i
2
2n
1
2n
2
T
3
S
11
S
11
0
S
11
S
11
0
00S
33
0
@
1
A
Conditions : S
11
> 0, S
33
> 0
T
a
3
: S
33
n
1
; T
b
3
: S
33
4n
1
1; T
c
3
: S
33
4n
1
1;
T
d
3
: S
33
2n
1
1; T
e
3
: S
33
2n
1
Table 13.1.2.2. Isomorphic subgroups of the space groups
(cont.)
Table 13.1.2.2. Isomorphic subgroups of the space groups
(cont.)
839
13.1. ISOMORPHIC SUBGROUPS
TRIGONAL AND HEXAGONAL SYSTEMS
Hexagonal axes
H
1
S
11
S
21
0
S
21
S
11
S
21
0
00 S
33
0
@
1
A
Conditions: S
11
> 0, 0 S
21
< S
11
, S
33
> 0, S
2
11
S
2
21
S
11
S
21
S
33
> 1
H
a
1
: S
33
n
1
; H
b
1
: S
33
3n
1
1; H
c
1
: S
33
3n
1
2;
H
d
1
: S
33
6n
1
1; H
e
1
: S
33
6n
1
5; H
f
1
: S
33
2n
3
1
Conditions: S
11
> 0, S
21
0, S
33
> 0, S
2
11
S
2
21
S
11
S
21
S
33
> 1
S
11
S
21
S
33
H
g
1
3n
1
13n
2
3n
3
1
H
h
1
3n
1
23n
2
23n
3
1
H
i
1
3n
1
3n
2
13n
3
1
H
j
1
3n
1
13n
2
13n
3
2
H
k
1
3n
1
23n
2
3n
3
2
H
l
1
3n
1
3n
2
23n
3
2
H
m
1
3n
1
13n
2
23n
3
H
n
1
3n
1
23n
2
13n
3
H
o
1
3n
1
3n
2
3n
3
H
2
S
11
00
0 S
11
0
00S
33
0
@
1
A
Conditions: S
11
> 0, S
33
> 0, S
11
S
33
> 1
S
11
S
33
H
a
2
n
1
n
2
H
b
2
n
1
3n
2
1
H
c
2
n
1
3n
2
2
H
d
2
3n
1
13n
2
1
H
e
2
3n
1
23n
2
2
H
f
2
3n
1
3n
2
H
g
2
n
1
2n
2
1
H
h
2
3n
1
16n
2
1
H
i
2
3n
1
26n
2
5
H
j
2
3n
1
6n
2
3
H
k
2
n
1
6n
2
1
H
l
2
n
1
6n
2
5
H
3
2S
11
S
11
0
S
11
S
11
0
00S
33
0
@
1
A
Conditions: S
11
> 0, S
33
> 0
H
a
3
: S
33
n
1
; H
b
3
: S
33
6n
1
1; H
c
3
: S
33
6n
1
5;
H
d
3
: S
33
3n
1
1; H
e
3
: S
33
3n
1
2; H
f
3
: S
33
2n
1
1
Rhombohedral axes
R
1
S
0
S
2
S
1
S
1
S
0
S
2
S
2
S
1
S
0
0
@
1
A
Conditions: S
0
> 0, S
1
S
0
, S
0
S
1
< S
2
< S
0
,
S
0
S
1
S
2
S
2
0
S
2
1
S
2
2
S
0
S
1
S
1
S
2
S
2
S
0
> 1
R
2
S
0
S
1
S
1
S
1
S
0
S
1
S
1
S
1
S
0
0
@
1
A
Conditions: R
a
2
: S
0
> 0, S
0
=2 < S
1
< S
0
;
R
2
b
: S
0
> 0, S
0
=2 < S
1
< S
0
, S
0
2S
1
2n 1
CUBIC SYSTEM
C
S
11
00
0 S
11
0
00S
11
0
@
1
A
Condition: S
11
> 1
C
a
: S
11
n
1
; C
b
: S
11
2n
1
1;
C
c
: S
11
4n
1
1; C
d
: S
11
4n
1
3
Table of space subgroups
No. 1 P1:S; No. 2 P
1 : S; No. 3 P112 : M
a
c
; P121 : M
b
a
;
No. 4 P112
1
: M
b
c
; P12
1
1 : M
b
b
; No. 5 A112 : M
c
c
, M
d
c
, M
e
c
;
C121 : M
c
b
, M
d
b
, M
e
b
; No. 6 P11m : M
a
c
; P1m1 : M
a
b
; No. 7 P11a : M
f
c
;
P1c1 : M
f
b
; No. 8 A11m : M
c
c
, M
d
c
, M
e
c
; C1m1 : M
c
b
, M
d
b
, M
e
b
;
No. 9 A11a : M
g
c
, M
h
c
, M
i
c
; C1c1 : M
g
b
, M
h
b
, M
i
b
; No. 10 P112=m : M
a
c
;
P12=m1 : M
a
b
; No. 11 P112
1
=m : M
b
c
; P12
1
=m1 : M
b
b
;
No. 12 A112=m : M
c
c
, M
d
c
, M
e
c
; C12=m1 : M
c
b
, M
d
b
, M
e
b
;
No. 13 P112=a : M
f
c
; P12=c1 : M
f
b
;
No. 14 P112
1
=a : M
j
c
; P12
1
=c1 : M
j
b
;
No. 15 A112=a : M
g
c
, M
h
c
, M
i
c
; C12=c1; M
g
b
, M
h
b
, M
i
b
; No. 16 P222 : O
a
1
;
No. 17 P222
1
: O
b
1
; No. 18 P2
1
2
1
2 : O
c
1
; No. 19 P2
1
2
1
2
1
: O
d
1
;
No. 20 C222
1
: O
d
1
, O
e
1
; No. 21 C222 : O
c
1
, O
f
1
, O
4
, O
5
;
No. 22 F222 : O
d
1
, O
g
1
; No. 23 I222 : O
d
1
, O
g
1
; No. 24 I2
1
2
1
2
1
: O
d
1
;
No. 25 Pmm2 : O
a
1
; No. 26 Pmc2
1
; O
b
1
; No. 27 Pcc2 : O
b
1
;
No. 28 Pma2 : O
h
1
; No. 29 Pca2
1
: O
i
1
; No. 30 Pnc2 : O
j
1
;
No. 31 Pmn2
1
: O
i
1
; No. 32 Pba2 : O
c
1
; No. 33 Pna2 : O
d
1
;
No. 34 Pnn2 : O
d
1
; No. 35 Cmm2 : O
c
1
, O
f
1
; No. 36 Cmc2
1
: O
d
1
;
No. 37 Ccc2 : O
d
1
, O
e
1
; No. 38 Amm2 : O
j
1
, O
k
1
, O
6
; No. 39 Abm2 : O
j
1
;
No. 40 Ama2 : O
d
1
, O
l
1
; No. 41 Aba2 : O
d
1
; No. 42 Fmm2 : O
d
1
, O
g
1
;
No. 43 Fdd2 : O
d
1
; No. 44 Imm2 : O
d
1
, O
g
1
; No. 45 Iba2 : O
d
1
;
No. 46 Ima2 : O
d
1
; No. 47 Pmmm : O
a
1
; No. 48 Pnnn : O
d
1
;
No. 49 Pccm : O
b
1
; No. 50 Pban : O
c
1
; No. 51 Pmma : O
h
1
No. 52 Pnna : O
d
1
; No. 53 Pmna : O
i
1
; No. 54 Pcca : O
i
1
;
No. 55 Pbam : O
c
1
; No. 56 Pccn : O
d
1
; No. 57 Pbcm : O
j
1
;
No. 58 Pnnm : O
d
1
; No. 59 Pmmn : O
c
1
; No. 60 Pbcn : O
d
1
;
No. 61 Pbca : O
d
1
; No. 62 Pnma : O
d
1
; No. 63 Cmcm : O
d
1
, O
e
1
;
No. 64 Cmca : O
d
1
; No. 65 Cmmm : O
c
1
, O
f
1
, O
4
, O
5
;
No. 66 Cccm : O
d
1
, O
e
1
; No. 67 Cmma : O
c
1
; No. 68 Ccca : O
d
1
;
No. 69 Fmmm : O
d
1
, O
g
1
; No. 70 Fddd : O
d
1
; No. 71 Immm : O
d
1
, O
g
1
;
No. 72 Ibam : O
d
1
; No. 73 Ibca : O
d
1
; No. 74 Imma : O
d
1
;
No. 75 P4 : T
a
1
; No. 76 P4
1
: T
b
1
(subgroups P4
1
, T
c
1
subgroups P4
3
;
No. 77 P4
2
: T
d
1
; No. 78 P4
3
: T
b
1
subgroups P4
3
, T
c
1
subgroups P4
1
;
No. 79 I4 : T
e
1
, T
f
1
, T
g
1
, T
h
1
; No. 80 I4
1
: T
e
1
, T
f
1
; No. 81 P
4 : T
a
1
;
No. 82 I
4 : T
e
1
, T
f
1
, T
g
1
, T
h
1
; No. 83 P4=m : T
a
1
; No. 84 P4
2
=m : T
d
1
;
No. 85 P4=n : T
i
1
, T
j
1
; No. 86 P4
2
=n : T
e
1
, T
f
1
;
No. 87 I4=m : T
e
1
, T
f
1
, T
g
1
, T
h
1
; No. 88 I4
1
=a : T
e
1
, T
f
1
;
No. 89 P422 : T
a
2
, T
a
3
; No. 90 P42
1
2 : T
b
2
;
No. 91 P4
1
22 : T
c
2
subgroups P4
1
22, T
d
2
subgroups P4
3
22,
T
b
3
subgroups P4
1
22, T
c
3
subgroups P4
3
22;
No. 92 P4
1
2
1
2 : T
e
2
subgroups P4
1
2
1
2, T
f
2
subgroups P4
3
2
1
2;
No. 93 P4
2
22 : T
g
2
, T
d
3
; No. 94 P4
2
2
1
2 : T
h
2
;
No. 95 P4
3
22 : T
c
2
subgroups P4
3
22, T
d
2
subgroups P4
1
22,
T
b
3
subgroups P4
3
22, T
c
3
subgroups P4
1
22;
No. 96 P4
3
2
1
2 : T
e
2
subgroups P4
3
2
1
2, T
f
2
subgroups P4
1
2
1
2;
No. 97 I422 : T
h
2
, T
i
2
, T
e
3
; No. 98 I4
1
22 : T
h
2
; No. 99 P4mm : T
a
2
, T
a
3
;
No. 100 P4bm : T
b
2
; No. 101 P4
2
cm : T
g
2
; No. 102 P4
2
nm : T
h
2
;
Table 13.1.2.2. Isomorphic subgroups of the space groups
(cont.)
Table 13.1.2.2. Isomorphic subgroups of the space groups
(cont.)
840
13. ISOMORPHIC SUBGROUPS OF SPACE GROUPS
S
1
2
1
2
1
2
0
B
@
1
C
A
1
2
S
11
S
21
1
2
S
11
S
21
1
2
S
33
0
B
@
1
C
A
n
1
n
2
n
3
0
B
@
1
C
A
or
n
1
1
2
n
2
1
2
n
3
1
2
0
B
@
1
C
A
13:1:1:4a
with n
1
, n
2
, n
3
integers.
From this it follows that either S
11
and S
21
have the same
parity and S
33
is even, or else S
11
and S
21
have opposite parities
and S
33
is odd.
If (W, w) and W
0
, w
0
represent the same operation 4
1
in the
two P lattices of G and H , with
^
w w
0
00
1
4
,fors 0,
equation (13.1.1.2b) reduces to
1
4
S
33
1n or S
33
4n 1, n integer:
There are similar conditions for glide and other screw operations.
The location part s, i.e. the relative positions of the origins O and
O
0
of group and subgroup, is another important problem. This
problem can be approached in two ways. First, only standard
settings and origins (as defined in the space-group tables of this
volume) of the group G and its subgroup H are considered. In this
case, the origin relation between G and H must be indicated by the
appropriate column s. For instance, in P2
1
2
1
2
1
the matrices
O
4
, O
5
, O
6
are only possible for s 6 o, say s
1
4
,
1
4
,
1
4
.
Second, one can describe H based on the same origin as G, i.e.
O and O
0
coincide. In this case, nonstandard descriptions of H
frequently result and one has to indicate the location of the
symmetry elements of H with respect to the origin of G.
Unfortunately, it was not possible to incorporate in the present
tables the implications that the choice of the origin has on the
coefficients S
ij
(cf. Bertaut, 1956; Billiet, 1978; Bertaut & Billiet,
1979).
The explicit forms of the matrices S for each space group and
each plane group are given in Tables 13.1.2.1 and 13.1.2.2 without
origin indications.
Example
Consider space groups P4=mmm and P4=mcc and envisage the
matrix T
3
which has the determinant 2S
33
S
2
11
. Its lowest value is 2
with S
11
S
33
1. The matrix T
3
then implies the axis
transformation
a
0
a b, b
0
a b, c
0
c:
Envisage also the matrix T
2
which has the determinant S
2
11
S
33
.
Its lowest value for a proper subgroup is 2 with S
11
1 and
S
33
2. The axis transformation is here
No. 103 P4cc : T
g
2
, T
d
3
; No. 104 P4nc : T
h
2
; No. 105 P4
2
mc : T
g
2
;
No. 106 P4
2
bc : T
h
2
; No. 107 I4mm : T
h
2
, T
i
2
, T
e
3
; No. 108 I4cm : T
h
2
;
No. 109 I4
1
md : T
h
2
; No. 110 I4
1
cd : T
h
2
; No. 111 P
42m : T
a
2
;
No. 112 P
42c : T
g
2
; No. 113 P
42
1
m : T
b
2
; No. 114 P
42
1
c : T
h
2
;
No. 115 P
4m2 : T
a
2
; No. 116 P
4c2 : T
g
2
; No. 117 P
4b2 : T
b
2
;
No. 118 P
4n2 : T
h
2
; No. 119 I
4m2 : T
h
2
, T
i
2
; No. 120 I
4c2 : T
h
2
;
No. 121 I
42m : T
h
2
, T
i
2
; No. 122 I
42d : T
h
2
; No. 123 P4=mmm : T
a
2
, T
a
3
;
No. 124 P4=mcc : T
g
2
, T
d
3
; No. 125 P4=nbm : T
b
2
; No. 126 P4=nnc : T
h
2
;
No. 127 P4=mbm : T
b
2
; No. 128 P4=mnc : T
h
2
; No. 129 P4=nmm : T
b
2
;
No. 130 P4=ncc : T
h
2
; No. 131 P4
2
=mmc : T
g
2
; No. 132 P4
2
=mcm : T
g
2
;
No. 133 P4
2
=nbc : T
h
2
; No. 134 P4
2
=nnm : T
h
2
; No. 135 P4
2
=mbc : T
h
2
;
No. 136 P4
2
=mnm : T
h
2
; No. 137 P4
2
=nmc : T
h
2
; No. 138 P4
2
=ncm : T
h
2
;
No. 139 I4=mmm : T
h
2
, T
i
2
, T
e
3
; No. 140 I4=mcm : T
h
2
;
No. 141 I4
1
=amd : T
h
2
; No. 142 I4
1
=acd : T
h
2
; No. 143 P3 : H
a
1
;
No. 144 P3
1
: H
b
1
subgroups P3
1
, H
c
1
subgroups P3
2
;
No. 145 P3
2
: H
b
1
subgroups P3
2
, H
c
1
subgroups P3
1
;
No. 146 R3 (hexagonal axes): H
g
1
, H
h
1
, H
i
1
, H
j
1
, H
k
1
, H
l
1
, H
m
1
, H
n
1
, H
o
1
;
R3 (rhombohedral axes): R
1
; No. 147 P
3 : H
a
1
;
No. 148 R
3 (hexagonal axes): H
g
1
, H
h
1
, H
i
1
, H
j
1
, H
k
1
, H
l
1
, H
m
1
, H
n
1
, H
o
1
;
R
3 (rhombohedral axes): R
1
; No. 149 P312 : H
a
2
; No. 150 P321 : H
a
2
;
No. 151 P3
1
12 : H
b
2
subgroups P3
1
12, H
c
2
subgroups P3
2
12;
No. 152 P3
1
21 : H
b
2
subgroups P3
1
21, H
c
2
subgroups P3
2
21;
No. 153 P3
2
12 : H
b
2
subgroups P3
2
12, H
c
2
subgroups P3
1
12;
No. 154 P3
2
21 : H
b
2
subgroups P3
2
21, H
c
2
subgroups P3
1
21;
No. 155 R32 (hexagonal axes): H
d
2
, H
e
2
, H
f
2
; R32 (rhombohedral axes):
R
a
2
; No. 156 P3m1 : H
a
2
; No. 157 P31m : H
a
2
; No. 158 P3c1 : H
g
2
;
No. 159 P31c : H
g
2
; No. 160 R3m (hexagonal axes): H
d
2
, H
e
2
, H
f
2
;
R3m (rhombohedral axes): R
a
2
; No. 161 R3c (hexagonal axes):
H
h
2
, H
i
2
, H
j
2
; R3c (rhombohedral axes): R
b
2
; No. 162 P
31m : H
a
2
;
No. 163 P
31c : H
g
2
; No. 164 P
3m1 : H
a
2
; No. 165 P
3c1 : H
g
2
;
No. 166 R
3m (hexagonal axes): H
d
2
, H
e
2
, H
f
2
; R
3m (rhombohedral
axes): R
a
2
; No. 167 R
3c (hexagonal axes): H
h
2
, H
i
2
, H
j
2
;
R
3c (rhombohedral axes): R
b
2
; No. 168 P6: H
a
1
;
No. 169 P6
1
: H
d
1
subgroups P6
1
, H
e
1
subgroups P6
5
;
No. 170 P6
5
: H
d
1
subgroups P6
5
, H
e
1
subgroups P6
1
;
No. 171 P6
2
: H
b
1
subgroups P6
2
, H
c
1
subgroups P6
4
;
No. 172 P6
4
: H
b
1
subgroups P6
4
, H
c
1
subgroups P6
2
;
No. 173 P6
3
: H
f
1
; No. 174 P
6 : H
a
1
; No. 175 P6=m : H
a
1
;
No. 176 P6
3
=m : H
f
1
; No. 177 P622 : H
a
2
; H
a
3
;
No. 178 P6
1
22 : H
k
2
subgroups P6
1
22, H
i
2
subgroups P6
5
22,
H
b
3
subgroups P6
1
22, H
c
3
subgroups P6
5
22;
No. 179 P6
5
22 : H
k
2
subgroups P6
5
22, H
l
2
subgroups P6
1
22,
H
b
3
subgroups P6
5
22, H
c
3
subgroups P6
1
22;
No. 180 P6
2
22 : H
b
2
subgroups P6
2
22, H
c
2
subgroups P6
4
22,
H
d
3
subgroups P6
2
22, H
e
3
subgroups P6
4
22;
No. 181 P6
4
22 : H
b
2
subgroups P6
4
22, H
c
2
subgroups P6
2
22,
H
d
3
subgroups P6
4
22, H
e
3
subgroups P6
2
22; No. 182 P6
3
22 : H
g
2
, H
f
3
;
No. 183 P6mm : H
a
2
, H
a
3
; No. 184 P6cc : H
g
2
, H
f
3
; No. 185 P6
3
cm : H
g
2
;
No. 186 P6
3
mc : H
g
2
; No. 187 P
6m2 : H
a
2
; No. 188 P
6c2 : H
g
2
;
No. 189 P
62m : H
a
2
; No. 190 P
62c : H
g
2
; No. 191 P6=mmm : H
a
2
, H
a
3
;
No. 192 P6=mcc : H
g
2
, H
f
3
; No. 193 P6
3
=mcm : H
g
2
;
No. 194 P6
3
=mmc : H
g
2
; No. 195 P23 : C
a
; No. 196 F23 : C
a
;
No. 197 I23 : C
a
; No. 198 P2
1
3 : C
b
; No. 199 I2
1
3 : C
b
;
No. 200 Pm
3 : C
a
; No. 201 Pn
3 : C
b
; No. 202 Fm
3 : C
a
;
No. 203 Fd
3 : C
b
; No. 204 Im
3 : C
a
; No. 205 Pa
3 : C
b
;
No. 206 Ia
3 : C
b
; No. 207 P432 : C
a
; No. 208 P4
2
32 : C
b
;
No. 209 F432 : C
a
; No. 210 F4
1
32 : C
b
; No. 211 I432 : C
a
;
No. 212 P4
3
32 : C
c
subgroups P4
3
32, C
d
subgroups P4
1
32;
No. 213 P4
1
32 : C
c
subgroups P4
1
32, C
d
subgroups P4
3
32;
No. 214 I4
1
32 : C
b
; No. 215 P
43m : C
a
; No. 216 F
43m : C
a
;
No. 217 I
43m : C
a
: No. 218 P
43n : C
b
; No. 219 F
43c : C
b
;
No. 220 I
43d : C
b
; No. 221 Pm
3m : C
a
; No. 222 Pn
3n : C
b
;
No. 223 Pm
3n : C
b
; No. 224 Pn
3m : C
b
; No. 225 Fm
3m : C
a
;
No. 226 Fm
3c : C
b
; No. 227 Fd
3m : C
b
; No. 228 Fd
3c : C
b
;
No. 229 Im
3m : C
a
; No. 230 Ia
3d : C
b
.
Table 13.1.2.2. Isomorphic subgroups of the space groups
(cont.)
Table of space subgroups (cont.)
Table 13.1.2.2. Isomorphic subgroups of the space groups
(cont.)
Table of space subgroups (cont.)
841
13.1. ISOMORPHIC SUBGROUPS
a
0
a, b
0
b, c
0
2c:
This transformation is permitted for P 4 = mmm but not for
P4=mcc where the parity rules require S
33
2n 1. Thus, the
lowest value of S
2
11
S
33
for a subgroup of space group P4=mcc is 3
and the axis transformation is
a
0
a, b
0
b, c
0
3c:
13.1.2.5. Plane groups
There is no difficulty in reducing the preceding considerations to
plane groups, where S is a 2 2 matrix and s a 2 1 matrix.
13.1.2.6. Tables of matrices for isomorphic subgroups
The matrices and the restrictions on the coefficients are listed for
the plane groups in Table 13.1.2.1 and for the space groups in Table
13.1.2.2.
For the triclinic and monoclinic systems, there is an infinite
choice of matrices for each index, owing to the infinite number of
equivalent unit cells. For the other space groups, several different
(but finitely many) choices of matrix occur. In all cases, we have
restricted this choice to one matrix for each group–subgroup
relation so that each subgroup is listed exactly once (apart from
origin choice).
Example
No. 178, P6
1
22, has the matrix H
k
2
for all isomorphic and
isosymbolic subgroups P6
1
22, having the lattices n
11
a, n
11
b,
6n
33
1c n
11
1, n
33
0, whilst H
b
3
is used for all
isomorphic and isosymbolic subgroups with the lattices
n
11
2a b, n
11
a b, 6n
33
1c n
11
1, n
33
0.
These two kinds of subgroups are obviously different, having
different translation lattices. The same group, P6
1
22, has the
matrix H
l
2
for the isomorphic and enantiomorphic subgroups
P6
5
22 of lattices n
11
a, n
11
b, 6n
33
5c whilst H
c
3
is used for the
isomorphic and enantiomorphic subgroups P6
5
22 of lattices
n
11
2a b, n
11
a b, 6n
33
5c.
In the tables, each system is preceded by the appropriate general
form of the matrix, which is also given in this chapter, followed by
the more specialized matrices such as T
2
, T
3
. Under Conditions,we
have listed the nonredundant inequalities and parity conditions*
that ensure the uniqueness of the matrix for each subgroup. Also, we
have used rules in order to avoid repetition of equivalent unit cells.
For instance, for trigonal and hexagonal groups (rhombohedral
groups excepted), we have restricted a
0
to lie between a and a b
excluding this last vector from the sector of 60
because there is a
repetition after a 60
rotation of the unit cell.
*
More exactly ‘congruence modulo Z conditions’.
842
13. ISOMORPHIC SUBGROUPS OF SPACE GROUPS
13.2. Derivative lattices
BY Y. BILLIET AND E. F. BERTAUT
13.2.1. Introduction
The three-dimensional subgroups of space group P1 and the two-
dimensional subgroups of plane group p1 are all isomorphic
subgroups; i.e. these subgroups are pure translation groups and
correspond to lattices. In the past, these lattices have often been
called ‘superlattices’ (the term ‘sublattice’ perhaps would be more
precise). To avoid confusion, the lattices that correspond to the
isomorphic subgroups of P1 and p1 are designated here as
derivative lattices.
The number of derivative lattices (both maximal and nonmax-
imal) of a lattice is infinite and always several derivative lattices of
index i2 exist. Only for prime indices are maximal derivative
lattices obtained; for any prime p, there are p
2
p 1 three-
dimensional derivative lattices of P1, whereas there are p 1
two-dimensional derivative lattices of p1. The number of
nonmaximal derivative lattices is given by more complicated
formulae (cf. Billiet & Rolley Le Coz, 1980).
13.2.2. Construction of three-dimensional derivative
lattices
It is possible to construct in a simple way all three-dimensional
derivative lattices of a lattice (Table 13.2.2.1). Starting from a
primitive unit cell defined by a, b, c, each derivative lattice
possesses exactly one primitive unit cell defined by a
0
, b
0
, c
0
by
means of the following relation
a
0
p
1
a, b
0
p
2
b q
1
a, c
0
p
3
c r
1
a q
2
b
p
1
, p
2
, p
3
positive integers, not necessarily prime;
index p
1
p
2
p
3
> 1; q
1
, q
2
, r
1
integers;
p
1
=2 < q
1
p
1
=2; p
2
=2 < q
2
p
2
=2;
p
1
=2 < r
1
p
1
=2:
Note that the vector a
0
has the same direction as the vector a and
the plane a
0
, b
0
is parallel to the plane (a, b), i.e. the matrix of the
transformation is triangular. Equivalent formulae can be derived by
permutations of the vectors a, b, c which keep the directions of b
0
or
c
0
and which preserve the parallelism of the planes b
0
, c
0
with
(b, c)ora
0
, c
0
with (a, c).
Table 13.2.2.1. Three-dimensional derivative lattices of indices
2to7
The entry for each derivative lattice starts with a running number which is
followed, between parentheses, by the appropriate basis-vector relations. It
should be noted that the seven derivative lattices of index 2 are also recorded in
the space-group table of P1 (No. 1) under Maximal isomorphic subgroups of
lowest index but in the slightly different sequence 1, 2, 3, 6, 5, 4, 7.
Index 2 12a, b, c;2a,2b, c;3 a, b,2c;42a, b a, c;
52a, b, c a;6a,2b, c b;72a, b a, c a
Index 3 13a, b, c;2a,3b, c;3a, b,3c;43a,
b a, c;
53a, b, c a;6a,3b, c b;73a, b a, c;
83a, b, c a;9a,3b, c b;103a, b a, c a;
113a, b a, c a;123a, b a, c a
;133a, b a, c a
Index 4 14a, b, c;2a,4b, c;3 a, b,4c;44a, b a, c;
54a, b, c a;6a,4b, c b;74a, b a, c;
84a, b, c a;9a,4b, c
b;104a, b 2a, c;
114a, b, c 2a;122a,2b a, c;13a,4b, c 2b;
142a, b,2c a;15a,2b,2c b;164a, b a, c a;
174a, b a, c a;18
4a, b a, c a;
194a, b a, c a;204a, b a, c 2a;
214a, b 2a, c a ;222a,2b a, c b;
234a, b a, c 2a;244a, b 2a, c a;
252a
,2b a, c a b;264a, b 2a, c 2a;
272a,2b a, c a ;282a, b a,2c a;292a,2b, c;
302a, b,2c;31a,2b,2c;322a,2b, c a;
332a,2b, c
b;342a, b a,2c;352a,2b, c a b
Index 5 15a, b, c;2a,5b, c;3a, b,5c;45a, b a, c;
55a, b, c a;6a,5b, c b;75a, b a, c;
85a,
b, c a;9a,5b, c b;105a, b 2a, c;
115a, b, c 2a;12a,5b, c 2b;135a, b 2a, c;
145a, b, c 2a;15a,5b, c 2b;165a, b a , c a
;
175a, b a, c a;185a, b a, c a;
195a, b a, c a;205a, b a, c 2a;
215a, b a, c 2a;225a, b 2a, c a;
235a, b 2a, c a;24
5a, b 2a, c 2a;
255a, b 2a, c 2a;265a, b 2a, c 2a;
275a, b 2a, c a ;285a, b a, c 2a;
295a, b 2a, c 2a;305a, b 2a, c
a;
315a, b a, c 2a
Index 6 16a, b, c;2a,6b, c;3a, b,6c;46a, b a, c;
56a, b, c a;6a,6b, c b;76a, b a, c;
86a, b, c a;9a,6b, c b;106a, b 2a,
c;
116a, b, c 2a;12a,6b, c 2b;133a,2b a, c;
14a,3b,2c b;153a, b,2c a;166a, b 2a, c;
176a, b, c 2a;18a,6b, c 2b;193a,2b
a, c;
20a,3b,2c b;213a, b,2c a;226a, b 3a, c;
236a, b, c 3a;24a,6b, c 3b;252a,3b a, c;
26a,2b,3c b;272a, b,3c a;286a,
b a, c a;
296a, b a, c a;306a, b a, c a;
316a, b a, c a;326a, b a, c 2a;
336a, b a, c 2a;343a,2b a, c b;
353a,2b a,
c b;366a, b 2a, c a;
376a, b 2a, c a;386a, b a, c 2a;
396a, b 2a, c a;403a,2b a, c a b;
416a, b a, c 2a;426a, b 2a
, c a;
433a,2b a, c a b;446a, b a, c 3a;
456a, b 3a, c a ;462a,3b a, c b;
476a, b a, c 3a;486a, b 3a, c a;
492a,3b a
, c b;506a, b 2a, c 2a;
513a,2b a, c a;52 3a, b a,2c a;
536a, b 2a, c 2a;543a,2b a, c a;
553a, b a,2c a;566a, b 2a, c
2a;
573a,2b a, c a;583a, b a,2c a;
596a, b 2a, c 2a;603a,2b a, c a;
613a, b a,2c a ;626a, b 2a, c 3a;
632a,3b a, c
a b;643a,2b, c a b;
656a, b 3a, c 2a;663a,2b a, c a b;
672a,3b, c a b;686a, b 2a, c 3a;
692a,3b a, c b a ;703a,2b, c
a b;
716a, b 3a, c 2a;723a,2b a, c a b;
732a,3b, c a b;746a, b 3a, c 3a;
752a,3b a, c a ;762a, b a,3c a;773a,2b, c
;
783a, b,2c;792a,3b, c;802a, b,3c;81a,3b,2c;
82a,2b,3c;833a,2b, c a;843a, b a,2c;
852a,3b, c b;863a,2b, c a;873a, b a,2c
;
882a,3b, c b;893a,2b, c b;902a,3b, c a;
912a, a b,3c
Table 13.2.2.1. Three-dimensional derivative lattices of indices
2to7(cont.)
843
International Tables for Crystallography (2006). Vol. A, Chapter 13.2, pp. 843–844.
Copyright © 2006 International Union of Crystallography
Example
One can derive easily the 7 derivative lattices of index 2.
12a, b, c; 2a,2b, c; 3a, b,2c; 42a, b a, c;
52a, b, c a; 6a,2b, c b; 72a, b a, c a:
Another primitive cell of a given derivative lattice is obtained if
one of the following three elementary transformations is performed
on the vectors of a primitive cell of this derivative lattice:
(i) the sign of a vector is changed;
(ii) two vectors are interchanged;
(iii) to a vector is added q times another vector (q integer).
(i) and (ii) are left-handed transformations, (iii) is right-handed.
Example
The primitive cell a
00
, b
00
, c
00
(a
00
2b a, b
00
9a 2c 2b,
c
00
c a 3b) belongs to the derivative lattice of index 10
given by the primitive cell a
0
, b
0
, c
0
a
0
5a, b
0
2b a, c
0
c 2a b because these two cells are related by the following
sequence of elementary transformations:
2b a,9a 2c 2b, c a 3b!
iii
2b a,9a 2c 2b, c 2a b!
iii
2b a,5a, c 2a b!
i
2b a,5a, c 2a b
!
ii
5a,2b a, c 2a b;
(a
00
, b
00
, c
00
) and (a
0
, b
0
, c
0
) have the same handedness.
13.2.3. Two-dimensional derivative lattices
All previous considerations are valid also for two-dimensional
lattices and their derivative lattices (Table 13.2.3.1). The relevant
formula for any index is
a
0
p
1
a, b
0
p
2
b qa
a direction is kept
p
1
, p
2
positive integers, not necessarily prime;
index p
1
p
2
> 1; q any integer; p
1
=2 < q p
1
=2:
A similar formula is obtained by interchange of a and b.
References
13.1
Bertaut, E. F. (1956). Structure de FeS stoechiome
´
trique. Bull. Soc.
Fr. Mine
´
ral. Cristallogr. 79, 276–292.
Bertaut, E. F. & Billiet, Y. (1979). On equivalent subgroups and
supergroups of the space groups. Acta Cryst. A35, 733–745.
Billiet, Y. (1973). Les sous-groupes isosymboliques des groupes
spatiaux. Bull. Soc. Fr. Mine
´
ral. Cristallogr. 96, 327–334.
Billiet, Y. (1978). Some remarks on the ‘family tree’ of
Ba
¨
rnighausen. Acta Cryst. A34, 1023–1025.
Billiet, Y. (1979). Le groupe P1 et ses sous-groupes. I. Outillage
mathe
´
matique: automorphisme et factorisation matricielle. Acta
Cryst. A35, 485–496.
Billiet, Y. & Rolley Le Coz, M. (1980). Le groupe P1 et ses sous-
groupes. II. Tables de sous-groupes. Acta Cryst. A36, 242–248.
13.2
Billiet, Y. & Rolley Le Coz, M. (1980). Le groupe P1 et ses sous-
groupes. II. Tables de sous-groupes. Acta Cryst. A36, 242–248.
Index 7 17a, b, c;2a,7b, c;3 a, b,7c;47a, b a, c;
57a, b, c b;6a,7b, c b;77a, b a, c;
87a, b, c a;9a,7b, c b;107a, b 2a ,
c;
117a, b, c 3a;12a,7b, c 2b;137a, b 3a, c;
147a, b, c 2a;15a,7b, c 3b;167a, b 2a, c;
177a, b, c 3a;18a,7b, c 2b;197
a, b 3a, c;
207a, b, c 2a;21a,7b, c 3b;227a, b a, c a;
237a, b a, c a;247a, b a, c a;
257a, b a, c a;267a, b a, c 2a
;
277a, b a, c 2a;287a, b 2a, c a;
297a, b 2a, c a;307a, b 3a, c 3a;
317a, b 3a, c 3a;327a, b a, c 3a;
337a, b a, c
3a;347a, b 3a, c a;
357a, b 3a, c a;367a, b 2a, c 2a;
377a, b 2a, c 2a;387a, b 2a, c 2a;
397a, b 3a, c a;407a, b a
, c 3a;
417a, b 2a, c 2a;427a, b 3a, c a;
437a, b a, c 3a;447a, b 2a, c 3a;
457a, b 3a, c 2a;467a, b 2a, c 3a;
477
a, b 3a, c 2a;487a, b 2a, c 3a;
497a, b 3a, c 2a;507a, b 2a, c 3a;
517a, b 3a, c 2a;527a, b 3a, c 3a;
537a, b 2a, c
a;547a, b a, c 2a;
557a, b 3a, c 3a;567a, b 2a, c a;
577a, b a, c 2a
Table 13.2.2.1. Three-dimensional derivative lattices of indices
2to7(cont.)
Table 13.2.3.1. Two-dimensional derivative lattices of indices
2to7
The entry for each derivative lattice starts with a running number which is
followed, between parentheses, by the appropriate basis-vector relations.
Index 2 12a, b; 2a,2b; 32a, b a
Index 3 13a, b; 2a,3b; 33a, b a; 43a, b a
Index 4 14a, b; 2a,4b; 34a, b a; 44a, b a; 54a, b 2
a;
62a,2b a; 72a,2b
Index 5 15a, b; 2a,5b; 35a, b a; 45a, b a; 55a, b 2a;
65a, b 2a
Index 6 16a, b; 2a,6b; 36a, b a;
46a, b a; 56a, b 2a;
63a,2b a; 76a, b 2a; 83a,2b a; 96a, b 3a;
102a,3b a; 113a,2b; 122a,3b
Index 7 17a, b; 2a,7b
; 37a, b a ; 47a, b a; 57a, b 2a;
67a, b 3a; 77a, b 2a; 87a, b 3a
844
13. ISOMORPHIC SUBGROUPS OF SPACE GROUPS
references
14.1. Introduction and definition
BY W. FISCHER AND E. KOCH
14.1.1. Introduction
In crystal structures belonging to different structure types and
showing different space-group symmetries, the relative locations of
symmetrically equivalent atoms nevertheless may be the same (e.g.
Cl in CsCl and F in CaF
2
). The concept of lattice complexes can be
used to reveal relationships between crystal structures even if they
belong to different space-group types.
14.1.2. Definition
The term lattice complex (Gitterkomplex) had originally been
created by P. Niggli (1919), but it was not used by him with an
unambiguous meaning. Later on, Hermann (1935) modified and
specified the concept of lattice complexes, but the rigorous
definition used here was proposed much later by Fischer & Koch
(1974) [cf. also Koch & Fischer (1978)]. An alternative definition
was given by Zimmermann & Burzlaff (1974) at the same time.
To introduce the concept of lattice complexes, relationships
between point configurations are regarded.
The set of all points that are symmetrically equivalent to a given
one with respect to a certain space group is called a point
configuration (cf. also crystallographic orbit; Section 8.3.2).
In each space group, there exist infinitely many point
configurations. Given a coordinate system, they may be obtained
by varying the coordinates x, y, z of a starting point and by
calculating all symmetrically equivalent points.
Point configurations refer to the arrangements of atoms in crystal
structures. They are analogous to the crystal forms in crystal
morphology, where a crystal form is a set of symmetrically
equivalent faces. Crystal forms are grouped into types designated
by names like ‘cube’, ‘tetragonal dipyramid’ etc. In a similar,
though not strictly analogous way, point configurations are also
grouped into types, called lattice complexes. For example, all point
configurations forming a cubic primitive point lattice belong to the
same lattice complex. Each lattice complex is thus a set of infinitely
many point configurations. The following stepwise procedure
describes all point configurations belonging to the same lattice
complex:
(i) Take all point configurations of a particular Wyckoff position
in a particular space group. Mathematically these point configura-
tions are distinguished by the following property: the site-symmetry
groups of two arbitrary points from any two of the point
configurations are conjugate in the space group (cf. Section 8.3.2).
(ii) Collect all point configurations of Wyckoff positions that
belong to the same Wyckoff set of a given space group (cf. Section
8.3.2). Such Wyckoff positions play an analogous role with respect
to this space group. Their point configurations have the following
property: the site-symmetry groups of two arbitrary points from any
two point configurations are conjugate in the affine normalizer (cf.
Section 15.3.2) of the space group considered.
(iii) Assemble all point configurations of corresponding Wyckoff
sets from all space groups of one of the 219 (affine) space-group
types, i.e. take all point configurations belonging to a particular type
of Wyckoff set (Section 8.3.2). Each affine mapping that maps two
space groups of the same type onto each other simultaneously maps
onto each other the site-symmetry groups of the points from the
point configurations of the corresponding Wyckoff sets.
According to (i), (ii) and (iii), a lattice complex* is defined as
follows:
A lattice complex is the set of all point configurations that may be
generated within one type of Wyckoff set.
Example
Take, in a particular space group of type P4 = mmm, the Wyckoff
position 4 lx00. The points of each corresponding point
configuration form squares that replace the points of the
tetragonal primitive lattice referring to Wyckoff position 1 a .
For all conceivable point configurations of 4l, the squares have
the same orientation, but their edges have different lengths.
Congruent arrangements of squares but shifted by
1
2
c or by
1
2
a b or by
1
2
a b c give the point configurations of the
Wyckoff positions 4m,4n and 4o, respectively, in the same space
group. The four Wyckoff positions 4l to 4o, all with site
symmetry m2m., make up a Wyckoff set (cf. Table 14.2.3.2).
They are mapped onto each other, for example, by the
translati ons
1
2
c,
1
2
a b and
1
2
a b c, which belong to the
Euclidean (and affine) normalizer of the group. If one space
group of type P4=mmm is mapped onto another space group of
the same type, the Wyckoff set 4l to 4o as a whole is transformed
to 4l to 4o. The individual Wyckoff positions may be inter-
changed, however. The set of all point configurations from the
Wyckoff positions 4l to 4o of all space groups of type P4=mmm
constitutes a lattice complex. Its point configurations may be
derived as described above, but now starting from all space
groups P4=mmm with all conceivable lengths and orientations of
the basis vectors instead of starting from just a particular group.
Accordingly, the point configurations may differ in their
orientation, in the size of their squares and in the distances
between the centres of their squares.
Just as all crystal forms of a particular type may be found in
different point-group types, the same lattice complex may occur in
different space-group types.
Example
The lattice complex ‘cubic primitive lattice’ may be generated,
among others, in Pm
3m 1a, b,inFm
3m 8c and in Ia
38a, b with
site symmetry m
3m,
43m and :
3:, respectively. The type of
Wyckoff set specified by Pm
3m 1a, b leads to the same set of
point configurations as Fm
3m 8c or Ia
38a, b. Each point
configuration of this lattice complex can be generated by a
properly chosen space group in each of these space-group types.
All Wyckoff positions, Wyckoff sets and types of Wyckoff set
that generate, as described above, the same set of point
configurations are assigned to the same lattice complex. Accord-
ingly, the following criterion holds: two Wyckoff positions are
*
This definition agrees with that given by Fischer & Koch (1974) and Koch &
Fischer (1978), but now the term Wyckoff position is used instead of Punktlage or
point position, Wyckoff set instead of Konfigurationslage or configuration set, type
of Wyckoff set instead of Klasse von Konfigurationslagen or class of configuration
sets. New aspects have been taken into account by Koch & Fischer (1985).
846
International Tables for Crystallography (2006). Vol. A, Chapter 14.1, pp. 846–847.
Copyright © 2006 International Union of Crystallography
assigned to the same lattice complex if there is a suitable
transformation that maps the point configurations of the two
Wyckoff positions onto each other and if their space groups belong
to the same crystal family (cf. Sections 8.2.7 and 8.2.8). Suitable
transformations are translations, proper or improper rotations,
isotropic or anisotropic expansions or more general affine mappings
(without violation of the metric conditions for the corresponding
crystal family) and all their products.
By this criterion, the Wyckoff positions of all space groups (1731
entries in the space-group tables, 1128 types of Wyckoff set) are
uniquely assigned to 402 lattice complexes.
The same concept has been used for the point configurations and
Wyckoff positions in the plane groups. Here the Wyckoff positions
(72 entries to the plane-group tables, 51 types of Wyckoff set) are
assigned to 30 plane lattice complexes or net complexes (cf.
Burzlaff et al., 1968).
847
14.1. INTRODUCTION AND DEFINITION
14.2. Symbols and properties of lattice complexes
BY W. FISCHER AND E. KOCH
14.2.1. Reference symbols and characteristic Wyckoff
positions
If a lattice complex can be generated in different space-group types,
one of them stands out because its corresponding Wyckoff positions
show the highest site symmetry. This is called the characteristic
space-group type of the lattice complex. The space groups of all the
other types in which the lattice complex may be generated are
subgroups of the space groups of the characteristic type.
Different lattice complexes may have the same characteristic
space-group type but in that case they differ in the oriented site
symmetry of their Wyckoff positions within the space groups of that
type.
The characteristic space-group type and the corresponding
oriented site symmetry express the common symmetry properties
of all point configurations of a lattice complex. Therefore, they can
be used to identify each lattice complex. Within the reference
symbols of lattice complexes, however, instead of the site symmetry
the Wyckoff letter of one of the Wyckoff positions with that site
symmetry is given, as was first done by Hermann (1935). This
Wyckoff position is called the characteristic Wyckoff position of the
lattice complex.
Examples
(1) Pm
3m is the characteristic space-group type for the lattice
complex of all cubic primitive point lattices. The Wyckoff
positions with the highest possible site symmetry m
3m are
la 000 and 1b
1
2
1
2
1
2
, from which 1a has been chosen as the
characteristic position. Thus, the lattice complex is designated
Pm
3ma.
(2) Pm
3m is also characteristic for another lattice complex that
corresponds to Wyckoff position 8g xxx .3m. Thus, the reference
symbol for this lattice complex is Pm
3mg. Each of its point
configurations may be derived by replacing each point of a
cubic primitive lattice by eight points arranged at the corners
of a cube.
In Tables 14.2.3.1 and 14.2.3.2, the reference symbols denote the
lattice complex of each Wyckoff position. The reference symbols of
characteristic Wyckoff positions are marked by asterisks (e.g. 2e in
P2=c). If in a particular space group several Wyckoff positions
belong to the same Wyckoff set (cf. Koch & Fischer, 1975), the
reference symbol is given only once (e.g. Wyckoff positions 4l to 4o
in P 4 = mmm). To enable this, the usual sequence of Wyckoff
positions had to be changed in a few cases (e.g. in P4
2
=mcm). For
Wyckoff positions assigned to the same lattice complex but
belonging to different Wyckoff sets, the reference symbol is
repeated. In I4=m, for example, Wyckoff positions 4c and 4d are
both assigned to the lattice complex P4=mmm a. They do not
belong, however, to the same Wyckoff set because the site-
symmetry groups 2=m.. of 4c and
4.. of 4d are different.
14.2.2. Additional properties of lattice complexes
14.2.2.1. The degrees of freedom
The number of coordinate parameters that can be varied
independently within a Wyckoff position is called its number of
degrees of freedom. For most lattice complexes, the number of
degrees of freedom is the same as for any of its Wyckoff positions.
The lattice complex with characteristic Wyckoff posi tion
Pm
312jm:: 0yz, for instance, has two degrees of freedom. If,
however, the variation of a coordinate corresponds to a shift of the
point configuration as a whole, one degree of freedom is lost.
Therefore, I4
1
8b xyz is the characteristic Wyckoff position of a
lattice complex with only two degrees of freedom, although position
8b itself has three degrees of freedom. Another example is given by
P4=m 4jm:: xy0 and P44d 1 xyz. Both Wyckoff positions belong to
lattice complex P4=mjwith two degrees of freedom.
According to its number of degrees of freedom, a lattice complex
is called invariant, univariant, bivariant or trivariant. In total, there
exist 402 lattice complexes, 36 of which are invariant, 106
univariant, 105 bivariant and 155 trivariant. The 30 plane lattice
complexes are made up of 7 invariant, 10 univariant and 13
bivariant ones.
Most of the invariant and univariant lattice complexes
correspond to several types of Wyckoff set. In contrast to that,
only one type of Wyckoff set belongs to each trivariant lattice
complex. A bivariant lattice complex may either correspond to one
type of Wyckoff set (e.g. Pm
3 j) or to two types (P4 d, for example,
belongs to the lattice complex with the characteristic position
P4=mj).
14.2.2.2. Limiting complexes and comprehensive complexes
For point groups, the occurrence of limiting crystal forms is well
known. In 4=m, for instance, any tetragonal prism fhk0g is a special
crystal form with face symmetry m.. . In point group 4, on the other
hand, the tetragonal prisms fhk0g belong, as special cases, to the set
of general crystal forms fhklg, the tetragonal pyramids, and there is
no difference between fhklg and fhk0g in either the number or the
symmetry of their faces. Therefore, the tetragonal prism is called a
‘limiting form’ of the tetragonal pyramid. In a case like this, all
possible sets of equivalent faces belonging to a special type of
crystal form (the tetragonal prism) may also be generated as a subset
of another more comprehensive type of crystal form (the tetragonal
pyramid). Of course, it is not possible, by considering a tetragonal
prism by itself, to decide whether it has been generated by point
group 4=m or by point group 4. This distinction can be made,
however, if the tetragonal prism shows the right striations or occurs
in combination with other appropriate crystal forms. Low quartz
(oriented point group 321) gives a well known example: the hex-
agonal prism f10
10g has the same site symmetry 1 as any trigonal
trapezohedron fhkilg. Therefore, f10
10g may be recognized as a
limiting form only if the crystal shows in addition at least one
trigonal trapezohedron.
A similar relation may exist between two lattice complexes. Let
L be a lattice complex generated by a Wyckoff position of a space
group G (e.g. by P4=mmm 4lm2m: x00). An appropriate Wyckoff
position of a subgroup H of G (e.g. P4=m 4jm:: xy0) may produce
not only all point configurations of L
but other point configurations
in addition (with different orientations of the squares in the
example). The complete set then forms a second lattice complex
M. Such relationships led to the following definition (Fischer &
Koch, 1974, 1978):
If a lattice complex L forms a true subset of another lattice
complex M, the lattice complex L is called a limiting complex of
M and the lattice complex M a comprehensive complex of L.
The point configurations of the limiting complex L are generated
within M by restrictions imposed on the coordinate or/and the
metrical parameters.
In the above example, such a restriction holds for the y
coordinate: the condition y 0 for Wyckoff position 4j of P4=m
filters out exactly those point configurations that constitute the
848
International Tables for Crystallography (2006). Vol. A, Chapter 14.2, pp. 848–872.
Copyright © 2006 International Union of Crystallography
lattice complex P4= mmm l. The latter complex is, therefore, a
limiting complex of the lattice complex P4=mj. In the present case
of restricted coordinates, both complexes belong to the same crystal
family and L has fewer degrees of freedom than M.
Another kind of limiting-complex relation is connected with
restrictions for metrical parameters. All point configurations of the
lattice complex Pm
3maare also generated by P4=mmm a under the
restriction a c, i.e. in special space groups of type P4=mmm. Here
L and M have the same number of degrees of freedom, but belong to
different crystal families.
Finally, the two types of parameter restrictions for limiting
complexes may also occur in combination. The trivariant lattice
complex with characteristic Wyckoff position P4
1
2
1
28b xyz,for
example, contains the invariant cubic lattice complex Fm
3maas
a limiting complex. The parameter restrictions necessary are
x
1
2
, y 0, z
1
16
, c=a 4
p
2.
As for a limiting form in crystal morphology, it is often
impossible to decide by which symmetry (space group and Wyckoff
set) a particular point configuration, regarded by itself, has been
generated. If a point configuration belongs to a lattice complex that
is part of a comprehensive complex, this point configuration is a
member of both complexes. As a consequence, the lattice
complexes do not form equivalence classes of point configurations.
Only if a point configuration is inspected in combination with a
sufficient number of other point configurations – like sets of
symmetrically equivalent atoms in a crystal structure – does it make
sense to assign this point configuration to a particular lattice
complex. An example is found in the crystal structures of the spinel
type. Here, the oxygen atoms occupy Wyckoff position 32e xxx in
Fd
3m with x
3
8
(referred to origin choice 1). If x is restricted to
3
8
, the point configurations generated are those of the lattice complex
Fm
3ma(formed by all face-centred cubic point lattices). If for a
spinel-type structure this restriction holds exactly, the point
configurations of the cations would, nevertheless, reveal the true
generating symmetry of the oxygen point configuration. It has,
therefore, to be considered a member of the comprehensive
complex Fd
3me rather than a member of the lattice complex
Fm
3ma(which includes among others the point configuration of
the copper atoms in the crystal structure of copper). For practical
applications, a point configuration contained in several lattice
complexes may be investigated within the complex that is the least
comprehensive but still allows the physical behaviour under
discussion. This corresponds to the definition of the symmetry of
a crystal generally used in crystallography: the highest symmetry
that can be assigned to a crystal as a whole is that of its least
symmetrical property known to date.
Even though limiting-complex relations are very useful for
establishing crystallochemical relationships between different
crystal structures, a complete study has not yet been carried out.
Apart from isolated examples in the literature, systematic
treatments have been given only for special aspects: plane lattice
complexes (Burzlaff et al., 1968); cubic lattice complexes (Koch,
1974); point complexes, rod complexes and layer complexes
(Fischer & Koch, 1978); extraordinary orbits for plane groups
(Lawrenson & Wondratschek, 1976); noncharacteristic orbits of
space groups except those that are due to metrical specialization
(Engel et al., 1984). The closely related concepts of limiting
complexes and noncharacteristic orbits have been compared by
Koch & Fischer (1985).
14.2.2.3. Weissenberg complexes
Depending on their site-symmetry groups, two kinds of Wyckoff
position may be distinguished:
(i) The site-symmetry group of any point is a proper subgroup of
another site-symmetry group from the same space group. Then, the
Wyckoff position contains, among others, point configurations with
the property that the distance between two suitable chosen points is
shorter than any small number ">0.
Example
Each point configuration of the lattice complex with the
characteristic Wyckoff position P4=mmm 4jm:2mxx0 may be
imagined as squares of four points surrounding the points of a
tetragonal primitive lattice. For x ! 0, the squares become
infinitesimally small. Point configurations with x 0 show site
symmetry 4=mmm, their multiplicity is decreased from 4 to 1,
and they belong to lattice complex P4=mmm a.
(ii) The site-symmetry group of any point belonging to the
regarded Wyckoff position is not a subgroup of any other site-
symmetry group from the same space group.
Example
In Pmma, there does not exist a site-symmetry group that is a
proper supergroup of mm2, the site-symmetry group of Wyckoff
position Pmma 2e
1
4
0z. As a consequence, the distance between
any two symmetrically equivalent points belonging to Pmma e
cannot become shorter than the minimum of
1
2
a, b and c.
A lattice complex contains either Wyckoff positions exclusively
of the first or exclusively of the second kind. Most lattice complexes
are made up from Wyckoff positions of the first kind.
There exist, however, 67 lattice complexes that do not contain
point configurations with infinitesimal short distances between
symmetry-related points [cf. Hauptgitter (Weissenberg, 1925)].
These lattice complexes have been called Weissenberg complexes
by Fischer et al. (1973). The 36 invariant lattice complexes are
trivial examples of Weissenberg complexes. In addition, there exist
24 univariant (monoclinic 2, orthorhombic 5, tetragonal 7,
hexagonal 5, cubic 5) and 6 bivariant Weissenberg complexes
(monoclinic 1, orthorhombic 2, tetragonal 1, hexagonal 2). The only
trivariant Weissenberg complex is P 2
1
2
1
2
1
a. All Weissenberg
complexes with degrees of freedom have the following common
property: each Weissenberg complex contains at least two invariant
limiting complexes belonging to the same crystal family.
Example
Pmma e is a comprehensive complex of Pmmm a and of Cmmm
a. Within the characteristic Wyckoff position,
1
4
00 refers to
Pmmm a and
1
4
0
1
4
to Cmmm a.
Except for the seven invariant plane lattice complexes, there
exists only one further Weissenberg complex within the plane
groups, namely the univariant rectangular complex p2 mg c.
14.2.3. Descriptive symbols
14.2.3.1. Introduction
For the study of relations between crystal structures, lattice-
complex symbols are desirable that show as many relations between
point configurations as possible. To this end, Hermann (1960)
derived descriptive lattice-complex symbols that were further
developed by Donnay et al. (1966) and completed by Fischer et
al. (1973). These symbols describe the arrangements of the points in
the point configurations and refer directly to the coordinate
descriptions of the Wyckoff positions. Since a lattice complex, in
general, contains Wyckoff positions with different coordinate
descriptions, it may be represented by several different descriptive
(continued on page 870)
849
14.2. SYMBOLS AND PROPERTIES OF LATTICE COMPLEXES