Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
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Ccc2 C
13
2v
mm2 Orthorhombic
No. 37 Ccc2
Patterson symmetry Cmmm
Origin on cc2
Asymmetric unit 0 ≤ x ≤
1
4
;0≤ y ≤
1
2
;0≤ z ≤ 1
Symmetry operations
For (0,0,0)+ set
(1) 1 (2) 2 0, 0,z (3) cx,0, z (4) c 0,y,z
For (
1
2
,
1
2
,0)+ set
(1) t(
1
2
,
1
2
,0) (2) 2
1
4
,
1
4
,z (3) n(
1
2
,0,
1
2
) x,
1
4
,z (4) n(0,
1
2
,
1
2
)
1
4
,y,z
242
International Tables for Crystallography (2006). Vol. A, Space group 37, pp. 242–243.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 37 Ccc2
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(
1
2
,
1
2
,0); (2); (3)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (
1
2
,
1
2
,0)+
Reflection conditions
General:
8 d 1(1)x,y,z (2) ¯x, ¯y,z (3) x, ¯y,z +
1
2
(4) ¯x,y, z +
1
2
hkl : h + k = 2n
0kl : k,l = 2n
h0l : h, l = 2n
hk0: h + k = 2n
h00 : h = 2n
0k0: k = 2n
00l : l = 2n
Special: as above, plus
4 c ..2
1
4
,
1
4
,z
1
4
,
3
4
,z +
1
2
hkl : k + l = 2n
4 b ..20,
1
2
,z 0,
1
2
,z +
1
2
hkl : l = 2n
4 a ..20,0,z 0,0,z +
1
2
hkl : l = 2n
Symmetry of special projections
Along [001] c2mm
a
= ab
= b
Origin at 0,0,z
Along [100] p1m1
a
=
1
2
bb
=
1
2
c
Origin at x,0,0
Along [010] p11m
a
=
1
2
cb
=
1
2
a
Origin at 0,y, 0
Maximal non-isomorphic subgroups
I
[2] C 1c1(Cc,9) (1; 3)+
[2] Cc11(Cc,9) (1; 4)+
[2] C 112(P2, 3) (1; 2)+
IIa [2] Pnn2 (34) 1; 2; (3; 4)+(
1
2
,
1
2
,0)
[2] Pnc2 (30) 1; 3; (2; 4)+(
1
2
,
1
2
,0)
[2] Pcn2(Pnc2, 30) 1; 4; (2; 3)+(
1
2
,
1
2
,0)
[2] Pcc2 (27) 1; 2; 3; 4
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] Ccc2(a
= 3a or b
= 3b) (37); [3] Ccc2(c
= 3c) (37)
Minimal non-isomorphic supergroups
I
[2] Cccm (66); [2] Ccce (68); [2] P4cc (103); [2] P4nc(104); [2] P4
2
mc(105); [2] P4
2
bc(106); [2] P
¯
42c (112);
[2] P
¯
42
1
c (114); [3] P6cc(184)
II [2] Fmm2 (42); [2] Pcc2(a
=
1
2
a,b
=
1
2
b) (27); [2] Cmm2(c
=
1
2
c) (35)
243
Amm2 C
14
2v
mm2 Orthorhombic
No. 38 Amm2
Patterson symmetry Ammm (Cmmm )
Origin on mm2
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤
1
2
Symmetry operations
For (0,0,0)+ set
(1) 1 (2) 2 0, 0,z (3) mx, 0,z (4) m 0, y,z
For (0,
1
2
,
1
2
)+ set
(1) t(0,
1
2
,
1
2
) (2) 2(0, 0,
1
2
) 0,
1
4
,z (3) cx,
1
4
,z (4) n(0,
1
2
,
1
2
) 0,y, z
244
International Tables for Crystallography (2006). Vol. A, Space group 38, pp. 244–245.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 38 Amm2
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(0,
1
2
,
1
2
); (2); (3)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (0,
1
2
,
1
2
)+
Reflection conditions
General:
8 f 1(1)x,y,z (2) ¯x, ¯y,z (3) x, ¯y,z (4) ¯x,y,zhkl: k + l = 2n
0kl : k + l = 2 n
h0l : l = 2n
hk0: k = 2n
0k0: k = 2n
00l : l = 2n
Special: no extra conditions
4 em..
1
2
,y,z
1
2
, ¯y, z
4 dm.. 0,y,z 0, ¯y,z
4 c . m . x,0,z ¯x,0,z
2 bmm2
1
2
,0, z
2 amm20, 0,z
Symmetry of special projections
Along [001] p2mm
a
= ab
=
1
2
b
Origin at 0,0,z
Along [100] c1m1
a
= bb
= c
Origin at x, 0,0
Along [010] p11m
a
=
1
2
cb
= a
Origin at 0, y,0
Maximal non-isomorphic subgroups
I
[2] A1m1(Cm,8) (1; 3)+
[2] Am11(Pm,6) (1; 4)+
[2] A112 (C2, 5) (1; 2)+
IIa [2] Pnm2
1
(Pmn2
1
, 31) 1; 3; (2; 4)+(0,
1
2
,
1
2
)
[2] Pnc2 (30) 1; 2; (3; 4)+(0,
1
2
,
1
2
)
[2] Pmc2
1
(26) 1; 4; (2; 3)+(0,
1
2
,
1
2
)
[2] Pmm2 (25) 1; 2; 3; 4
IIb [2] Ima2(a
= 2a) (46); [2] Imm2(a
= 2a) (44); [2] Ama2(a
= 2a) (40)
Maximal isomorphic subgroups of lowest index
IIc
[2] Amm2(a
= 2a) (38); [3] Amm2(b
= 3b) (38); [3] Amm2(c
= 3c) (38)
Minimal non-isomorphic supergroups
I
[2] Cmcm(63); [2]Cmmm (65); [3] P
¯
6m2 (187); [3] P
¯
62m (189)
II [2] Fmm2 (42); [2] Pmm2(b
=
1
2
b,c
=
1
2
c) (25)
245
Aem2 C
15
2v
mm2 Orthorhombic
No. 39 Aem2
Patterson symmetry Ammm (Cmmm )
Former space-group symbol Abm2; cf. Chapter 1.3
Origin on ec2
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
4
;0≤ z ≤ 1
Symmetry operations
For (0,0,0)+ set
(1) 1 (2) 2 0, 0,z (3) mx,
1
4
,z (4) b 0,y, z
For (0,
1
2
,
1
2
)+ set
(1) t(0,
1
2
,
1
2
) (2) 2(0, 0,
1
2
) 0,
1
4
,z (3) cx, 0,z (4) c 0,y,z
246
International Tables for Crystallography (2006). Vol. A, Space group 39, pp. 246–247.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 39 Aem2
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(0,
1
2
,
1
2
); (2); (3)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (0,
1
2
,
1
2
)+
Reflection conditions
General:
8 d 1(1)x,y,z (2) ¯x, ¯y,z (3) x, ¯y+
1
2
,z (4) ¯x,y +
1
2
,zhkl: k + l = 2n
0kl : k,l = 2n
h0l : l = 2n
hk0: k = 2n
0k0: k = 2n
00l : l = 2n
Special: as above, plus
4 c . m . x,
1
4
,z ¯x,
3
4
,z no extra conditions
4 b ..2
1
2
,0, z
1
2
,
1
2
,zhkl: k = 2n
4 a ..20,0, z 0,
1
2
,zhkl: k = 2n
Symmetry of special projections
Along [001] p2mm
a
= ab
=
1
2
b
Origin at 0,0,z
Along [100] p1m1
a
=
1
2
bb
=
1
2
c
Origin at x,0,0
Along [010] p11m
a
=
1
2
cb
= a
Origin at 0,y, 0
Maximal non-isomorphic subgroups
I
[2] A1m1(Cm,8) (1; 3)+
[2] Ae11(Pc,7) (1; 4)+
[2] A112 (C2, 5) (1; 2)+
IIa [2] Pbc2
1
(Pca2
1
, 29) 1; 4; (2; 3)+(0,
1
2
,
1
2
)
[2] Pbm2(Pma2, 28) 1; 2; 3; 4
[2] Pcc2 (27) 1; 2; (3; 4)+(0,
1
2
,
1
2
)
[2] Pcm2
1
(Pmc2
1
, 26) 1; 3; (2; 4)+(0,
1
2
,
1
2
)
IIb [2] Ibm2(a
= 2a)(Ima2, 46); [2] Iba2(a
= 2a) (45); [2] Aea2(a
= 2a) (41)
Maximal isomorphic subgroups of lowest index
IIc
[2] Aem2(a
= 2a) (39); [3] Aem2(b
= 3b) (39); [3] Aem2(c
= 3c) (39)
Minimal non-isomorphic supergroups
I
[2] Cmce(64); [2]Cmme (67)
II [2] Fmm2 (42); [2] Pmm2(b
=
1
2
b,c
=
1
2
c) (25)
247
Ama2 C
16
2v
mm2 Orthorhombic
No. 40 Ama2
Patterson symmetry Ammm (Cmmm )
Origin on 1a2
Asymmetric unit 0 ≤ x ≤
1
4
;0≤ y ≤
1
2
;0≤ z ≤ 1
Symmetry operations
For (0,0,0)+ set
(1) 1 (2) 2 0, 0,z (3) ax,0,z (4) m
1
4
,y,z
For (0,
1
2
,
1
2
)+ set
(1) t(0,
1
2
,
1
2
) (2) 2(0, 0,
1
2
) 0,
1
4
,z (3) n(
1
2
,0,
1
2
) x,
1
4
,z (4) n(0,
1
2
,
1
2
)
1
4
,y,z
248
International Tables for Crystallography (2006). Vol. A, Space group 40, pp. 248–249.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 40 Ama2
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(0,
1
2
,
1
2
); (2); (3)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (0,
1
2
,
1
2
)+
Reflection conditions
General:
8 c 1(1)x,y,z (2) ¯x, ¯y,z (3) x +
1
2
, ¯y, z (4) ¯x +
1
2
,y,zhkl: k + l = 2n
0kl : k + l = 2 n
h0l : h, l = 2n
hk0: k = 2n
h00 : h = 2n
0k0: k = 2n
00l : l = 2n
Special: as above, plus
4 bm..
1
4
,y,z
3
4
, ¯y, z no extra conditions
4 a ..20,0, z
1
2
,0, zhkl: h = 2n
Symmetry of special projections
Along [001] p2mg
a
= ab
=
1
2
b
Origin at 0,0,z
Along [100] c1m1
a
= bb
= c
Origin at x,0,0
Along [010] p11m
a
=
1
2
cb
=
1
2
a
Origin at 0,y, 0
Maximal non-isomorphic subgroups
I
[2] A1a1(Cc,9) (1; 3)+
[2] Am11(Pm,6) (1; 4)+
[2] A112 (C2, 5) (1; 2)+
IIa [2] Pnn2 (34) 1; 2; (3; 4)+(0,
1
2
,
1
2
)
[2] Pna2
1
(33) 1; 3; (2; 4)+(0,
1
2
,
1
2
)
[2] Pmn2
1
(31) 1; 4; (2; 3)+(0,
1
2
,
1
2
)
[2] Pma2 (28) 1; 2; 3; 4
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] Ama2(a
= 3a) (40); [3] Ama2(b
= 3b) (40); [3] Ama2(c
= 3c) (40)
Minimal non-isomorphic supergroups
I
[2] Cmcm(63); [2]Cccm (66); [3] P
¯
6c2 (188); [3] P
¯
62c (190)
II [2] Fmm2 (42); [2] Pma2(b
=
1
2
b,c
=
1
2
c) (28); [2] Amm2(a
=
1
2
a) (38)
249
Aea2 C
17
2v
mm2 Orthorhombic
No. 41 Aea2
Patterson symmetry Ammm (Cmmm )
Former space-group symbol Aba2; cf. Chapter 1.3
Origin on 1n2
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤
1
2
Symmetry operations
For (0,0,0)+ set
(1) 1 (2) 2 0, 0,z (3) ax,
1
4
,z (4) b
1
4
,y,z
For (0,
1
2
,
1
2
)+ set
(1) t(0,
1
2
,
1
2
) (2) 2(0, 0,
1
2
) 0,
1
4
,z (3) n(
1
2
,0,
1
2
) x,0,z (4) c
1
4
,y,z
250
International Tables for Crystallography (2006). Vol. A, Space group 41, pp. 250–251.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 41 Aea2
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(0,
1
2
,
1
2
); (2); (3)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (0,
1
2
,
1
2
)+
Reflection conditions
General:
8 b 1(1)x, y,z (2) ¯x, ¯y,z (3) x +
1
2
, ¯y +
1
2
,z (4) ¯x+
1
2
,y +
1
2
,zhkl: k + l = 2n
0kl : k,l = 2n
h0l : h, l = 2n
hk0: k = 2n
h00 : h = 2n
0k0: k = 2n
00l : l = 2n
Special: as above, plus
4 a ..20,0,z
1
2
,
1
2
,zhkl: h + k = 2n
Symmetry of special projections
Along [001] p2mg
a
= ab
=
1
2
b
Origin at 0,0,z
Along [100] p1m1
a
=
1
2
bb
=
1
2
c
Origin at x,0,0
Along [010] p11m
a
=
1
2
cb
=
1
2
a
Origin at 0,y, 0
Maximal non-isomorphic subgroups
I
[2] A1a1(Cc,9) (1; 3)+
[2] Ae11(Pc,7) (1; 4)+
[2] A112 (C2, 5) (1; 2)+
IIa [2] Pbn2
1
(Pna2
1
, 33) 1; 4; (2; 3)+(0,
1
2
,
1
2
)
[2] Pba2 (32) 1; 2; 3; 4
[2] Pcn2(Pnc2, 30) 1; 2; (3; 4)+(0,
1
2
,
1
2
)
[2] Pca2
1
(29) 1; 3; (2; 4)+(0 ,
1
2
,
1
2
)
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] Aea2(a
= 3a) (41); [3] Aea2(b
= 3b) (41); [3] Aea2(c
= 3c) (41)
Minimal non-isomorphic supergroups
I
[2] Cmce(64); [2]Ccce (68)
II [2] Fmm2 (42); [2] Pma2(b
=
1
2
b,c
=
1
2
c) (28); [2] Aem2(a
=
1
2
a) (39)
251