Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
Подождите немного. Документ загружается.
CONTINUED No. 68 Ccce
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(
1
2
,
1
2
,0); (2); (3); (5)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (
1
2
,
1
2
,0)+
Reflection conditions
General:
16 i 1(1)x,y,z (2) ¯x +
1
2
, ¯y +
1
2
,z (3) ¯x,y, ¯z (4) x +
1
2
, ¯y +
1
2
, ¯z
(5) ¯x, ¯y +
1
2
, ¯z +
1
2
(6) x +
1
2
,y, ¯z +
1
2
(7) x, ¯y+
1
2
,z +
1
2
(8) ¯x+
1
2
,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
8 h ..2
1
4
,
1
4
,z
3
4
,
1
4
, ¯z
3
4
,
1
4
, ¯z +
1
2
1
4
,
1
4
,z +
1
2
hkl : l = 2n
8 g ..20, 0,z 0,0, ¯z 0,
1
2
, ¯z +
1
2
0,
1
2
,z +
1
2
hkl : k + l = 2n
8 f . 2 . 0, y,0
1
2
, ¯y +
1
2
,00, ¯y +
1
2
,
1
2
1
2
,y,
1
2
hkl : k + l = 2n
8 e 2 .. x,0, 0¯x +
1
2
,
1
2
,0¯x,
1
2
,
1
2
x +
1
2
,0,
1
2
hkl : k + l = 2n
8 d
¯
10,
1
4
,
1
4
1
2
,
1
4
,
1
4
0,
1
4
,
3
4
1
2
,
1
4
,
3
4
hkl : k,l = 2n
8 c
¯
1
1
4
,0,
1
4
1
4
,
1
2
,
1
4
3
4
,0,
3
4
3
4
,
1
2
,
3
4
hkl : k,l = 2n
4 b 222 0,0,
1
2
0,
1
2
,0 hk l : k + l = 2n
4 a 222 0,0,00,
1
2
,
1
2
hkl : k + l = 2n
Symmetry of special projections
Along [001] p2mm
a
=
1
2
ab
=
1
2
b
Origin at 0,0,z
Along [100] p2mm
a
=
1
2
bb
=
1
2
c
Origin at x,0,0
Along [010] p2mm
a
=
1
2
cb
=
1
2
a
Origin at 0,y, 0
Maximal non-isomorphic subgroups
I
[2] Cc2e (Aea2, 41) (1; 3; 6; 8)+
[2] C 2ce (Aea2, 41) (1; 4; 6; 7)+
[2] Ccc2 (37) (1; 2; 7; 8)+
[2] C 222(21) (1; 2; 3; 4)+
[2] C 12/c1(C2/c, 15) (1; 3; 5; 7)+
[2] C 2/c11(C2 /c, 15) (1; 4; 5; 8)+
[2] C 112/e (P2/c, 13) (1; 2; 5; 6)+
IIa [2] Pcnb( Pbcn, 60) 1; 4; 5; 8; (2; 3; 6; 7)+(
1
2
,
1
2
,0)
[2] Pnca (Pbcn, 60) 1; 4; 6; 7; (2; 3; 5; 8)+(
1
2
,
1
2
,0)
[2] Pcca (54) 1; 2; 3; 4; 5; 6; 7; 8
[2] Pccb (Pcca, 54) 1; 2; 7; 8; (3; 4; 5; 6)+(
1
2
,
1
2
,0)
[2] Pnnb(Pnna, 52) 1; 2; 3; 4; (5; 6; 7; 8)+(
1
2
,
1
2
,0)
[2] Pnna(52) 1; 2; 5; 6; (3; 4; 7; 8)+(
1
2
,
1
2
,0)
[2] Pncb (Pban, 50) 1; 3; 5; 7; (2; 4; 6; 8)+(
1
2
,
1
2
,0)
[2] Pcna (Pban, 50) 1; 3; 6; 8; (2; 4; 5; 7)+(
1
2
,
1
2
,0)
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] Ccce (a
= 3a or b
= 3b) (68); [3] Ccce (c
= 3c) (68)
Minimal non-isomorphic supergroups
I
[2] P4/nnc(126); [2] P4/ncc(130); [2] P4
2
/nbc(133); [2] P4
2
/nmc(137)
II [2] F mmm(69); [2] Pccm (a
=
1
2
a,b
=
1
2
b) (49); [2] Cmme (c
=
1
2
c) (67)
313
Ccce D
22
2h
mmm Orthorhombic
No. 68 C 2/c 2/c 2/e
Patterson symmetry Cmmm
Former space-group symbol Ccca; cf. Chapter 1.3
ORIGIN CHOICE 2
Origin at
¯
1atnce,at0,−
1
4
,−
1
4
from 222
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
4
;0≤ z ≤
1
2
Symmetry operations
For (0,0,0)+ set
(1) 1 (2) 2
1
4
,0, z (3) 2 0,y,
1
4
(4) 2(
1
2
,0, 0) x, 0,
1
4
(5)
¯
10, 0,0(6)ax,y, 0(7)cx,0,z (8) c
1
4
,y,z
For (
1
2
,
1
2
,0)+ set
(1) t(
1
2
,
1
2
,0) (2) 2 0,
1
4
,z (3) 2(0,
1
2
,0)
1
4
,y,
1
4
(4) 2 x,
1
4
,
1
4
(5)
¯
1
1
4
,
1
4
,0(6)bx,y, 0(7)n(
1
2
,0,
1
2
) x,
1
4
,z (8) n(0,
1
2
,
1
2
) 0,y, z
314
CONTINUED No. 68 Ccce
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(
1
2
,
1
2
,0); (2); (3); (5)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (
1
2
,
1
2
,0)+
Reflection conditions
General:
16 i 1(1)x,y,z (2) ¯x+
1
2
, ¯y, z (3) ¯x, y, ¯z +
1
2
(4) x +
1
2
, ¯y, ¯z +
1
2
(5) ¯x, ¯y, ¯z (6) x +
1
2
,y, ¯z (7) x, ¯y,z +
1
2
(8) ¯x +
1
2
,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
8 h ..2
1
4
,0, z
3
4
,0, ¯z +
1
2
3
4
,0, ¯z
1
4
,0, z +
1
2
hkl : l = 2n
8 g ..20,
1
4
,z 0,
1
4
, ¯z +
1
2
0,
3
4
, ¯z 0,
3
4
,z +
1
2
hkl : k + l = 2n
8 f . 2 . 0, y,
1
4
1
2
, ¯y,
1
4
0, ¯y,
3
4
1
2
,y,
3
4
hkl : k + l = 2n
8 e 2 .. x,
1
4
,
1
4
¯x+
1
2
,
3
4
,
1
4
¯x,
3
4
,
3
4
x +
1
2
,
1
4
,
3
4
hkl : k + l = 2n
8 d
¯
10,0,0
1
2
,0, 00, 0,
1
2
1
2
,0,
1
2
hkl : k,l = 2n
8 c
¯
1
1
4
,
3
4
,0
1
4
,
1
4
,0
3
4
,
3
4
,
1
2
3
4
,
1
4
,
1
2
hkl : k,l = 2n
4 b 222 0,
1
4
,
3
4
0,
3
4
,
1
4
hkl : k + l = 2n
4 a 222 0,
1
4
,
1
4
0,
3
4
,
3
4
hkl : k + l = 2n
Symmetry of special projections
Along [001] p2mm
a
=
1
2
ab
=
1
2
b
Origin at 0,0,z
Along [100] p2mm
a
=
1
2
bb
=
1
2
c
Origin at x,0,0
Along [010] p2mm
a
=
1
2
cb
=
1
2
a
Origin at 0,y, 0
Maximal non-isomorphic subgroups
I
[2] Cc2e (Aea2, 41) (1; 3; 6; 8)+
[2] C 2ce (Aea2, 41) (1; 4; 6; 7)+
[2] Ccc2 (37) (1; 2; 7; 8)+
[2] C 222(21) (1; 2; 3; 4)+
[2] C 12/c1(C2/c, 15) (1; 3; 5; 7)+
[2] C 2/c11(C2 /c, 15) (1; 4; 5; 8)+
[2] C 112/e (P2/c, 13) (1; 2; 5; 6)+
IIa [2] Pcnb( Pbcn, 60) 1; 4; 5; 8; (2; 3; 6; 7)+(
1
2
,
1
2
,0)
[2] Pnca (Pbcn, 60) 1; 4; 6; 7; (2; 3; 5; 8)+(
1
2
,
1
2
,0)
[2] Pcca (54) 1; 2; 3; 4; 5; 6; 7; 8
[2] Pccb (Pcca, 54) 1; 2; 7; 8; (3; 4; 5; 6)+(
1
2
,
1
2
,0)
[2] Pnnb(Pnna, 52) 1; 2; 3; 4; (5; 6; 7; 8)+(
1
2
,
1
2
,0)
[2] Pnna(52) 1; 2; 5; 6; (3; 4; 7; 8)+(
1
2
,
1
2
,0)
[2] Pncb (Pban, 50) 1; 3; 5; 7; (2; 4; 6; 8)+(
1
2
,
1
2
,0)
[2] Pcna (Pban, 50) 1; 3; 6; 8; (2; 4; 5; 7)+(
1
2
,
1
2
,0)
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] Ccce (a
= 3a or b
= 3b) (68); [3] Ccce (c
= 3c) (68)
Minimal non-isomorphic supergroups
I
[2] P4/nnc(126); [2] P4/ncc(130); [2] P4
2
/nbc(133); [2] P4
2
/nmc(137)
II [2] F mmm(69); [2] Pccm (a
=
1
2
a,b
=
1
2
b) (49); [2] Cmme (c
=
1
2
c) (67)
315
Fmmm D
23
2h
mmm Orthorhombic
No. 69 F 2/m 2/ m 2/m
Patterson symmetry F mmm
Origin at centre (mmm)
Asymmetric unit 0 ≤ x ≤
1
4
;0≤ y ≤
1
4
;0≤ z ≤
1
2
Symmetry operations
For (0,0,0)+ set
(1) 1 (2) 2 0, 0,z (3) 2 0,y,0(4)2x,0,0
(5)
¯
10, 0,0(6)mx,y,0(7)mx,0 , z (8) 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) 2(0,
1
2
,0) 0,y,
1
4
(4) 2 x,
1
4
,
1
4
(5)
¯
10,
1
4
,
1
4
(6) bx,y,
1
4
(7) cx,
1
4
,z (8) n(0,
1
2
,
1
2
) 0,y,z
For (
1
2
,0,
1
2
)+ set
(1) t(
1
2
,0,
1
2
) (2) 2(0, 0,
1
2
)
1
4
,0, z (3) 2
1
4
,y,
1
4
(4) 2(
1
2
,0, 0) x,0,
1
4
(5)
¯
1
1
4
,0,
1
4
(6) ax,y,
1
4
(7) n(
1
2
,0,
1
2
) x,0,z (8) c
1
4
,y,z
For (
1
2
,
1
2
,0)+ set
(1) t(
1
2
,
1
2
,0) (2) 2
1
4
,
1
4
,z (3) 2(0,
1
2
,0)
1
4
,y,0(4)2(
1
2
,0, 0) x,
1
4
,0
(5)
¯
1
1
4
,
1
4
,0(6)n(
1
2
,
1
2
,0) x,y,0(7)ax,
1
4
,z (8) b
1
4
,y,z
316
International Tables for Crystallography (2006). Vol. A, Space group 69, pp. 316–318.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 69 Fmmm
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(0,
1
2
,
1
2
); t(
1
2
,0,
1
2
); (2); (3); (5)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (0,
1
2
,
1
2
)+ (
1
2
,0,
1
2
)+ (
1
2
,
1
2
,0)+
Reflection conditions
General:
32 p 1(1)x,y,z (2) ¯x, ¯y,z (3) ¯x,y, ¯z (4) x, ¯y, ¯z
(5) ¯x, ¯y, ¯z (6) x, y, ¯z (7) x, ¯y, z (8) ¯x,y,z
hkl : h + k,h + l,k + l = 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
16 o ..mx,y, 0¯x, ¯y,0¯x,y,0 x, ¯y,0 no extra conditions
16 n . m . x,0,z ¯x,0, z ¯x, 0, ¯zx,0, ¯z no extra conditions
16 mm.. 0,y, z 0, ¯y,z 0,y, ¯z
0, ¯y, ¯z no extra conditions
16 l 2 .. x,
1
4
,
1
4
¯x,
3
4
,
1
4
¯x,
3
4
,
3
4
x,
1
4
,
3
4
hkl : h = 2n
16 k . 2 .
1
4
,y,
1
4
3
4
, ¯y,
1
4
3
4
, ¯y,
3
4
1
4
,y,
3
4
hkl : h = 2n
16 j ..2
1
4
,
1
4
,z
3
4
,
1
4
, ¯z
3
4
,
3
4
, ¯z
1
4
,
3
4
,zhkl: h = 2n
8 imm20,0, z 0,0, ¯z no extra conditions
8 hm2 m 0, y,00, ¯y,0 no extra conditions
8 g 2 mm x,0,0¯x,0,0 no extra conditions
8 f 222
1
4
,
1
4
,
1
4
3
4
,
3
4
,
3
4
hkl : h = 2n
8 e ..2/m
1
4
,
1
4
,0
3
4
,
1
4
,0 hkl : h = 2n
8 d . 2/m .
1
4
,0,
1
4
3
4
,0,
1
4
hkl : h = 2n
8 c 2/m .. 0,
1
4
,
1
4
0,
3
4
,
1
4
hkl : h = 2n
4 bmmm 0, 0,
1
2
no extra conditions
4 ammm 0, 0,0 no extra conditions
Symmetry of special projections
Along [001] p2mm
a
=
1
2
ab
=
1
2
b
Origin at 0,0,z
Along [100] p2mm
a
=
1
2
bb
=
1
2
c
Origin at x,0,0
Along [010] p2mm
a
=
1
2
cb
=
1
2
a
Origin at 0,y, 0
317
Fmmm No. 69 CONTINUED
Maximal non-isomorphic subgroups
I
[2] Fmm2 (42) (1; 2; 7; 8)+
[2] Fm2m (Fmm2, 42) (1; 3; 6; 8)+
[2] F 2mm (Fmm2, 42) (1; 4; 6; 7)+
[2] F 222(22) (1; 2; 3; 4)+
[2] F 112/m (C 2/m, 12) (1; 2; 5; 6)+
[2] F 12/m1(C 2/m, 12) (1; 3; 5; 7)+
[2] F 2/m11(C2/m, 12) (1; 4; 5; 8)+
IIa [2] Aeaa(Ccce, 68) 1; 2; 3; 4; (1; 2; 3; 4)+(0,
1
2
,
1
2
); (5; 6; 7; 8)+(
1
2
,0,
1
2
); (5; 6; 7; 8)+(
1
2
,
1
2
,0)
[2] Bbeb (Ccce, 68) 1; 2; 3; 4; (1; 2; 3; 4)+(
1
2
,0,
1
2
); (5; 6; 7; 8)+(0,
1
2
,
1
2
); (5; 6; 7; 8)+(
1
2
,
1
2
,0)
[2] Ccce (68) 1; 2; 3; 4; (1; 2; 3; 4)+(
1
2
,
1
2
,0); (5; 6; 7; 8)+(0,
1
2
,
1
2
); (5; 6; 7; 8)+(
1
2
,0,
1
2
)
[2] Cmme (67) 1; 2; 7; 8; (1; 2; 7; 8)+(
1
2
,
1
2
,0); (3; 4; 5; 6)+(0,
1
2
,
1
2
); (3; 4; 5; 6)+(
1
2
,0,
1
2
)
[2] Bmem (Cmme, 67) 1; 3; 6; 8; (1; 3; 6; 8)+(
1
2
,0,
1
2
); (2; 4; 5; 7)+(0,
1
2
,
1
2
); (2; 4; 5; 7)+(
1
2
,
1
2
,0)
[2] Aemm (Cmme, 67) 1; 4; 6; 7; (1; 4; 6; 7)+(0,
1
2
,
1
2
); (2; 3; 5; 8)+(
1
2
,0,
1
2
); (2; 3; 5; 8)+(
1
2
,
1
2
,0)
[2] Cccm (66) 1; 2; 5; 6; (1; 2; 5; 6)+(
1
2
,
1
2
,0); (3; 4; 7; 8)+(0,
1
2
,
1
2
); (3; 4; 7; 8)+(
1
2
,0,
1
2
)
[2] Bbmb (Cccm, 66) 1; 3; 5; 7; (1; 3; 5; 7)+(
1
2
,0,
1
2
); (2; 4; 6; 8)+(0,
1
2
,
1
2
); (2; 4; 6; 8)+(
1
2
,
1
2
,0)
[2] Amaa (Cccm, 66) 1; 4; 5; 8; (1; 4; 5; 8)+(0,
1
2
,
1
2
); (2; 3; 6; 7)+(
1
2
,0,
1
2
); (2; 3; 6; 7)+(
1
2
,
1
2
,0)
[2] Ammm (Cmmm, 65) 1; 2; 3; 4; 5; 6; 7; 8; (1; 2; 3; 4; 5; 6; 7; 8)+(0,
1
2
,
1
2
)
[2] Bmmm (Cmmm, 65) 1; 2; 3; 4; 5; 6; 7; 8; (1; 2; 3; 4; 5; 6; 7; 8)+(
1
2
,0,
1
2
)
[2] Cmmm (65) 1; 2; 3; 4; 5; 6; 7; 8; (1; 2; 3; 4; 5; 6; 7; 8)+(
1
2
,
1
2
,0)
[2] Aeam(Cmce, 64) 1; 2; 5; 6; (1; 2; 5; 6)+(0,
1
2
,
1
2
); (3; 4; 7; 8)+(
1
2
,0,
1
2
); (3; 4; 7; 8)+(
1
2
,
1
2
,0)
[2] Bbem (Cmce, 64) 1; 2; 5; 6; (1; 2; 5; 6)+(
1
2
,0,
1
2
); (3; 4; 7; 8)+(0,
1
2
,
1
2
); (3; 4; 7; 8)+(
1
2
,
1
2
,0)
[2] Aema (Cmce, 64) 1; 3; 5; 7; (1; 3; 5; 7)+(0,
1
2
,
1
2
); (2; 4; 6; 8)+(
1
2
,0,
1
2
); (2; 4; 6; 8)+(
1
2
,
1
2
,0)
[2] Ccme(Cmce, 64) 1; 3; 5; 7; (1; 3; 5; 7)+(
1
2
,
1
2
,0); (2; 4; 6; 8)+(0,
1
2
,
1
2
); (2; 4; 6; 8)+(
1
2
,0,
1
2
)
[2] Bmeb (Cmce, 64) 1; 4; 5; 8; (1; 4; 5; 8)+(
1
2
,0,
1
2
); (2; 3; 6; 7)+(0,
1
2
,
1
2
); (2; 3; 6; 7)+(
1
2
,
1
2
,0)
[2] Cmce(64) 1; 4; 5; 8; (1; 4; 5; 8)+(
1
2
,
1
2
,0); (2; 3; 6; 7)+(0,
1
2
,
1
2
); (2; 3; 6; 7)+(
1
2
,0,
1
2
)
[2] Amam(Cmcm, 63) 1; 3; 6; 8; (1; 3; 6; 8)+(0,
1
2
,
1
2
); (2; 4; 5; 7)+(
1
2
,0,
1
2
); (2; 4; 5; 7)+(
1
2
,
1
2
,0)
[2] Amma (Cmcm, 63) 1; 2; 7; 8; (1; 2; 7; 8)+(0,
1
2
,
1
2
); (3; 4; 5; 6)+(
1
2
,0,
1
2
); (3; 4; 5; 6)+(
1
2
,
1
2
,0)
[2] Bmmb (Cmcm, 63) 1; 2; 7; 8; (1; 2; 7; 8)+(
1
2
,0,
1
2
); (3; 4; 5; 6)+(0,
1
2
,
1
2
); (3; 4; 5; 6)+(
1
2
,
1
2
,0)
[2] Bbmm (Cmcm, 63) 1; 4; 6; 7; (1; 4; 6; 7)+(
1
2
,0,
1
2
); (2; 3; 5; 8)+(0,
1
2
,
1
2
); (2; 3; 5; 8)+(
1
2
,
1
2
,0)
[2] Cmcm(63) 1; 3; 6; 8; (1; 3; 6; 8)+(
1
2
,
1
2
,0); (2; 4; 5; 7)+(0,
1
2
,
1
2
); (2; 4; 5; 7)+(
1
2
,0,
1
2
)
[2] Ccmm(Cmcm, 63) 1; 4; 6; 7; (1; 4; 6; 7)+(
1
2
,
1
2
,0); (2; 3; 5; 8)+(0,
1
2
,
1
2
); (2; 3; 5; 8)+(
1
2
,0,
1
2
)
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] F mmm(a
= 3a or b
= 3b or c
= 3c) (69)
Minimal non-isomorphic supergroups
I
[2] I 4/mmm(139); [2] I 4/mcm(140); [3] Fm
¯
3 (202)
II [2] Pmmm (a
=
1
2
a,b
=
1
2
b,c
=
1
2
c) (47)
318
Fddd D
24
2h
mmm Orthorhombic
No. 70 F 2/d 2 /d 2/d
Patterson symmetry F mmm
ORIGIN CHOICE 1
Origin at 222, at −
1
8
,−
1
8
,−
1
8
from
¯
1
Asymmetric unit 0 ≤ x ≤
1
8
;0≤ y ≤
1
4
;0≤ z ≤ 1
Symmetry operations
For (0,0,0)+ set
(1) 1 (2) 2 0, 0,z (3) 2 0,y,0(4)2x,0,0
(5)
¯
1
1
8
,
1
8
,
1
8
(6) d(
1
4
,
1
4
,0) x, y,
1
8
(7) d(
1
4
,0,
1
4
) x,
1
8
,z (8) d(0,
1
4
,
1
4
)
1
8
,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) 2(0,
1
2
,0) 0,y,
1
4
(4) 2 x,
1
4
,
1
4
(5)
¯
1
1
8
,
3
8
,
3
8
(6) d(
1
4
,
3
4
,0) x, y,
3
8
(7) d(
1
4
,0,
3
4
) x,
3
8
,z (8) d(0,
3
4
,
3
4
)
1
8
,y,z
For (
1
2
,0,
1
2
)+ set
(1) t(
1
2
,0,
1
2
) (2) 2(0, 0,
1
2
)
1
4
,0, z (3) 2
1
4
,y,
1
4
(4) 2(
1
2
,0, 0) x,0,
1
4
(5)
¯
1
3
8
,
1
8
,
3
8
(6) d(
3
4
,
1
4
,0) x, y,
3
8
(7) d(
3
4
,0,
3
4
) x,
1
8
,z (8) d(0,
1
4
,
3
4
)
3
8
,y,z
For (
1
2
,
1
2
,0)+ set
(1) t(
1
2
,
1
2
,0) (2) 2
1
4
,
1
4
,z (3) 2(0,
1
2
,0)
1
4
,y,0(4)2(
1
2
,0, 0) x,
1
4
,0
(5)
¯
1
3
8
,
3
8
,
1
8
(6) d(
3
4
,
3
4
,0) x, y,
1
8
(7) d(
3
4
,0,
1
4
) x,
3
8
,z (8) d(0,
3
4
,
1
4
)
3
8
,y,z
320
International Tables for Crystallography (2006). Vol. A, Space group 70, pp. 320–323.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 70 Fddd
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(0,
1
2
,
1
2
); t(
1
2
,0,
1
2
); (2); (3); (5)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (0,
1
2
,
1
2
)+ (
1
2
,0,
1
2
)+ (
1
2
,
1
2
,0)+
Reflection conditions
General:
32 h 1(1)x,y,z (2) ¯x, ¯y,z (3) ¯x,y, ¯z (4) x, ¯y, ¯z
(5) ¯x +
1
4
, ¯y +
1
4
, ¯z +
1
4
(6) x +
1
4
,y +
1
4
, ¯z +
1
4
(7) x +
1
4
, ¯y +
1
4
,z +
1
4
(8) ¯x +
1
4
,y +
1
4
,z +
1
4
hkl : h + k = 2n and
h + l,k + l = 2n
0kl : k + l = 4 n and
k,l = 2n
h0l : h + l = 4n and
h,l = 2n
hk0: h + k = 4n and
h,k = 2n
h00 : h = 4n
0k0: k = 4n
00l : l = 4n
Special: as above, plus
16 g ..20,0,z 0,0, ¯
z
1
4
,
1
4
, ¯z +
1
4
1
4
,
1
4
,z +
1
4
16 f . 2 . 0,y, 00, ¯y,0
1
4
, ¯y +
1
4
,
1
4
1
4
,y +
1
4
,
1
4
16 e 2 .. x,0,0¯x, 0,0¯x +
1
4
,
1
4
,
1
4
x +
1
4
,
1
4
,
1
4
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
hkl : h = 2n + 1
or h + k + l = 4n
16 d
¯
1
5
8
,
5
8
,
5
8
3
8
,
3
8
,
5
8
3
8
,
5
8
,
3
8
5
8
,
3
8
,
3
8
16 c
¯
1
1
8
,
1
8
,
1
8
7
8
,
7
8
,
1
8
7
8
,
1
8
,
7
8
1
8
,
7
8
,
7
8
hkl : h = 2n + 1
or h, k, l = 4n + 2
or h, k, l = 4n
8 b 222 0,0,
1
2
1
4
,
1
4
,
3
4
8 a 222 0,0,0
1
4
,
1
4
,
1
4
hkl : h = 2n + 1
or h + k + l = 4n
Symmetry of special projections
Along [001] c2mm
a
=
1
2
ab
=
1
2
b
Origin at 0,0,z
Along [100] c2mm
a
=
1
2
bb
=
1
2
c
Origin at x,0,0
Along [010] c2mm
a
=
1
2
cb
=
1
2
a
Origin at 0, y,0
Maximal non-isomorphic subgroups
I
[2] Fdd2 (43) (1; 2; 7; 8)+
[2] Fd2d (Fdd2, 43) (1; 3; 6; 8)+
[2] F 2dd(Fdd2, 43) (1; 4; 6; 7)+
[2] F 222(22) (1; 2; 3; 4)+
[2] F 112/d (C 2/c, 15) (1; 2; 5; 6)+
[2] F 12/d 1(C 2/c, 15) (1; 3; 5; 7)+
[2] F 2/d 11(C2 /c, 15) (1; 4; 5; 8)+
IIa none
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] Fddd(a
= 3a or b
= 3b or c
= 3c) (70)
Minimal non-isomorphic supergroups
I
[2] I 4
1
/amd (141); [2] I 4
1
/acd (142); [3] Fd
¯
3 (203)
II [2] Pnnn(a
=
1
2
a,b
=
1
2
b,c
=
1
2
c) (48)
321
Fddd D
24
2h
mmm Orthorhombic
No. 70 F 2/d 2 /d 2/d
Patterson symmetry F mmm
ORIGIN CHOICE 2
Origin at
¯
1atddd,at
1
8
,
1
8
,
1
8
from 222
Asymmetric unit 0 ≤ x ≤
1
8
; −
1
8
≤ y ≤
1
8
;0≤ z ≤ 1
Symmetry operations
For (0,0,0)+ set
(1) 1 (2) 2
3
8
,
3
8
,z (3) 2
3
8
,y,
3
8
(4) 2 x,
3
8
,
3
8
(5)
¯
10, 0,0(6)d(
1
4
,
1
4
,0) x, y,0(7)d(
1
4
,0,
1
4
) x,0, z (8) d(0,
1
4
,
1
4
) 0,y, z
For (0,
1
2
,
1
2
)+ set
(1) t(0,
1
2
,
1
2
) (2) 2(0, 0,
1
2
)
3
8
,
1
8
,z (3) 2(0,
1
2
,0)
3
8
,y,
1
8
(4) 2 x,
1
8
,
1
8
(5)
¯
10,
1
4
,
1
4
(6) d(
1
4
,
3
4
,0) x, y,
1
4
(7) d(
1
4
,0,
3
4
) x,
1
4
,z (8) d(0,
3
4
,
3
4
) 0,y, z
For (
1
2
,0,
1
2
)+ set
(1) t(
1
2
,0,
1
2
) (2) 2(0, 0,
1
2
)
1
8
,
3
8
,z (3) 2
1
8
,y,
1
8
(4) 2(
1
2
,0, 0) x,
3
8
,
1
8
(5)
¯
1
1
4
,0,
1
4
(6) d(
3
4
,
1
4
,0) x, y,
1
4
(7) d(
3
4
,0,
3
4
) x,0, z (8) d(0,
1
4
,
3
4
)
1
4
,y,z
For (
1
2
,
1
2
,0)+ set
(1) t(
1
2
,
1
2
,0) (2) 2
1
8
,
1
8
,z (3) 2(0,
1
2
,0)
1
8
,y,
3
8
(4) 2(
1
2
,0, 0) x,
1
8
,
3
8
(5)
¯
1
1
4
,
1
4
,0(6)d(
3
4
,
3
4
,0) x, y,0(7)d(
3
4
,0,
1
4
) x,
1
4
,z (8) d(0,
3
4
,
1
4
)
1
4
,y,z
322
CONTINUED No. 70 Fddd
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(0,
1
2
,
1
2
); t(
1
2
,0,
1
2
); (2); (3); (5)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates
(0,0, 0)+ (0,
1
2
,
1
2
)+ (
1
2
,0,
1
2
)+ (
1
2
,
1
2
,0)+
Reflection conditions
General:
32 h 1(1)x,y,z (2) ¯x +
3
4
, ¯y +
3
4
,z (3) ¯x +
3
4
,y, ¯z +
3
4
(4) x, ¯y +
3
4
, ¯z +
3
4
(5) ¯x, ¯y, ¯z (6) x +
1
4
,y +
1
4
, ¯z (7) x +
1
4
, ¯y, z +
1
4
(8) ¯x,y +
1
4
,z +
1
4
hkl : h + k,h + l,k + l = 2n
0kl : k + l = 4 n, k,l = 2n
h0l : h + l = 4n, h, l = 2n
hk0: h + k = 4n, h, k = 2n
h00 : h = 4n
0k0: k = 4n
00l : l = 4n
Special: as above, plus
16 g ..2
1
8
,
1
8
,z
5
8
,
1
8
, ¯z +
3
4
7
8
,
7
8
, ¯z
3
8
,
7
8
,z +
1
4
16 f . 2 .
1
8
,y,
1
8
5
8
, ¯y +
3
4
,
1
8
7
8
, ¯y,
7
8
3
8
,y +
1
4
,
7
8
16 e 2 .. x,
1
8
,
1
8
¯x+
3
4
,
5
8
,
1
8
¯x,
7
8
,
7
8
x +
1
4
,
3
8
,
7
8
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
hkl : h = 2n + 1
or h + k + l = 4n
16 d
¯
1
1
2
,
1
2
,
1
2
1
4
,
1
4
,
1
2
1
4
,
1
2
,
1
4
1
2
,
1
4
,
1
4
16 c
¯
10,0,0
3
4
,
3
4
,0
3
4
,0,
3
4
0,
3
4
,
3
4
hkl : h = 2n + 1
or h, k, l = 4n + 2
or h, k, l = 4n
8 b 222
1
8
,
1
8
,
5
8
7
8
,
7
8
,
3
8
8 a 222
1
8
,
1
8
,
1
8
7
8
,
7
8
,
7
8
hkl : h = 2n + 1
or h + k + l = 4n
Symmetry of special projections
Along [001] c2mm
a
=
1
2
ab
=
1
2
b
Origin at
1
8
,
1
8
,z
Along [100] c2mm
a
=
1
2
bb
=
1
2
c
Origin at x,
1
8
,
1
8
Along [010] c2mm
a
=
1
2
cb
=
1
2
a
Origin at
1
8
,y,
1
8
Maximal non-isomorphic subgroups
I
[2] Fdd2 (43) (1; 2; 7; 8)+
[2] Fd2d (Fdd2, 43) (1; 3; 6; 8)+
[2] F 2dd(Fdd2, 43) (1; 4; 6; 7)+
[2] F 222(22) (1; 2; 3; 4)+
[2] F 112/d (C 2/c, 15) (1; 2; 5; 6)+
[2] F 12/d 1(C 2/c, 15) (1; 3; 5; 7)+
[2] F 2/d 11(C2 /c, 15) (1; 4; 5; 8)+
IIa none
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] Fddd(a
= 3a or b
= 3b or c
= 3c) (70)
Minimal non-isomorphic supergroups
I
[2] I 4
1
/amd (141); [2] I 4
1
/acd (142); [3] Fd
¯
3 (203)
II [2] Pnnn(a
=
1
2
a,b
=
1
2
b,c
=
1
2
c) (48)
323