Hahn Th. (ed.) International tables for crystallography. Vol. A. Space-group symmetry

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P222

222 Orthorhombic

No. 17 P222

Patterson symmetry Pmmm

Origin at 212

Asymmetric unit 0 ≤ x ≤

;0≤ y ≤

;0≤ z ≤ 1

Symmetry operations

(1) 1 (2) 2(0, 0,

) 0,0,z (3) 2 0,y,

(4) 2 x , 0,0

202

International Tables for Crystallography (2006). Vol. A, Space group 17, pp. 202–203.

CONTINUED No. 17 P222

Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3)

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates Reﬂection conditions

General:

4 e 1(1)x,y,z (2) ¯x, ¯y,z +

(3) ¯x,y, ¯z +

(4) x, ¯y, ¯z 00l : l = 2n

Special: as above, plus

2 d . 2 .

,y,

, ¯y,

h0l : l = 2n

2 c . 2 . 0,y,

0, ¯y,

h0l : l = 2n

2 b 2 .. x,

,0¯x,

0kl : l = 2n

2 a 2 .. x, 0,0¯x,0,

0kl : l = 2n

Symmetry of special projections

Along [001] p2mm



= ab



= b

Origin at 0,0,z

Along [100] p2gm



= bb



= c

Origin at x,0,0

Along [010] p2mg



= cb



= a

Origin at 0,y,

Maximal non-isomorphic subgroups

[2] P112

(P2

,4) 1; 2

[2] P121 (P2, 3) 1; 3

[2] P211 (P2, 3) 1; 4

IIa none

IIb [2] P2



= 2a)(P2

2, 18); [2] P22



= 2b)(P2

2, 18); [2]C 222



= 2a,b



= 2b) (20)

Maximal isomorphic subgroups of lowest index

IIc

[2] P222



= 2a or b



= 2b) (17); [3] P222



= 3c) (17)

Minimal non-isomorphic supergroups

[2] Pmma (51); [2] Pnna (52); [2] Pmna (53); [2] Pcca (54); [2] P4

22 (91); [2] P4

22 (95)

II [2] C 222

(20); [2] A222 (C222, 21); [2] B222(C222, 21); [2] I 2

(24); [2] P222(c



c) (16)

203

2 D

222 Orthorhombic

No. 18 P2

2 Patterson symmetry Pmmm

Origin at intersection of 2 with perpendicular plane co ntaining 2

axes

Asymmetric unit 0 ≤ x ≤

;0≤ y ≤

;0≤ z ≤ 1

Symmetry operations

(1) 1 (2) 2 0,0,z (3) 2(0,

,0)

,y,0(4)2(

,0, 0) x,

204

International Tables for Crystallography (2006). Vol. A, Space group 18, pp. 204–205.

CONTINUED No. 18 P2

Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3)

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates Reﬂection conditions

General:

4 c 1(1)x,y,z (2) ¯x, ¯y,z (3) ¯x+

,y +

, ¯z (4) x +

, ¯y +

, ¯zh00 : h = 2n

0k0: k = 2n

Special: as above, plus

2 b ..20,

,0, ¯zhk0: h + k = 2n

2 a ..20,0,z

, ¯zhk0: h + k = 2n

Symmetry of special projections

Along [001] p2gg



= ab



= b

Origin at 0,0,z

Along [100] p2mg



= bb



= c

Origin at x,

Along [010] p2gm



= cb



= a

Origin at

,y,0

Maximal non-isomorphic subgroups

[2] P12

1(P2

,4) 1; 3

[2] P2

11(P2

,4) 1; 4

[2] P112 (P2, 3) 1; 2

IIa none

IIb [2] P2



= 2c) (19)

Maximal isomorphic subgroups of lowest index

IIc

[2] P2

2(c



= 2c) (18); [3] P2

2(a



= 3a or b



= 3b) (18)

Minimal non-isomorphic supergroups

[2] Pbam(55); [2] Pccn (56); [2] Pbcm(57); [2] Pnnm(58); [2] Pmmn (59); [2] Pbcn(60); [2] P42

2 (90); [2] P4

2 (94);

[2] P

m (113); [2] P

c (114)

II [2] A2

22(C222

, 20); [2] B22

2(C 222

, 20); [2]C 222 (21); [2] I 222 (23); [2] P22

2(a



a)(P222

, 17);

[2] P2

22(b



b)(P222

, 17)

205

222 Orthorhombic

No. 19 P2

Patterson symmetry Pmmm

Origin at midpoint of three non-intersecting pairs of parallel 2

axes

Asymmetric unit 0 ≤ x ≤

;0≤ y ≤

;0≤ z ≤ 1

Symmetry operations

(1) 1 (2) 2(0, 0,

)

,0, z (3) 2(0,

,0) 0, y,

(4) 2(

,0, 0) x,

206

International Tables for Crystallography (2006). Vol. A, Space group 19, pp. 206–207.

CONTINUED No. 19 P2

Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3)

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates Reﬂection conditions

General:

4 a 1(1)x, y,z (2) ¯x +

, ¯y, z +

(3) ¯x,y +

, ¯z +

(4) x +

, ¯y +

, ¯zh00 : h = 2n

0k0: k = 2n

00l : l = 2n

Symmetry of special projections

Along [001] p2gg



= ab



= b

Origin at

,0, z

Along [100] p2gg



= bb



= c

Origin at x,

Along [010] p2gg



= cb



= a

Origin at 0,y,

Maximal non-isomorphic subgroups

[2] P112

(P2

,4) 1; 2

[2] P12

1(P2

,4) 1; 3

[2] P2

11(P2

,4) 1; 4

IIa none

IIb none

Maximal isomorphic subgroups of lowest index

IIc

[3] P2



= 3a or b



= 3b or c



= 3c) (19)

Minimal non-isomorphic supergroups

[2] Pbca (61); [2] Pnma (62); [2] P4

2 (92); [2] P4

2 (96); [3] P2

3 (198)

II [2] A2

22(C222

, 20); [2] B22

2(C 222

, 20); [2]C 222

(20); [2] I 2

(24); [2] P22



a)(P2

2, 18);

[2] P2



b)(P2

2, 18); [2] P2

2(c



c) (18)

207

C222

222 Orthorhombic

No. 20 C222

Patterson symmetry Cmmm

Origin at 212

Asymmetric unit 0 ≤ x ≤

;0≤ y ≤

;0≤ z ≤

Symmetry operations

For (0,0,0)+ set

(1) 1 (2) 2(0,0,

) 0,0,z (3) 2 0,y,

(4) 2 x,0,0

For (

,0)+ set

(1) t(

,0) (2) 2(0,0,

)

,z (3) 2(0,

,0)

,y,

(4) 2(

,0, 0) x,

208

International Tables for Crystallography (2006). Vol. A, Space group 20, pp. 208–209.

CONTINUED No. 20 C 222

Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(

,0); (2); (3)

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates

(0,0, 0)+ (

,0)+

Reﬂection conditions

General:

8 c 1(1)x,y,z (2) ¯x, ¯y,z +

(3) ¯x,y, ¯z +

(4) x, ¯y, ¯zhkl: h + k = 2n

0kl : k = 2n

h0l : h = 2n

hk0: h + k = 2n

h00 : h = 2n

0k0: k = 2n

00l : l = 2n

Special: as above, plus

4 b . 2 . 0,y,

0, ¯y,

h0l : l = 2n

4 a 2 .. x, 0,0¯x,0,

0kl : l = 2n

Symmetry of special projections

Along [001] c2mm



= ab



= b

Origin at 0,0,z

Along [100] p2gm



= c

Origin at x,0,0

Along [010] p2mg



= cb



Origin at 0, y,

Maximal non-isomorphic subgroups

[2] C 121(C2, 5) (1; 3)+

[2] C 211(C2, 5) (1; 4)+

[2] C 112

(P2

,4) (1; 2)+

IIa [2] P2

(19) 1; 2; (3; 4 )+(

,0)

[2] P2

(P2

2, 18) 1; 3; (2; 4)+(

,0)

[2] P22

(P2

2, 18) 1; 4; (2; 3)+(

,0)

[2] P222

(17) 1; 2; 3; 4

IIb none

Maximal isomorphic subgroups of lowest index

IIc

[3] C 222



= 3a or b



= 3b) (20); [3]C 222



= 3c) (20)

Minimal non-isomorphic supergroups

[2] Cmcm(63); [2]Cmce(64); [2] P4

22 (91); [2] P4

2 (92); [2] P4

22 (95); [2] P4

2 (96); [3] P6

22 (178);

[3] P6

22 (179); [3] P6

22 (182)

II [2] F 222 (22); [2] P222



a,b



b) (17); [2] C222(c



c) (21)

209

C222 D

222 Orthorhombic

No. 21 C222

Patterson symmetry Cmmm

Origin at 222

Asymmetric unit 0 ≤ x ≤

;0≤ y ≤

;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

For (

,0)+ set

(1) t(

,0) (2) 2

,z (3) 2(0,

,0)

,y,0(4)2(

,0, 0) x,

210

International Tables for Crystallography (2006). Vol. A, Space group 21, pp. 210–211.

CONTINUED No. 21 C222

Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); t(

,0); (2); (3)

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates

(0,0, 0)+ (

,0)+

Reﬂection conditions

General:

8 l 1(1)x,y,z (2) ¯x, ¯y,z (3) ¯x,y, ¯z (4) x, ¯y, ¯zhkl: h + k = 2n

0kl : k = 2n

h0l : h = 2n

hk0: h + k = 2n

h00 : h = 2n

0k0: k = 2n

Special: as above, plus

4 k ..2

, ¯zhk0: h = 2n

4 j ..20,

,z 0,

, ¯z no extra conditions

4 i ..20,0, z 0,0, ¯z no extra conditions

4 h . 2 . 0, y,

0, ¯y,

no extra conditions

4 g . 2 . 0, y,00, ¯y, 0 no extra conditions

4 f 2 .. x,0,

¯x,0,

no extra conditions

4 e 2 .. x,0,0¯x, 0,0 no extra conditions

2 d 222 0,0,

no extra conditions

2 c 222

,0,

no extra conditions

2 b 222 0,

,0 no extra conditions

2 a 222 0,0,0 no extra conditions

Symmetry of special projections

Along [001] c2mm



= ab



= b

Origin at 0,0,z

Along [100] p2mm



= c

Origin at x,0,0

Along [010] p2mm



= cb



Origin at 0,y, 0

Maximal non-isomorphic subgroups

[2] C 121(C2, 5) (1; 3)+

[2] C 211(C2, 5) (1; 4)+

[2] C 112(P2, 3) (1; 2)+

IIa [2] P2

2 (18) 1; 2; (3; 4)+(

,0)

[2] P2

22(P222

, 17) 1; 3; (2; 4)+(

,0)

[2] P22

2(P222

, 17) 1; 4; (2; 3)+(

,0)

[2] P222 (16) 1; 2; 3; 4

IIb [2] I 2



= 2c) (24); [2] I 222 (c



= 2c) (23); [2] C222



= 2c) (20)

Maximal isomorphic subgroups of lowest index

IIc

[2] C 222(c



= 2c) (21); [3] C222 (a



= 3a or b



= 3b) (21)

Minimal non-isomorphic supergroups

[2] Cmmm (65); [2] Cccm (66); [2] Cmme (67); [2] Ccce (68); [2] P422 (89); [2] P42

2 (90); [2] P4

22 (93); [2] P4

2 (94);

[2] P

4m2 (115); [2] P

4c2 (116); [2] P

4b2 (117); [2] P

4n2 (118); [3] P622 (177); [3] P6

22 (180); [3] P6

22 (181)

II [2] F 222 (22); [2] P222(a



a,b



b) (16)

211