Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry
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Pban D
4
2h
mmm Orthorhombic
No. 50 P 2/b 2/a 2/n
Patterson symmetry Pmmm
ORIGIN CHOICE 2
Origin at
¯
1atban,at−
1
4
,−
1
4
,0 from 222
Asymmetric unit 0 ≤ x ≤
1
4
;0≤ y ≤ 1; 0 ≤ z ≤
1
2
Symmetry operations
(1) 1 (2) 2
1
4
,
1
4
,z (3) 2
1
4
,y,0(4)2x,
1
4
,0
(5)
¯
10, 0,0(6)n(
1
2
,
1
2
,0) x,y,0(7)ax,0,z (8) b 0,y,z
Minimal non-isomorphic supergroups
I
[2] P4/nbm(125); [2] P4
2
/nbc(133)
II [2] Cmmm (65); [2] Aeaa(Ccce, 68); [2] Bbeb (Ccce, 68); [2] Ibam(72); [2] Pbmb (a
=
1
2
a)(Pccm, 49);
[2] Pmaa (b
=
1
2
b)(Pccm, 49)
272
CONTINUED No. 50 Pban
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
8 m 1(1)x,y,z (2) ¯x+
1
2
, ¯y +
1
2
,z (3) ¯x +
1
2
,y, ¯z (4) x, ¯y +
1
2
, ¯z
(5) ¯x, ¯y, ¯z (6) x +
1
2
,y +
1
2
, ¯z (7) x +
1
2
, ¯y, z (8) ¯x,y +
1
2
,z
0kl : k = 2n
h0l : h = 2n
hk0: h + k = 2n
h00 : h = 2n
0k0: k = 2n
Special: as above, plus
4 l ..2
1
4
,
3
4
,z
1
4
,
3
4
, ¯z
3
4
,
1
4
, ¯z
3
4
,
1
4
,zhkl: h + k = 2n
4 k ..2
1
4
,
1
4
,z
1
4
,
1
4
, ¯z
3
4
,
3
4
, ¯z
3
4
,
3
4
,zhkl: h + k = 2n
4 j . 2 .
1
4
,y,
1
2
1
4
, ¯y +
1
2
,
1
2
3
4
, ¯y,
1
2
3
4
,y +
1
2
,
1
2
hkl : h + k = 2n
4 i . 2 .
1
4
,y,0
1
4
, ¯y +
1
2
,0
3
4
, ¯y, 0
3
4
,y +
1
2
,0 hkl : h + k = 2n
4 h 2 .. x,
1
4
,
1
2
¯x +
1
2
,
1
4
,
1
2
¯x,
3
4
,
1
2
x +
1
2
,
3
4
,
1
2
hkl : h + k = 2n
4 g 2 .. x,
1
4
,0¯x +
1
2
,
1
4
,0¯x,
3
4
,0 x +
1
2
,
3
4
,0 hkl : h + k = 2n
4 f
¯
10,0,
1
2
1
2
,
1
2
,
1
2
1
2
,0,
1
2
0,
1
2
,
1
2
hkl : h,k = 2n
4 e
¯
10,0,0
1
2
,
1
2
,0
1
2
,0, 00,
1
2
,0 hkl : h,k = 2n
2 d 222
1
4
,
1
4
,
1
2
3
4
,
3
4
,
1
2
hkl : h + k = 2n
2 c 222
3
4
,
1
4
,
1
2
1
4
,
3
4
,
1
2
hkl : h + k = 2n
2 b 222
3
4
,
1
4
,0
1
4
,
3
4
,0 hk l : h + k = 2n
2 a 222
1
4
,
1
4
,0
3
4
,
3
4
,0 hk l : h + k = 2n
Symmetry of special projections
Along [001] c2mm
a
= ab
= b
Origin at
1
4
,
1
4
,z
Along [100] p2mm
a
=
1
2
bb
= c
Origin at x,0,0
Along [010] p2mm
a
= cb
=
1
2
a
Origin at 0,y, 0
Maximal non-isomorphic subgroups
I
[2] Pba2 (32) 1; 2; 7; 8
[2] Pb2n (Pnc2, 30) 1; 3; 6; 8
[2] P2an (Pnc2, 30) 1; 4; 6; 7
[2] P222 (16) 1; 2; 3; 4
[2] P112/n (P2/c, 13) 1; 2; 5; 6
[2] P12/a1(P2/c, 13) 1; 3; 5; 7
[2] P2/b11(P2/c, 13) 1; 4; 5; 8
IIa none
IIb [2] Pnan(c
= 2c)(Pnna, 52); [2] Pbnn(c
= 2c)(Pnna, 52); [2] Pnnn (c
= 2c) (48)
Maximal isomorphic subgroups of lowest index
IIc
[2] Pban(c
= 2c) (50); [3] Pban (a
= 3a or b
= 3b) (50)
(Continued on preceding page)
273
Pmma D
5
2h
mmm Orthorhombic
No. 51 P 2
1
/m 2/m 2/a Patterson symmetry Pmmm
Origin at centre (2/m) at 2
1
2/ma
Asymmetric unit 0 ≤ x ≤
1
4
;0≤ y ≤
1
2
;0≤ z ≤ 1
Symmetry operations
(1) 1 (2) 2
1
4
,0, z (3) 2 0,y,0(4)2(
1
2
,0, 0) x, 0,0
(5)
¯
10, 0,0(6)ax,y,0(7)mx,0,z (8) m
1
4
,y,z
274
International Tables for Crystallography (2006). Vol. A, Space group 51, pp. 274–275.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 51 Pmma
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
8 l 1(1)x,y,z (2) ¯x+
1
2
, ¯y, z (3) ¯x, y, ¯z (4) x +
1
2
, ¯y, ¯z
(5) ¯x, ¯y, ¯z (6) x +
1
2
,y, ¯z (7) x, ¯y,z (8) ¯x+
1
2
,y,z
hk0: h = 2n
h00 : h = 2n
Special: as above, plus
4 km..
1
4
,y,z
1
4
, ¯y, z
3
4
,y, ¯z
3
4
, ¯y, ¯z no extra conditions
4 j . m . x,
1
2
,z ¯x +
1
2
,
1
2
,z ¯x,
1
2
, ¯zx+
1
2
,
1
2
, ¯z no extra conditions
4 i . m . x,0,z ¯x +
1
2
,0, z ¯x, 0, ¯zx+
1
2
,0, ¯z no extra conditions
4 h . 2 . 0, y,
1
2
1
2
, ¯y,
1
2
0, ¯y,
1
2
1
2
,y,
1
2
hkl : h = 2n
4 g . 2 . 0, y,0
1
2
, ¯y, 00, ¯y,0
1
2
,y,0 hkl : h = 2n
2 fmm2
1
4
,
1
2
,z
3
4
,
1
2
, ¯z no extra conditions
2 emm2
1
4
,0, z
3
4
,0, ¯z no extra conditions
2 d . 2/m . 0,
1
2
,
1
2
1
2
,
1
2
,
1
2
hkl : h = 2n
2 c . 2/m . 0,0,
1
2
1
2
,0,
1
2
hkl : h = 2n
2 b . 2/m . 0,
1
2
,0
1
2
,
1
2
,0 hkl : h = 2n
2 a . 2/m . 0, 0,0
1
2
,0, 0 hkl : h = 2n
Symmetry of special projections
Along [001] p2mm
a
=
1
2
ab
= b
Origin at 0,0,z
Along [100] p2mm
a
= bb
= c
Origin at x,0,0
Along [010] p2gm
a
= cb
= a
Origin at 0,y, 0
Maximal non-isomorphic subgroups
I
[2] Pm2a (Pma2, 28) 1; 3; 6; 8
[2] P2
1
ma (Pmc2
1
, 26) 1; 4; 6; 7
[2] Pmm2 (25) 1; 2; 7; 8
[2] P2
1
22(P222
1
, 17) 1; 2; 3; 4
[2] P112/a (P2/c, 13) 1; 2; 5; 6
[2] P2
1
/m11 (P2
1
/m, 11) 1; 4; 5; 8
[2] P12/m1(P2/m, 10) 1; 3; 5; 7
IIa none
IIb [2] Pmmn (b
= 2b) (59); [2] Pbma (b
= 2b)(Pbcm, 57); [2] Pbmn (b
= 2b)(Pmna, 53); [2] Pmca (c
= 2c)(Pbcm, 57);
[2] Pcma (c
= 2c)(Pbam, 55); [2] Pcca (c
= 2c) (54); [2] Aema (b
= 2b,c
= 2c)(Cmce, 64);
[2] Amma (b
= 2b,c
= 2c)(Cmcm, 63)
Maximal isomorphic subgroups of lowest index
IIc
[2] Pmma (b
= 2b) (51); [2] Pmma (c
= 2c) (51); [3] Pmma (a
= 3a) (51)
Minimal non-isomorphic supergroups
I
none
II [2] Amma (Cmcm, 63); [2] Bmmm (Cmmm, 65); [2] Cmme (67); [2] Imma (74); [2] Pmmm (a
=
1
2
a) (47)
275
Pnna D
6
2h
mmm Orthorhombic
No. 52 P 2/n 2
1
/n 2/a Patterson symmetry Pmmm
Origin at
¯
1onn1a
Asymmetric unit 0 ≤ x ≤ 1; 0 ≤ y ≤
1
4
;0≤ z ≤
1
2
Symmetry operations
(1) 1 (2) 2
1
4
,0, z (3) 2(0,
1
2
,0)
1
4
,y,
1
4
(4) 2 x,
1
4
,
1
4
(5)
¯
10, 0,0(6)ax,y,0(7)n(
1
2
,0,
1
2
) x,
1
4
,z (8) n(0,
1
2
,
1
2
) 0,y, z
276
International Tables for Crystallography (2006). Vol. A, Space group 52, pp. 276–277.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 52 Pnna
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
8 e 1(1)x,y,z (2) ¯x+
1
2
, ¯y, z (3) ¯x +
1
2
,y +
1
2
, ¯z +
1
2
(4) x, ¯y +
1
2
, ¯z +
1
2
(5) ¯x, ¯y, ¯z (6) x +
1
2
,y, ¯z (7) x +
1
2
, ¯y +
1
2
,z +
1
2
(8) ¯x,y +
1
2
,z +
1
2
0kl : k + l = 2 n
h0l : h + l = 2n
hk0: h = 2n
h00 : h = 2n
0k0: k = 2n
00l : l = 2n
Special: as above, plus
4 d 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 : h + l = 2n
4 c ..2
1
4
,0, z
1
4
,
1
2
, ¯z +
1
2
3
4
,0, ¯z
3
4
,
1
2
,z +
1
2
hkl : h + k + l = 2n
4 b
¯
10, 0,
1
2
1
2
,0,
1
2
1
2
,
1
2
,00,
1
2
,0 hkl : h,k + l = 2n
4 a
¯
10, 0,0
1
2
,0, 0
1
2
,
1
2
,
1
2
0,
1
2
,
1
2
hkl : h,k + l = 2n
Symmetry of special projections
Along [001] p2gm
a
=
1
2
ab
= b
Origin at 0,0,z
Along [100] c2mm
a
= bb
= c
Origin at x, 0,0
Along [010] c2mm
a
= cb
= a
Origin at
1
4
,y,
1
4
Maximal non-isomorphic subgroups
I
[2] Pnn2 (34) 1; 2; 7; 8
[2] Pn2
1
a (Pna2
1
, 33) 1; 3; 6; 8
[2] P2na (Pnc2, 30) 1; 4; 6; 7
[2] P22
1
2(P222
1
, 17) 1; 2; 3; 4
[2] P12
1
/n1(P2
1
/c, 14) 1; 3; 5; 7
[2] P112/a (P2/ c, 13) 1; 2; 5; 6
[2] P2/n11(P2/c, 13) 1; 4; 5; 8
IIa none
IIb none
Maximal isomorphic subgroups of lowest index
IIc
[3] Pnna(a
= 3a) (52); [3] Pnna (b
= 3b) (52); [3] Pnna (c
= 3c) (52)
Minimal non-isomorphic supergroups
I
none
II [2] Bbmm (Cmcm, 63); [2] Amaa (Cccm, 66); [2]Ccce (68); [2] Imma (74); [2] Pncm (a
=
1
2
a)(Pmna, 53);
[2] Pcna (b
=
1
2
b)(Pban, 50); [2] Pbaa(c
=
1
2
c)(Pcca, 54)
277
Pmna D
7
2h
mmm Orthorhombic
No. 53 P 2/m 2/n 2
1
/a Patterson symmetry Pmmm
Origin at centre (2/m) at 2/mn1
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤ 1; 0 ≤ z ≤
1
4
Symmetry operations
(1) 1 (2) 2(0, 0,
1
2
)
1
4
,0, z (3) 2
1
4
,y,
1
4
(4) 2 x,0,0
(5)
¯
10, 0,0(6)ax,y,
1
4
(7) n(
1
2
,0,
1
2
) x,0,z (8) m 0,y,z
278
International Tables for Crystallography (2006). Vol. A, Space group 53, pp. 278–279.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 53 Pmna
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
8 i 1(1)x,y,z (2) ¯x+
1
2
, ¯y, z +
1
2
(3) ¯x+
1
2
,y, ¯z +
1
2
(4) x, ¯y, ¯z
(5) ¯x, ¯y, ¯z (6) x +
1
2
,y, ¯z +
1
2
(7) x +
1
2
, ¯y, z +
1
2
(8) ¯x,y,z
h0l : h + l = 2n
hk0: h = 2n
h00 : h = 2n
00l : l = 2n
Special: as above, plus
4 hm.. 0,y,z
1
2
, ¯y, z +
1
2
1
2
,y, ¯z +
1
2
0, ¯y, ¯z no extra conditions
4 g . 2 .
1
4
,y,
1
4
1
4
, ¯y,
3
4
3
4
, ¯y,
3
4
3
4
,y,
1
4
hkl : h = 2n
4 f 2 .. x,
1
2
,0¯x +
1
2
,
1
2
,
1
2
¯x,
1
2
,0 x +
1
2
,
1
2
,
1
2
hkl : h + l = 2n
4 e 2 .. x,0, 0¯x +
1
2
,0,
1
2
¯x,0, 0 x +
1
2
,0,
1
2
hkl : h + l = 2n
2 d 2/m .. 0,
1
2
,0
1
2
,
1
2
,
1
2
hkl : h + l = 2n
2 c 2/m ..
1
2
,
1
2
,00,
1
2
,
1
2
hkl : h + l = 2n
2 b 2/ m ..
1
2
,0, 00,0,
1
2
hkl : h + l = 2n
2 a 2/ m .. 0,0,0
1
2
,0,
1
2
hkl : h + l = 2n
Symmetry of special projections
Along [001] p2mm
a
=
1
2
ab
= b
Origin at 0,0,z
Along [100] p2gm
a
= bb
= c
Origin at x,0,0
Along [010] c2mm
a
= cb
= a
Origin at 0,y, 0
Maximal non-isomorphic subgroups
I
[2] Pmn2
1
(31) 1; 2; 7; 8
[2] P2na (Pnc2, 30) 1; 4; 6; 7
[2] Pm2a (Pma2, 28) 1; 3; 6; 8
[2] P222
1
(17) 1; 2; 3; 4
[2] P112
1
/a (P2
1
/c, 14) 1; 2; 5; 6
[2] P12/n1(P2/c, 13) 1; 3; 5; 7
[2] P2/m11(P2/m, 10) 1; 4; 5; 8
IIa none
IIb [2] Pbna(b
= 2b)(Pbcn, 60); [2] Pmnn (b
= 2b)(Pnnm, 58); [2] Pbnn(b
= 2b)(Pnna, 52)
Maximal isomorphic subgroups of lowest index
IIc
[2] Pmna (b
= 2b) (53); [3] Pmna (a
= 3a) (53); [3] Pmna (c
= 3c) (53)
Minimal non-isomorphic supergroups
I
none
II [2] Cmce(64); [2] Bmmm (Cmmm, 65); [2] Amaa (Cccm, 66); [2] Imma (74); [2] Pmaa(c
=
1
2
c)(Pccm, 49);
[2] Pmcm (a
=
1
2
a)(Pmma, 51)
279
Pcca D
8
2h
mmm Orthorhombic
No. 54 P 2
1
/c 2/ c 2/a Patterson symmetry Pmmm
Origin at
¯
1on1ca
Asymmetric unit 0 ≤ x ≤
1
2
;0≤ y ≤
1
2
;0≤ z ≤
1
2
Symmetry operations
(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
280
International Tables for Crystallography (2006). Vol. A, Space group 54, pp. 280–281.
Copyright © 2006 International Union of Crystallography
CONTINUED No. 54 Pcca
Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5)
Positions
Multiplicity,
Wyckoff letter,
Site symmetry
Coordinates Reflection conditions
General:
8 f 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
0kl : l = 2n
h0l : l = 2n
hk0: h = 2n
h00 : h = 2n
00l : l = 2n
Special: as above, plus
4 e ..2
1
4
,
1
2
,z
3
4
,
1
2
, ¯z +
1
2
3
4
,
1
2
, ¯z
1
4
,
1
2
,z +
1
2
hkl : l = 2n
4 d ..2
1
4
,0, z
3
4
,0, ¯z +
1
2
3
4
,0, ¯z
1
4
,0, z +
1
2
hkl : l = 2n
4 c . 2 . 0,y,
1
4
1
2
, ¯y,
1
4
0, ¯y,
3
4
1
2
,y,
3
4
hkl : h + l = 2n
4 b
¯
10,
1
2
,0
1
2
,
1
2
,00,
1
2
,
1
2
1
2
,
1
2
,
1
2
hkl : h,l = 2n
4 a
¯
10, 0,0
1
2
,0, 00,0,
1
2
1
2
,0,
1
2
hkl : h,l = 2n
Symmetry of special projections
Along [001] p2mm
a
=
1
2
ab
= b
Origin at 0,0,z
Along [100] p2mm
a
= bb
=
1
2
c
Origin at x,0,0
Along [010] p2gm
a
=
1
2
cb
= a
Origin at 0,y, 0
Maximal non-isomorphic subgroups
I
[2] Pc2a (Pba2, 32) 1; 3; 6; 8
[2] P2
1
ca (Pca2
1
, 29) 1; 4; 6; 7
[2] Pcc2 (27) 1; 2; 7; 8
[2] P2
1
22(P222
1
, 17) 1; 2; 3; 4
[2] P2
1
/c11 (P2
1
/c, 14) 1; 4; 5; 8
[2] P112/a (P2/ c, 13) 1; 2; 5; 6
[2] P12/c1(P2/c, 13) 1; 3; 5; 7
IIa none
IIb [2] Pnca (b
= 2b)(Pbcn, 60); [2] Pccn(b
= 2b) (56); [2] Pncn (b
= 2b)(Pnna, 52)
Maximal isomorphic subgroups of lowest index
IIc
[2] Pcca (b
= 2b) (54); [3] Pcca (a
= 3a) (54); [3] Pcca (c
= 3c) (54)
Minimal non-isomorphic supergroups
I
none
II [2] Aema (Cmce, 64); [2] Bmem (Cmme, 67); [2] Ccce (68); [2] Ibca(73); [2] Pccm (a
=
1
2
a) (49); [2] Pmma (c
=
1
2
c) (51)
281