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

P4

2

/ncm D

16

4h

4/mmm Tetragonal

No. 138 P 4

2

/n 2

1

/c 2/m Patterson symmetry P4/mmm

ORIGIN CHOICE 1

Origin at

¯

4cg,at−

1

4

,

1

4

,−

1

4

from centre (2/m)

Asymmetric unit 0 ≤ x ≤

1

4

;0≤ y ≤

1

2

;0≤ z ≤ 1; x ≤ y; y ≤

1

2

− x

Symmetry operations

(1) 1 (2) 2 0, 0,z (3) 4

+

(0,0,

1

2

) 0,

1

2

,z (4) 4

−

(0,0,

1

2

)

1

2

,0, z

(5) 2(0,

1

2

,0)

1

4

,y,0(6)2(

1

2

,0, 0) x,

1

4

,0(7)2x,x,

1

4

(8) 2 x , ¯x,

1

4

(9)

¯

1

4

,

1

4

,

1

4

(10) n(

1

2

,

1

2

,0) x,y,

1

4

(11)

¯

4

+

0,0,z;0,0, 0 (12)

¯

4

−

0,0,z;0,0,0

(13) cx,0,z (14) c 0,y,z (15) mx+

1

2

, ¯x, z (16) g(

1

2

,

1

2

,0) x,x,z

474

International Tables for Crystallography (2006). Vol. A, Space group 138, pp. 474–477.

CONTINUED No. 138 P4

2

/ncm

Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5); (9)

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates Reﬂection conditions

General:

16 j 1(1)x, y,z (2) ¯x, ¯y, z (3) ¯y +

1

2

,x +

1

2

,z +

1

2

(4) y +

1

2

, ¯x +

1

2

,z +

1

2

(5) ¯x+

1

2

,y +

1

2

, ¯z (6) x +

1

2

, ¯y +

1

2

, ¯z (7) y,x, ¯z +

1

2

(8) ¯y, ¯x, ¯z +

1

2

(9) ¯x+

1

2

, ¯y +

1

2

, ¯z +

1

2

(10) x +

1

2

,y +

1

2

, ¯z +

1

2

(11) y, ¯x, ¯z (12) ¯y,x, ¯z

(13) x, ¯y,z +

1

2

(14) ¯x,y, z +

1

2

(15) ¯y +

1

2

, ¯x +

1

2

,z (16) y +

1

2

,x +

1

2

,z

hk0: h + k = 2n

0kl : l = 2n

00l : l = 2n

h00 : h = 2n

Special: as above, plus

8 i ..mx,x +

1

2

,z ¯x, ¯x +

1

2

,z ¯x,x +

1

2

,z +

1

2

x, ¯x +

1

2

,z +

1

2

¯x+

1

2

,x, ¯zx+

1

2

, ¯x, ¯zx+

1

2

,x, ¯z +

1

2

¯x +

1

2

, ¯x, ¯z +

1

2

no extra conditions

8 h ..2 x,x,

3

4

¯x, ¯x,

3

4

¯x+

1

2

,x +

1

2

,

1

4

x +

1

2

, ¯x +

1

2

,

1

4

¯x+

1

2

, ¯x +

1

2

,

3

4

x +

1

2

,x +

1

2

,

3

4

x, ¯x,

1

4

¯x,x,

1

4

hkl : h + k = 2n

8 g ..2 x,x,

1

4

¯x, ¯x,

1

4

¯x+

1

2

,x +

1

2

,

3

4

x +

1

2

, ¯x +

1

2

,

3

4

¯x+

1

2

, ¯x +

1

2

,

1

4

x +

1

2

,x +

1

2

,

1

4

x, ¯x,

3

4

¯x,x,

3

4

hkl : h + k = 2n

8 f 2 .. 0,0,z

1

2

,

1

2

,z +

1

2

1

2

,

1

2

, ¯z 0,0, ¯z +

1

2

1

2

,

1

2

, ¯z +

1

2

0,0, ¯z 0 , 0,z +

1

2

1

2

,

1

2

,z

hkl : h + k,l = 2n

4 e 2 . mm 0,

1

2

,z 0,

1

2

,z +

1

2

1

2

,0, ¯z

1

2

,0, ¯z +

1

2

hkl : l = 2n

4 d ..2/m

1

4

,

1

4

,

3

4

3

4

,

3

4

,

3

4

1

4

,

3

4

,

1

4

3

4

,

1

4

,

1

4

hkl : h + k,h + l,k + l = 2n

4 c ..2/m

1

4

,

1

4

,

1

4

3

4

,

3

4

,

1

4

1

4

,

3

4

,

3

4

3

4

,

1

4

,

3

4

hkl : h + k,h + l,k + l = 2n

4 b

¯

4 .. 0, 0,0

1

2

,

1

2

,

1

2

1

2

,

1

2

,00, 0,

1

2

hkl : h + k,l = 2n

4 a 2 . 22 0,0,

1

4

1

2

,

1

2

,

3

4

1

2

,

1

2

,

1

4

0,0,

3

4

hkl : h + k,l = 2n

Symmetry of special projections

Along [001] p4mm

a



=

1

2

(a − b) b



=

1

2

(a + b)

Origin at 0,0,z

Along [100] p2mg

a



= bb



=

1

2

c

Origin at x,

1

4

,0

Along [110] p2mm

a



=

1

2

(−a + b) b



= c

Origin at x, x,

1

4

Maximal non-isomorphic subgroups

I

[2] P

¯

4c2 (116) 1; 2; 7; 8; 11; 12; 13; 14

[2] P

¯

42

1

m (113) 1; 2; 5; 6; 11; 12; 15; 16

[2] P4

2

cm (101) 1; 2; 3; 4; 13; 14; 15; 16

[2] P4

2

1

2 (94) 1; 2; 3; 4; 5; 6; 7; 8

[2] P4

2

/n11(P4

2

/n, 86) 1; 2; 3; 4; 9; 10; 11; 12

[2] P2/n12/m (Cmme, 67) 1; 2; 7; 8; 9; 10; 15; 16

[2] P2/n2

1

/c1(Pccn, 56) 1; 2; 5; 6; 9; 10; 13; 14

IIa none

IIb none

Maximal isomorphic subgroups of lowest index

IIc

[3] P4

2

/ncm( c



= 3c) (138); [9] P4

2

/ncm(a



= 3a,b



= 3b) (138)

Minimal non-isomorphic supergroups

I

none

II [2] C 4

2

/mcm (P4

2

/mmc, 131); [2] I 4 /mcm(140); [2] P4/nmm(c



=

1

2

c) (129)

475

P4

2

/ncm D

16

4h

4/mmm Tetragonal

No. 138 P 4

2

/n 2

1

/c 2/m Patterson symmetry P4/mmm

ORIGIN CHOICE 2

Origin at centre (2/m) at n1(2/m,2

1

/g),at

1

4

,−

1

4

,

1

4

from

¯

4

Asymmetric unit −

1

4

≤ x ≤

1

4

; −

1

4

≤ y ≤

1

4

;0≤ z ≤

1

2

; x ≤ y

Symmetry operations

(1) 1 (2) 2

1

4

,

1

4

,z (3) 4

+

(0,0,

1

2

)

1

4

,

1

4

,z (4) 4

−

(0,0,

1

2

)

1

4

,

1

4

,z

(5) 2(0,

1

2

,0) 0,y,

1

4

(6) 2(

1

2

,0, 0) x,0,

1

4

(7) 2(

1

2

,

1

2

,0) x, x,0(8)2x, ¯x,0

(9)

¯

10, 0,0 (10) n(

1

2

,

1

2

,0) x,y,0 (11)

¯

4

+

1

4

,−

1

4

,z;

1

4

,−

1

4

,

1

4

(12)

¯

4

−

1

4

,

1

4

,z; −

1

4

,

1

4

,

1

4

(13) cx,

1

4

,z (14) c

1

4

,y,z (15) mx+

1

2

, ¯x, z (16) mx,x, z

476

CONTINUED No. 138 P4

2

/ncm

Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5); (9)

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates Reﬂection conditions

General:

16 j 1(1)x,y,z (2) ¯x+

1

2

, ¯y +

1

2

,z (3) ¯y +

1

2

,x,z +

1

2

(4) y, ¯x +

1

2

,z +

1

2

(5) ¯x,y +

1

2

, ¯z +

1

2

(6) x +

1

2

, ¯y, ¯z +

1

2

(7) y +

1

2

,x +

1

2

, ¯z (8) ¯y, ¯x, ¯z

(9) ¯x, ¯y, ¯z (10) x +

1

2

,y +

1

2

, ¯z (11) y +

1

2

, ¯x, ¯z +

1

2

(12) ¯y,x +

1

2

, ¯z +

1

2

(13) x, ¯y +

1

2

,z +

1

2

(14) ¯x +

1

2

,y,z +

1

2

(15) ¯y +

1

2

, ¯x +

1

2

,z (16) y,x,z

hk0: h + k = 2n

0kl : l = 2n

00l : l = 2n

h00 : h = 2n

Special: as above, plus

8 i ..mx,x,z ¯x +

1

2

, ¯x +

1

2

,z ¯x +

1

2

,x,z +

1

2

x, ¯x +

1

2

,z +

1

2

¯x,x +

1

2

, ¯z +

1

2

x +

1

2

, ¯x, ¯z +

1

2

x +

1

2

,x +

1

2

, ¯z ¯x, ¯x, ¯z

no extra conditions

8 h ..2 x, ¯x, 0¯x +

1

2

,x +

1

2

,0 x +

1

2

,x,

1

2

¯x, ¯x +

1

2

,

1

2

¯x,x,0 x+

1

2

, ¯x +

1

2

,0¯x +

1

2

, ¯x,

1

2

x,x +

1

2

,

1

2

hkl : h + k = 2n

8 g ..2 x, ¯x,

1

2

¯x +

1

2

,x +

1

2

,

1

2

x +

1

2

,x,0¯x, ¯x +

1

2

,0

¯x,x,

1

2

x +

1

2

, ¯x +

1

2

,

1

2

¯x +

1

2

, ¯x, 0 x,x +

1

2

,0

hkl : h + k = 2n

8 f 2 ..

3

4

,

1

4

,z

1

4

,

3

4

,z +

1

2

1

4

,

3

4

, ¯z +

1

2

3

4

,

1

4

, ¯z

1

4

,

3

4

, ¯z

3

4

,

1

4

, ¯z +

1

2

3

4

,

1

4

,z +

1

2

1

4

,

3

4

,z

hkl : h + k,l = 2n

4 e 2 . mm

1

4

,

1

4

,z

1

4

,

1

4

,z +

1

2

3

4

,

3

4

, ¯z +

1

2

3

4

,

3

4

, ¯zhkl: l = 2n

4 d ..2/m 0,0, 0

1

2

,

1

2

,0

1

2

,0,

1

2

0,

1

2

,

1

2

hkl : h + k,h + l,k + l = 2n

4 c ..2/m 0,0,

1

2

1

2

,

1

2

,

1

2

1

2

,0, 00,

1

2

,0 hkl : h + k,h + l, k + l = 2n

4 b

¯

4 ..

3

4

,

1

4

,

3

4

1

4

,

3

4

,

1

4

1

4

,

3

4

,

3

4

3

4

,

1

4

,

1

4

hkl : h + k,l = 2n

4 a 2 . 22

3

4

,

1

4

,0

1

4

,

3

4

,

1

2

1

4

,

3

4

,0

3

4

,

1

4

,

1

2

hkl : h + k,l = 2n

Symmetry of special projections

Along [001] p4mm

a



=

1

2

(a − b) b



=

1

2

(a + b)

Origin at

1

4

,

1

4

,z

Along [100] p2mg

a



= bb



=

1

2

c

Origin at x,0,0

Along [110] p2mm

a



=

1

2

(−a + b) b



= c

Origin at x, x,0

Maximal non-isomorphic subgroups

I

[2] P

¯

4c2 (116) 1; 2; 7; 8; 11; 12; 13; 14

[2] P

¯

42

1

m (113) 1; 2; 5; 6; 11; 12; 15; 16

[2] P4

2

cm (101) 1; 2; 3; 4; 13; 14; 15; 16

[2] P4

2

1

2 (94) 1; 2; 3; 4; 5; 6; 7; 8

[2] P4

2

/n11(P4

2

/n, 86) 1; 2; 3; 4; 9; 10; 11; 12

[2] P2/n12/m (Cmme, 67) 1; 2; 7; 8; 9; 10; 15; 16

[2] P2/n2

1

/c1(Pccn, 56) 1; 2; 5; 6; 9; 10; 13; 14

IIa none

IIb none

Maximal isomorphic subgroups of lowest index

IIc

[3] P4

2

/ncm( c



= 3c) (138); [9] P4

2

/ncm(a



= 3a,b



= 3b) (138)

Minimal non-isomorphic supergroups

I

none

II [2] C 4

2

/mcm (P4

2

/mmc, 131); [2] I 4 /mcm(140); [2] P4/nmm(c



=

1

2

c) (129)

477

I 4/mmm D

17

4h

4/mmm Tetragonal

No. 139 I 4/m 2/m 2/m

Patterson symmetry I 4/mmm

Origin at centre (4/mmm)

Asymmetric unit 0 ≤ x ≤

1

2

;0≤ y ≤

1

2

;0≤ z ≤

1

4

; x ≤ y

Symmetry operations

For (0,0,0)+ set

(1) 1 (2) 2 0,0,z (3) 4

+

0,0,z (4) 4

−

0,0,z

(5) 2 0,y, 0(6)2x,0, 0(7)2x,x, 0(8)2x, ¯x,0

(9)

¯

10, 0,0 (10) mx, y,0 (11)

¯

4

+

0,0,z;0,0,0 (12)

¯

4

−

0,0,z;0,0, 0

(13) mx,0,z (14) m 0, y,z (15) mx, ¯x,z (16) mx,x,z

For (

1

2

,

1

2

,

1

2

)+ set

(1) t(

1

2

,

1

2

,

1

2

) (2) 2(0,0,

1

2

)

1

4

,

1

4

,z (3) 4

+

(0,0,

1

2

) 0,

1

2

,z (4) 4

−

(0,0,

1

2

)

1

2

,0, z

(5) 2(0,

1

2

,0)

1

4

,y,

1

4

(6) 2(

1

2

,0, 0) x,

1

4

,

1

4

(7) 2(

1

2

,

1

2

,0) x,x,

1

4

(8) 2 x, ¯x +

1

2

,

1

4

(9)

¯

1

4

,

1

4

,

1

4

(10) n(

1

2

,

1

2

,0) x,y,

1

4

(11)

¯

4

+

1

2

,0, z;

1

2

,0,

1

4

(12)

¯

4

−

0,

1

2

,z;0,

1

2

,

1

4

(13) n(

1

2

,0,

1

2

) x,

1

4

,z (14) n(0,

1

2

,

1

2

)

1

4

,y,z (15) cx+

1

2

, ¯x, z (16) n(

1

2

,

1

2

,

1

2

) x,x,z

Maximal non-isomorphic subgroups (continued)

IIa

[2] P4

2

/nmc(137) 1; 2; 7; 8; 11; 12; 13; 14; (3; 4; 5; 6; 9; 10; 15; 16)+(

1

2

,

1

2

,

1

2

)

[2] P4

2

/mnm (136) 1; 2; 7; 8; 9; 10; 15; 16; (3; 4; 5; 6; 11; 12; 13; 14)+(

1

2

,

1

2

,

1

2

)

[2] P4

2

/nnm(134) 1; 2; 5; 6; 11; 12; 15; 16; (3; 4; 7; 8; 9; 10; 13; 14)+(

1

2

,

1

2

,

1

2

)

[2] P4

2

/mmc (131) 1; 2; 5; 6; 9; 10; 13; 14; (3; 4; 7; 8; 11; 12; 15; 16)+(

1

2

,

1

2

,

1

2

)

[2] P4/nmm(129) 1; 2; 3; 4; 13; 14; 15; 16; (5; 6; 7; 8; 9; 10; 11; 12)+(

1

2

,

1

2

,

1

2

)

[2] P4/mnc(128) 1; 2; 3; 4; 9; 10; 11; 12; (5; 6; 7; 8; 13; 14; 15; 16)+(

1

2

,

1

2

,

1

2

)

[2] P4/nnc(126) 1; 2; 3; 4; 5; 6; 7; 8; (9; 10; 11; 12; 13; 14; 15; 16)+(

1

2

,

1

2

,

1

2

)

[2] P4/mmm(123) 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16

IIb none

Maximal isomorphic subgroups of lowest index

IIc

[3] I 4/mmm(c



= 3c) (139); [9] I 4/mmm(a



= 3a,b



= 3b) (139)

Minimal non-isomorphic supergroups

I

[3] Fm

¯

3m (225); [3] Im

¯

3m (229)

II [2] C 4/mmm (c



=

1

2

c)(P4/mmm, 123)

478

International Tables for Crystallography (2006). Vol. A, Space group 139, pp. 478–479.

CONTINUED No. 139 I 4/mmm

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

1

2

,

1

2

,

1

2

); (2); (3); (5); (9)

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates

(0,0, 0)+ (

1

2

,

1

2

,

1

2

)+

Reﬂection conditions

General:

32 o 1(1)x,y, z (2) ¯x, ¯y,z (3) ¯y,x,z (4) y, ¯x,z

(5) ¯x,y, ¯z (6) x, ¯y, ¯z (7) y, x, ¯z (8) ¯y, ¯x, ¯z

(9) ¯x, ¯y, ¯z (10) x, y, ¯z (11) y, ¯x, ¯z (12) ¯y,x, ¯z

(13) x, ¯y,z (14) ¯x

,y,z (15) ¯y, ¯x,z (16) y,x,z

hkl : h + k + l = 2n

hk0: h + k = 2n

0kl : k + l = 2n

hhl : l = 2n

00l : l = 2n

h00 : h = 2n

Special: as above, plus

16 n . m . 0,y, z 0, ¯y,z ¯y,0, zy, 0,z

0,y, ¯z 0, ¯y,

¯zy, 0, ¯z ¯y,0, ¯z

no extra conditions

16 m ..mx, x,z ¯x, ¯x , z ¯x,x,zx, ¯x,z

¯x,x, ¯zx, ¯x, ¯zx, x, ¯z ¯x, ¯x, ¯z

no extra conditions

16 lm.. x,y,0¯x, ¯y,0¯y,x,0 y, ¯x, 0

¯x,y,0 x ,

¯y,0 y, x,0¯y, ¯x,0

no extra conditions

16 k ..2 x,x +

1

2

,

1

4

¯x, ¯x +

1

2

,

1

4

¯x+

1

2

,x,

1

4

x +

1

2

, ¯x,

1

4

¯x, ¯x +

1

2

,

3

4

x,x +

1

2

,

3

4

x +

1

2

, ¯x,

3

4

¯x +

1

2

,x,

3

4

hkl : l = 2n

8 jm2 m. x,

1

2

,0¯x,

1

2

,0

1

2

,x,0

1

2

, ¯x, 0 no extra conditions

8 im2 m. x, 0,0¯x,0,00,x, 00, ¯x,0 no extra conditions

8 hm. 2mx,x, 0¯x, ¯x,0¯x,x,0 x, ¯x, 0 no extra conditions

8 g 2 mm. 0,

1

2

,z

1

2

,0, z 0,

1

2

, ¯z

1

2

,0, ¯zhkl: l = 2n

8 f ..2/m

1

4

,

1

4

,

1

4

3

4

,

3

4

,

1

4

3

4

,

1

4

,

1

4

1

4

,

3

4

,

1

4

hkl : k,l = 2n

4 e 4 mm 0, 0,z 0,0, ¯z no extra conditions

4 d

¯

4 m 20,

1

2

,

1

4

1

2

,0,

1

4

hkl : l = 2n

4 cmmm. 0,

1

2

,0

1

2

,0, 0 hkl : l = 2n

2 b 4/ mmm 0,0,

1

2

no extra conditions

2 a 4/ mmm 0,0,0 no extra conditions

Symmetry of special projections

Along [001] p4mm

a



=

1

2

(a − b) b



=

1

2

(a + b)

Origin at 0,0,z

Along [100] c2mm

a



= bb



= c

Origin at x,0,0

Along [110] p2mm

a



=

1

2

(−a + b) b



=

1

2

c

Origin at x,x , 0

Maximal non-isomorphic subgroups

I

[2] I

¯

42m (121) (1; 2; 5; 6; 11; 12; 15; 16)+

[2] I

¯

4m2 (119) (1; 2; 7; 8; 11; 12; 13; 14)+

[2] I 4mm (107) (1; 2; 3; 4; 13; 14; 15; 16)+

[2] I 422(97) (1; 2; 3; 4; 5; 6; 7; 8)+

[2] I 4/m11(I 4/m, 87) (1; 2; 3; 4; 9; 10; 11; 12)+

[2] I 2/m2/m1(Immm, 71) (1; 2; 5; 6; 9; 10; 13; 14)+

[2] I 2/m12/m (F mmm, 69) (1; 2; 7; 8; 9; 10; 15; 16)+

(Continued on preceding page)

479

I 4/mcm D

18

4h

4/mmm Tetragonal

No. 140 I 4/m 2/c 2/m

Patterson symmetry I 4/mmm

Origin at centre (4/m) at 4/mc2

1

/e

Asymmetric unit 0 ≤ x ≤

1

2

;0≤ y ≤

1

2

;0≤ z ≤

1

4

; y ≤

1

2

− x

Symmetry operations

For (0,0,0)+ set

(1) 1 (2) 2 0, 0,z (3) 4

+

0,0,z (4) 4

−

0,0,z

(5) 2 0,y,

1

4

(6) 2 x,0,

1

4

(7) 2 x,x,

1

4

(8) 2 x , ¯x,

1

4

(9)

¯

10, 0,0 (10) mx,y,0 (11)

¯

4

+

0,0,z;0,0, 0 (12)

¯

4

−

0,0,z;0,0,0

(13) cx,0,z (14) c 0,y,z (15) cx, ¯x,z (16) cx, x,z

For (

1

2

,

1

2

,

1

2

)+ set

(1) t(

1

2

,

1

2

,

1

2

) (2) 2(0, 0,

1

2

)

1

4

,

1

4

,z (3) 4

+

(0,0,

1

2

) 0,

1

2

,z (4) 4

−

(0,0,

1

2

)

1

2

,0, z

(5) 2(0,

1

2

,0)

1

4

,y,0(6)2(

1

2

,0, 0) x,

1

4

,0(7)2(

1

2

,

1

2

,0) x, x,0(8)2x, ¯x +

1

2

,0

(9)

¯

1

4

,

1

4

,

1

4

(10) n(

1

2

,

1

2

,0) x,y,

1

4

(11)

¯

4

+

1

2

,0, z;

1

2

,0,

1

4

(12)

¯

4

−

0,

1

2

,z;0,

1

2

,

1

4

(13) ax,

1

4

,z (14) b

1

4

,y,z (15) mx+

1

2

, ¯x, z (16) g(

1

2

,

1

2

,0) x,x,z

Maximal non-isomorphic subgroups (continued)

IIa

[2] P4

2

/ncm(138) 1; 2; 7; 8; 11; 12; 13; 14; (3; 4; 5; 6; 9; 10; 15; 16)+(

1

2

,

1

2

,

1

2

)

[2] P4

2

/mbc (135) 1; 2; 7; 8; 9; 10; 15; 16; (3; 4; 5; 6; 11; 12; 13; 14)+(

1

2

,

1

2

,

1

2

)

[2] P4

2

/nbc(133) 1; 2; 5; 6; 11; 12; 15; 16; (3; 4; 7; 8; 9; 10; 13; 14)+(

1

2

,

1

2

,

1

2

)

[2] P4

2

/mcm (132) 1; 2; 5; 6; 9; 10; 13; 14; (3; 4; 7; 8; 11; 12; 15; 16)+(

1

2

,

1

2

,

1

2

)

[2] P4/ncc(130) 1; 2; 3; 4; 13; 14; 15; 16; (5; 6; 7; 8; 9; 10; 11; 12)+(

1

2

,

1

2

,

1

2

)

[2] P4/mbm(127) 1; 2; 3; 4; 9; 10; 11; 12; (5; 6; 7; 8; 13; 14; 15; 16)+(

1

2

,

1

2

,

1

2

)

[2] P4/nbm(125) 1; 2; 3; 4; 5; 6; 7; 8; (9; 10; 11; 12; 13; 14; 15; 16)+(

1

2

,

1

2

,

1

2

)

[2] P4/mcc(124) 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16

IIb none

Maximal isomorphic subgroups of lowest index

IIc

[3] I 4/mcm(c



= 3c) (140); [9] I 4/mcm(a



= 3a,b



= 3b) (140)

Minimal non-isomorphic supergroups

I

[3] Fm

¯

3c (226)

II [2] C 4/mmm (c



=

1

2

c)(P4/mmm, 123)

480

International Tables for Crystallography (2006). Vol. A, Space group 140, pp. 480–481.

CONTINUED No. 140 I 4/mcm

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

1

2

,

1

2

,

1

2

); (2); (3); (5); (9)

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates

(0,0, 0)+ (

1

2

,

1

2

,

1

2

)+

Reﬂection conditions

General:

32 m 1(1)x,y,z (2) ¯x, ¯y,z (3) ¯y,x,z (4) y, ¯x,z

(5) ¯x,y, ¯z +

1

2

(6) x, ¯y, ¯z +

1

2

(7) y,x, ¯z +

1

2

(8) ¯y, ¯x, ¯z +

1

2

(9) ¯x, ¯y, ¯z (10) x,y, ¯z (11) y, ¯x, ¯z (12) ¯y,x, ¯z

(13) x, ¯y,z +

1

2

(14) ¯x,y, z +

1

2

(15) ¯y, ¯x,z +

1

2

(16) y,x , z +

1

2

hkl : h + k + l = 2n

hk0: h + k = 2n

0kl : k,l = 2n

hhl : l = 2n

00l : l = 2n

h00 : h = 2n

Special: as above, plus

16 l ..mx,x +

1

2

,z ¯x, ¯x +

1

2

,z ¯x +

1

2

,x,zx+

1

2

, ¯x, z

¯x,x +

1

2

, ¯z +

1

2

x, ¯x +

1

2

, ¯z +

1

2

x +

1

2

,x, ¯z +

1

2

¯x +

1

2

, ¯x, ¯z +

1

2

no extra conditions

16 km.. x,y,0¯x, ¯y, 0¯y,x, 0 y, ¯x,0

¯x,y,

1

2

x, ¯y,

1

2

y, x,

1

2

¯y, ¯x,

1

2

no extra conditions

16 j . 2 . x,0,

1

4

¯x,0,

1

4

0,x,

1

4

0, ¯x,

1

4

¯x,0,

3

4

x,0,

3

4

0, ¯x,

3

4

0,x,

3

4

hkl : l = 2n

16 i ..2 x,x,

1

4

¯x, ¯x,

1

4

¯x,x,

1

4

x, ¯x,

1

4

¯x, ¯x,

3

4

x,x,

3

4

x, ¯x,

3

4

¯x,x,

3

4

hkl : l = 2n

8 hm. 2mx,x +

1

2

,0¯x, ¯x +

1

2

,0¯x +

1

2

,x,0 x+

1

2

, ¯x,0 no extra conditions

8 g 2 . mm 0,

1

2

,z

1

2

,0, z 0,

1

2

, ¯z +

1

2

1

2

,0, ¯z +

1

2

hkl : l = 2n

8 f 4 .. 0,0 , z 0,0, ¯z +

1

2

0,0, ¯z 0,0 , z +

1

2

hkl : l = 2n

8 e ..2/m

1

4

,

1

4

,

1

4

3

4

,

3

4

,

1

4

3

4

,

1

4

,

1

4

1

4

,

3

4

,

1

4

hkl : k,l = 2n

4 dm. mm 0,

1

2

,0

1

2

,0, 0 hkl : l = 2n

4 c 4/m .. 0,0,00,0,

1

2

hkl : l = 2n

4 b

¯

42m 0,

1

2

,

1

4

1

2

,0,

1

4

hkl : l = 2n

4 a 422 0,0,

1

4

0,0,

3

4

hkl : l = 2n

Symmetry of special projections

Along [001] p4mm

a



=

1

2

(a − b) b



=

1

2

(a + b)

Origin at 0,0,z

Along [100] p2mm

a



=

1

2

bb



=

1

2

c

Origin at x,0,0

Along [110] p2mm

a



=

1

2

(−a + b) b



=

1

2

c

Origin at x, x,0

Maximal non-isomorphic subgroups

I

[2] I

¯

42m (121) (1; 2; 5; 6; 11; 12; 15; 16)+

[2] I

¯

4c2 (120) (1; 2; 7; 8; 11; 12; 13; 14)+

[2] I 4cm (108) (1; 2; 3; 4; 13; 14; 15; 16)+

[2] I 422(97) (1; 2; 3; 4; 5; 6; 7; 8)+

[2] I 4/m11(I 4/m, 87) (1; 2; 3; 4; 9; 10; 11; 12)+

[2] I 2/m2/c1(Ibam, 72) (1; 2; 5; 6; 9; 10; 13; 14)+

[2] I 2/m12/m (F mmm, 69) (1; 2; 7; 8; 9; 10; 15; 16)+

(Continued on preceding page)

481

I 4

1

/amd D

19

4h

4/mmm Tetragonal

No. 141 I 4

1

/a 2/m 2/d Patterson symmetry I 4/mmm

ORIGIN CHOICE 1

Origin at

¯

4m2, at 0,

1

4

,−

1

8

from centre (2/m)

Asymmetric unit 0 ≤ x ≤

1

2

;0≤ y ≤

1

2

;0≤ z ≤

1

8

Symmetry operations

For (0,0,0)+ set

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

1

2

)

1

4

,

1

4

,z (3) 4

+

(0,0,

1

4

) −

1

4

,

1

4

,z (4) 4

−

(0,0,

3

4

)

1

4

,−

1

4

,z

(5) 2

1

4

,y,

3

8

(6) 2 x,

1

4

,

1

8

(7) 2(

1

2

,

1

2

,0) x, x,

1

4

(8) 2 x, ¯x,0

(9)

¯

10,

1

4

,

1

8

(10) ax,y,

3

8

(11)

¯

4

+

0,0,z;0,0, 0 (12)

¯

4

−

0,

1

2

,z;0,

1

2

,

1

4

(13) n(

1

2

,0,

1

2

) x,

1

4

,z (14) m 0,y,z (15) d(

1

4

,−

1

4

,

3

4

) x +

1

4

, ¯x, z (16) d(

1

4

,

1

4

,

1

4

) x −

1

4

,x,z

For (

1

2

,

1

2

,

1

2

)+ set

(1) t(

1

2

,

1

2

,

1

2

) (2) 2 0,0,z (3) 4

+

(0,0,

3

4

)

1

4

,

1

4

,z (4) 4

−

(0,0,

1

4

)

1

4

,

1

4

,z

(5) 2(0,

1

2

,0) 0,y,

1

8

(6) 2(

1

2

,0, 0) x,0,

3

8

(7) 2 x,x,0(8)2x, ¯x +

1

2

,

1

4

(9)

¯

1

4

,0,

3

8

(10) bx,y,

1

8

(11)

¯

4

+

1

2

,0, z;

1

2

,0,

1

4

(12)

¯

4

−

0,0,z;0,0,0

(13) mx,0,z (14) n(0,

1

2

,

1

2

)

1

4

,y,z (15) d(−

1

4

,

1

4

,

1

4

) x +

1

4

, ¯x, z (16) d(

1

4

,

1

4

,

3

4

) x +

1

4

,x,z

482

International Tables for Crystallography (2006). Vol. A, Space group 141, pp. 482–485.

CONTINUED No. 141 I 4

1

/amd

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

1

2

,

1

2

,

1

2

); (2); (3); (5); (9)

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates

(0,0, 0)+ (

1

2

,

1

2

,

1

2

)+

Reﬂection conditions

General:

32 i 1(1)x, y,z (2) ¯x +

1

2

, ¯y +

1

2

,z +

1

2

(3) ¯y,x +

1

2

,z +

1

4

(4) y +

1

2

, ¯x, z +

3

4

(5) ¯x+

1

2

,y, ¯z +

3

4

(6) x, ¯y +

1

2

, ¯z +

1

4

(7) y +

1

2

,x +

1

2

, ¯z +

1

2

(8) ¯y, ¯x, ¯z

(9) ¯x, ¯y +

1

2

, ¯z +

1

4

(10) x +

1

2

,y, ¯z +

3

4

(11) y, ¯x, ¯z (12) ¯y +

1

2

,x +

1

2

, ¯z +

1

2

(13) x +

1

2

, ¯y +

1

2

,z +

1

2

(14) ¯x,y, z (15) ¯y +

1

2

, ¯x,z +

3

4

(16) y,x +

1

2

,z +

1

4

hkl : h + k + l = 2n

hk0: h,k = 2 n

0kl : k + l = 2n

hhl :2h + l = 4n

00l : l = 4n

h00 : h = 2n

h

¯

h0: h = 2n

Special: as above, plus

16 h . m . 0, y,z

1

2

, ¯y +

1

2

,z +

1

2

¯y,

1

2

,z +

1

4

y +

1

2

,0, z +

3

4

1

2

,y, ¯z +

3

4

0, ¯y+

1

2

, ¯z +

1

4

y +

1

2

,

1

2

, ¯z +

1

2

¯y,0, ¯z

no extra conditions

16 g ..2 x, x,0¯x +

1

2

, ¯x +

1

2

,

1

2

¯x,x +

1

2

,

1

4

x +

1

2

, ¯x,

3

4

¯x, ¯x +

1

2

,

1

4

x +

1

2

,x,

3

4

x, ¯x,0¯x +

1

2

,x +

1

2

,

1

2

hkl : l = 2n + 1

or 2h + l = 4n

16 f . 2 . x,

1

4

,

1

8

¯x +

1

2

,

1

4

,

5

8

3

4

,x +

1

2

,

3

8

3

4

, ¯x,

7

8

¯x,

1

4

,

1

8

x +

1

2

,

1

4

,

5

8

1

4

, ¯x,

7

8

1

4

,x +

1

2

,

3

8

hkl : l = 2n + 1

or h = 2n

8 e 2 mm. 0,0,z 0,

1

2

,z +

1

4

1

2

,0, ¯z +

3

4

1

2

,

1

2

, ¯z +

1

2

hkl : l = 2n + 1

or 2h + l = 4n

8 d . 2/m . 0,

1

4

,

5

8

1

2

,

1

4

,

1

8

3

4

,

1

2

,

7

8

3

4

,0,

3

8

8 c . 2/m . 0,

1

4

,

1

8

1

2

,

1

4

,

5

8

3

4

,

1

2

,

3

8

3

4

,0,

7

8



hkl : l = 2n + 1

or h, k = 2n, h + k + l = 4n

4 b

¯

4 m 20,0,

1

2

0,

1

2

,

3

4

4 a

¯

4 m 20,0,00,

1

2

,

1

4



hkl : l = 2n + 1

or 2h + l = 4n

Symmetry of special projections

Along [001] p4mm

a



=

1

2

ab



=

1

2

b

Origin at 0,0,z

Along [100] c2mm

a



= bb



= c

Origin at x,0,

3

8

Along [110] c2mm

a



=

1

2

(−a + b) b



=

1

2

c

Origin at x,x, 0

Maximal non-isomorphic subgroups

I

[2] I

¯

42d (122) (1; 2; 5; 6; 11; 12; 15; 16)+

[2] I

¯

4m2 (119) (1; 2; 7; 8; 11; 12; 13; 14)+

[2] I 4

1

md (109) (1; 2; 3; 4; 13; 14; 15; 16)+

[2] I 4

1

22 (98) (1; 2; 3; 4; 5; 6; 7; 8)+

[2] I 4

1

/a11(I 4

1

/a, 88) (1; 2; 3; 4; 9; 10; 11; 12)+

[2] I 2/a2/ m1(Imma, 74) (1; 2; 5; 6; 9; 10; 13; 14)+

[2] I 2/a12/d (Fddd, 70) (1; 2; 7; 8; 9; 10; 15; 16)+

IIa none

IIb none

Maximal isomorphic subgroups of lowest index

IIc

[3] I 4

1

/amd (c



= 3c) (141); [9] I 4

1

/amd (a



= 3a,b



= 3b) (141)

Minimal non-isomorphic supergroups

I

[3] Fd

¯

3m (227)

II [2] C 4

2

/amd (c



=

1

2

c)(P4

2

/nnm, 134)

483

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

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