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

P4/nbm D

3

4h

4/mmm Tetragonal

No. 125 P 4/n 2 /b 2/m

Patterson symmetry P4/mmm

ORIGIN CHOICE 1

Origin at 422 at 4/n22/g,at−

1

4

,−

1

4

,0 from centre (2/m)

Asymmetric unit 0 ≤ x ≤

1

2

;0≤ y ≤

1

2

;0≤ z ≤

1

2

; y ≤

1

2

− x

Symmetry operations

(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)

¯

1

4

,

1

4

,0 (10) n(

1

2

,

1

2

,0) x, y,0 (11)

¯

4

+

1

2

,0, z;

1

2

,0, 0 (12)

¯

4

−

0,

1

2

,z;0,

1

2

,0

(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 isomorphic subgroups of lowest index

IIc

[2] P4/nbm(c



= 2c) (125); [9] P4/nbm(a



= 3a,b



= 3b) (125)

Minimal non-isomorphic supergroups

I

none

II [2] C 4/mmm (P4/mmm, 123); [2] I 4/mcm(140)

434

International Tables for Crystallography (2006). Vol. A, Space group 125, pp. 434–437.

CONTINUED No. 125 P4/nbm

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 n 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 +

1

2

, ¯y +

1

2

, ¯z (10) x +

1

2

,y +

1

2

, ¯z (11) y +

1

2

, ¯x +

1

2

, ¯z (12) ¯y+

1

2

,x +

1

2

, ¯z

(13) x +

1

2

, ¯y +

1

2

,z (14) ¯x+

1

2

,y +

1

2

,z (15) ¯y +

1

2

, ¯x +

1

2

,z (16) y +

1

2

,x +

1

2

,z

hk0: h + k = 2n

0kl : k = 2n

h00 : h = 2n

Special: as above, plus

8 m ..mx,x +

1

2

,z ¯x, ¯x +

1

2

,z ¯x +

1

2

,x,zx+

1

2

, ¯x,z

¯x,x +

1

2

, ¯zx, ¯x +

1

2

, ¯zx+

1

2

,x, ¯z ¯x +

1

2

, ¯x, ¯z

no extra conditions

8 l . 2 . x,0,

1

2

¯x,0,

1

2

0,x,

1

2

0, ¯x,

1

2

¯x+

1

2

,

1

2

,

1

2

x +

1

2

,

1

2

,

1

2

1

2

, ¯x +

1

2

,

1

2

1

2

,x +

1

2

,

1

2

hkl : h + k = 2n

8 k . 2 . x, 0,0¯x,0, 00,x, 00, ¯x,0

¯x+

1

2

,

1

2

,0 x +

1

2

,

1

2

,0

1

2

, ¯x +

1

2

,0

1

2

,x +

1

2

,0

hkl : h + k = 2n

8 j ..2 x,x,

1

2

¯x, ¯x,

1

2

¯x,x,

1

2

x, ¯x,

1

2

¯x+

1

2

, ¯x +

1

2

,

1

2

x +

1

2

,x +

1

2

,

1

2

x +

1

2

, ¯x +

1

2

,

1

2

¯x +

1

2

,x +

1

2

,

1

2

hkl : h + k = 2n

8 i ..2 x,x,0¯x, ¯x,0¯x,x,0 x, ¯x,0

¯x+

1

2

, ¯x +

1

2

,0 x +

1

2

,x +

1

2

,0 x +

1

2

, ¯x +

1

2

,0¯x +

1

2

,x +

1

2

,0

hkl : h + k = 2n

4 h 2 . mm 0,

1

2

,z

1

2

,0, z 0,

1

2

, ¯z

1

2

,0, ¯zhkl: h + k = 2n

4 g 4 .. 0,0, z 0,0, ¯z

1

2

,

1

2

, ¯z

1

2

,

1

2

,zhkl: h + k = 2n

4 f ..2/m

1

4

,

1

4

,

1

2

3

4

,

3

4

,

1

2

3

4

,

1

4

,

1

2

1

4

,

3

4

,

1

2

hkl : h,k = 2n

4 e ..2/m

1

4

,

1

4

,0

3

4

,

3

4

,0

3

4

,

1

4

,0

1

4

,

3

4

,0 hkl : h, k = 2n

2 d

¯

42m 0,

1

2

,

1

2

1

2

,0,

1

2

hkl : h + k = 2n

2 c

¯

42m 0,

1

2

,0

1

2

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

2 b 422 0,0,

1

2

1

2

,

1

2

,

1

2

hkl : h + k = 2n

2 a 422 0,0, 0

1

2

,

1

2

,0 hkl : h + k = 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



= 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

¯

4b2 (117) 1; 2; 7; 8; 11; 12; 13; 14

[2] P

¯

42m (111) 1; 2; 5; 6; 11; 12; 15; 16

[2] P4bm (100) 1; 2; 3; 4; 13; 14; 15; 16

[2] P422 (89) 1; 2; 3; 4; 5; 6; 7; 8

[2] P4/n11(P4/n, 85) 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/b1(Pban, 50) 1; 2; 5; 6; 9; 10; 13; 14

IIa none

IIb [2] P4

2

/nnm(c



= 2c) (134); [2] P4

2

/nbc(c



= 2c) (133); [2] P4/nnc(c



= 2c) (126)

(Continued on preceding page)

435

P4/nbm D

3

4h

4/mmm Tetragonal

No. 125 P 4/n 2 /b 2/m

Patterson symmetry P4/mmm

ORIGIN CHOICE 2

Origin at centre (2/m) at n(b, a)(2

1

/g,2/m),at

1

4

,

1

4

,0 from 422

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

+

1

4

,

1

4

,z (4) 4

−

1

4

,

1

4

,z

(5) 2

1

4

,y,0(6)2x,

1

4

,0(7)2x,x,0(8)2x, ¯x +

1

2

,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

,0 (12)

¯

4

−

1

4

,

1

4

,z; −

1

4

,

1

4

,0

(13) ax,0,z (14) b 0, y,z (15) mx, ¯x,z (16) g(

1

2

,

1

2

,0) x,x,z

Maximal isomorphic subgroups of lowest index

IIc

[2] P4/nbm(c



= 2c) (125); [9] P4/nbm(a



= 3a,b



= 3b) (125)

Minimal non-isomorphic supergroups

I

none

II [2] C 4/mmm (P4/mmm, 123); [2] I 4/mcm(140)

436

CONTINUED No. 125 P4/nbm

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 n 1(1)x,y, z (2) ¯x +

1

2

, ¯y +

1

2

,z (3) ¯y +

1

2

,x,z (4) y, ¯x +

1

2

,z

(5) ¯x +

1

2

,y, ¯z (6) x, ¯y +

1

2

, ¯z (7) y,x, ¯z (8) ¯y +

1

2

, ¯x +

1

2

, ¯z

(9) ¯x, ¯y, ¯z (10) x +

1

2

,y +

1

2

, ¯z (11) y +

1

2

, ¯x, ¯z (12) ¯y,x +

1

2

, ¯z

(13) x +

1

2

, ¯y, z (14) ¯x,y +

1

2

,z (15) ¯y, ¯x,z (16) y +

1

2

,x +

1

2

,z

hk0: h + k = 2n

0kl : k = 2n

h00 : h = 2n

Special: as above, plus

8 m ..mx, ¯x,z ¯x +

1

2

,x +

1

2

,zx+

1

2

,x,z ¯x, ¯x +

1

2

,z

¯x+

1

2

, ¯x, ¯zx,x +

1

2

, ¯z ¯x, x, ¯zx+

1

2

, ¯x +

1

2

, ¯z

no extra conditions

8 l . 2 . x,

1

4

,

1

2

¯x +

1

2

,

1

4

,

1

2

1

4

,x,

1

2

1

4

, ¯x +

1

2

,

1

2

¯x,

3

4

,

1

2

x +

1

2

,

3

4

,

1

2

3

4

, ¯x,

1

2

3

4

,x +

1

2

,

1

2

hkl : h + k = 2n

8 k . 2 . x,

1

4

,0¯x +

1

2

,

1

4

,0

1

4

,x,0

1

4

, ¯x +

1

2

,0

¯x,

3

4

,0 x +

1

2

,

3

4

,0

3

4

, ¯x, 0

3

4

,x +

1

2

,0

hkl : h + k = 2n

8 j ..2 x,x,

1

2

¯x +

1

2

, ¯x +

1

2

,

1

2

¯x +

1

2

,x,

1

2

x, ¯x +

1

2

,

1

2

¯x, ¯x,

1

2

x +

1

2

,x +

1

2

,

1

2

x +

1

2

, ¯x,

1

2

¯x,x +

1

2

,

1

2

hkl : h + k = 2n

8 i ..2 x,x,0¯x +

1

2

, ¯x +

1

2

,0¯x +

1

2

,x,0 x, ¯x +

1

2

,0

¯x, ¯x, 0 x +

1

2

,x +

1

2

,0 x +

1

2

, ¯x,0¯x, x +

1

2

,0

hkl : h + k = 2n

4 h 2 . mm

3

4

,

1

4

,z

1

4

,

3

4

,z

3

4

,

1

4

, ¯z

1

4

,

3

4

, ¯zhkl: h + k = 2n

4 g 4 ..

1

4

,

1

4

,z

1

4

,

1

4

, ¯z

3

4

,

3

4

, ¯z

3

4

,

3

4

,zhkl: h + k = 2n

4 f ..2/m 0,0,

1

2

1

2

,

1

2

,

1

2

1

2

,0,

1

2

0,

1

2

,

1

2

hkl : h,k = 2n

4 e ..2/m 0,0,0

1

2

,

1

2

,0

1

2

,0, 00,

1

2

,0 hkl : h,k = 2n

2 d

¯

42m

3

4

,

1

4

,

1

2

1

4

,

3

4

,

1

2

hkl : h + k = 2n

2 c

¯

42m

3

4

,

1

4

,0

1

4

,

3

4

,0 hkl : h + k = 2n

2 b 422

1

4

,

1

4

,

1

2

3

4

,

3

4

,

1

2

hkl : h + k = 2n

2 a 422

1

4

,

1

4

,0

3

4

,

3

4

,0 hkl : h + k = 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] p2mm

a



=

1

2

bb



= 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

¯

4b2 (117) 1; 2; 7; 8; 11; 12; 13; 14

[2] P

¯

42m (111) 1; 2; 5; 6; 11; 12; 15; 16

[2] P4bm (100) 1; 2; 3; 4; 13; 14; 15; 16

[2] P422 (89) 1; 2; 3; 4; 5; 6; 7; 8

[2] P4/n11(P4/n, 85) 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/b1(Pban, 50) 1; 2; 5; 6; 9; 10; 13; 14

IIa none

IIb [2] P4

2

/nnm(c



= 2c) (134); [2] P4

2

/nbc(c



= 2c) (133); [2] P4/nnc(c



= 2c) (126)

(Continued on preceding page)

437

P4/nnc D

4

4h

4/mmm Tetragonal

No. 126 P 4/n 2 /n 2/c

Patterson symmetry P4/mmm

ORIGIN CHOICE 1

Origin at 422/n,at−

1

4

,−

1

4

,−

1

4

from

¯

1

Asymmetric unit 0 ≤ x ≤

1

2

;0≤ y ≤

1

2

;0≤ z ≤

1

4

Symmetry operations

(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)

¯

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

438

International Tables for Crystallography (2006). Vol. A, Space group 126, pp. 438–441.

CONTINUED No. 126 P4/nnc

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 k 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+

1

2

, ¯y +

1

2

, ¯z +

1

2

(10) x +

1

2

,y +

1

2

, ¯z +

1

2

(11) y +

1

2

, ¯x +

1

2

, ¯z +

1

2

(12) ¯y +

1

2

,x +

1

2

, ¯z +

1

2

(13) x +

1

2

, ¯y +

1

2

,z +

1

2

(14) ¯x +

1

2

,y +

1

2

,z +

1

2

(15) ¯y +

1

2

, ¯x +

1

2

,z +

1

2

(16) y +

1

2

,x +

1

2

,z +

1

2

hk0: h + k = 2n

0kl : k + l = 2n

hhl : l = 2n

00l : l = 2n

h00 : h = 2n

Special: as above, plus

8 j . 2 . x,0,

1

2

¯x,0,

1

2

0,x,

1

2

0, ¯x,

1

2

¯x +

1

2

,

1

2

,0 x +

1

2

,

1

2

,0

1

2

, ¯x +

1

2

,0

1

2

,x +

1

2

,0

hkl : h + k + l = 2n

8 i . 2 . x,0,0¯x,0, 00,x, 00, ¯x,0

¯x +

1

2

,

1

2

,

1

2

x +

1

2

,

1

2

,

1

2

1

2

, ¯x +

1

2

,

1

2

1

2

,x +

1

2

,

1

2

hkl : h + k + l = 2n

8 h ..2 x,x,0¯x, ¯x,0¯x,x,0 x, ¯x,0

¯x +

1

2

, ¯x +

1

2

,

1

2

x +

1

2

,x +

1

2

,

1

2

x +

1

2

, ¯x +

1

2

,

1

2

¯x+

1

2

,x +

1

2

,

1

2

hkl : h + k + l = 2n

8 g 2 ..

1

2

,0, z 0,

1

2

,z

1

2

,0, ¯z 0,

1

2

, ¯z

0,

1

2

, ¯z +

1

2

1

2

,0, ¯z +

1

2

0,

1

2

,z +

1

2

1

2

,0, z +

1

2

hkl : h + k,l = 2n

8 f

¯

1

4

,

1

4

,

1

4

3

4

,

3

4

,

1

4

3

4

,

1

4

,

1

4

1

4

,

3

4

,

1

4

3

4

,

1

4

,

3

4

1

4

,

3

4

,

3

4

1

4

,

1

4

,

3

4

3

4

,

3

4

,

3

4

hkl : h,k,l = 2n

4 e 4 .. 0, 0,z 0, 0, ¯z

1

2

,

1

2

, ¯z +

1

2

1

2

,

1

2

,z +

1

2

hkl : h + k + l = 2n

4 d

¯

4 ..

1

2

,0,

1

4

0,

1

2

,

1

4

1

2

,0,

3

4

0,

1

2

,

3

4

hkl : h + k,l = 2n

4 c 222.

1

2

,0, 00,

1

2

,00,

1

2

,

1

2

1

2

,0,

1

2

hkl : h + k,l = 2n

2 b 422 0,0,

1

2

1

2

,

1

2

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

2 a 422 0,0,0

1

2

,

1

2

,

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 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] P

¯

4n2 (118) 1; 2; 7; 8; 11; 12; 13; 14

[2] P

¯

42c (112) 1; 2; 5; 6; 11; 12; 15; 16

[2] P4nc (104) 1; 2; 3; 4; 13; 14; 15; 16

[2] P422 (89) 1; 2; 3; 4; 5; 6; 7; 8

[2] P4/n11(P4/n, 85) 1; 2; 3; 4; 9; 10; 11; 12

[2] P2/n12/c (Ccce, 68) 1; 2; 7; 8; 9; 10; 15; 16

[2] P2/n2/n1(Pnnn, 48) 1; 2; 5; 6; 9; 10; 13; 14

IIa none

IIb none

Maximal isomorphic subgroups of lowest index

IIc

[3] P4/nnc(c



= 3c) (126); [9] P4/nnc(a



= 3a,b



= 3b) (126)

Minimal non-isomorphic supergroups

I

[3] Pn

¯

3n (222)

II [2] I 4/mmm(139); [2] C 4/mcc (P4/mcc, 124); [2] P4/nbm(c



=

1

2

c) (125)

439

P4/nnc D

4

4h

4/mmm Tetragonal

No. 126 P 4/n 2 /n 2/c

Patterson symmetry P4/mmm

ORIGIN CHOICE 2

Origin at

¯

1atnn(n, c),at

1

4

,

1

4

,

1

4

from 422

Asymmetric unit −

1

4

≤ x ≤

1

4

; −

1

4

≤ y ≤

1

4

;0≤ z ≤

1

4

Symmetry operations

(1) 1 (2) 2

1

4

,

1

4

,z (3) 4

+

1

4

,

1

4

,z (4) 4

−

1

4

,

1

4

,z

(5) 2

1

4

,y,

1

4

(6) 2 x,

1

4

,

1

4

(7) 2 x,x,

1

4

(8) 2 x, ¯x +

1

2

,

1

4

(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

,0 (12)

¯

4

−

1

4

,

1

4

,z; −

1

4

,

1

4

,0

(13) n(

1

2

,0,

1

2

) x,0,z (14) n(0,

1

2

,

1

2

) 0,y, z (15) cx, ¯x,z (16) n(

1

2

,

1

2

,

1

2

) x,x,z

440

CONTINUED No. 126 P4/nnc

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 k 1(1)x,y, z (2) ¯x +

1

2

, ¯y +

1

2

,z (3) ¯y +

1

2

,x,z (4) y, ¯x +

1

2

,z

(5) ¯x +

1

2

,y, ¯z +

1

2

(6) x, ¯y +

1

2

, ¯z +

1

2

(7) y,x, ¯z +

1

2

(8) ¯y +

1

2

, ¯x +

1

2

, ¯z +

1

2

(9) ¯x, ¯y, ¯z (10) x +

1

2

,y +

1

2

, ¯z (11) y +

1

2

, ¯x, ¯z (12) ¯y,x +

1

2

, ¯z

(13) x +

1

2

, ¯y,z +

1

2

(14) ¯x,y +

1

2

,z +

1

2

(15) ¯y, ¯x,z +

1

2

(16) y +

1

2

,x +

1

2

,z +

1

2

hk0: h + k = 2n

0kl : k + l = 2n

hhl : l = 2n

00l : l = 2n

h00 : h = 2n

Special: as above, plus

8 j . 2 . x,

3

4

,

1

4

¯x +

1

2

,

3

4

,

1

4

3

4

,x,

1

4

3

4

, ¯x +

1

2

,

1

4

¯x,

1

4

,

3

4

x +

1

2

,

1

4

,

3

4

1

4

, ¯x,

3

4

1

4

,x +

1

2

,

3

4

hkl : h + k + l = 2n

8 i . 2 . x,

1

4

,

1

4

¯x +

1

2

,

1

4

,

1

4

1

4

,x,

1

4

1

4

, ¯x +

1

2

,

1

4

¯x,

3

4

,

3

4

x +

1

2

,

3

4

,

3

4

3

4

, ¯x,

3

4

3

4

,x +

1

2

,

3

4

hkl : h + k + l = 2n

8 h ..2 x,x,

1

4

¯x+

1

2

, ¯x +

1

2

,

1

4

¯x +

1

2

,x,

1

4

x, ¯x +

1

2

,

1

4

¯x, ¯x,

3

4

x +

1

2

,x +

1

2

,

3

4

x +

1

2

, ¯x,

3

4

¯x,x +

1

2

,

3

4

hkl : h + k + l = 2n

8 g 2 ..

1

4

,

3

4

,z

3

4

,

1

4

,z

1

4

,

3

4

, ¯z +

1

2

3

4

,

1

4

, ¯z +

1

2

3

4

,

1

4

, ¯z

1

4

,

3

4

, ¯z

3

4

,

1

4

,z +

1

2

1

4

,

3

4

,z +

1

2

hkl : h + k,l = 2n

8 f

¯

10, 0,0

1

2

,

1

2

,0

1

2

,0, 00,

1

2

,0

1

2

,0,

1

2

0,

1

2

,

1

2

0,0,

1

2

1

2

,

1

2

,

1

2

hkl : h,k,l = 2n

4 e 4 ..

1

4

,

1

4

,z

1

4

,

1

4

, ¯z +

1

2

3

4

,

3

4

, ¯z

3

4

,

3

4

,z +

1

2

hkl : h + k + l = 2n

4 d

¯

4 ..

1

4

,

3

4

,0

3

4

,

1

4

,0

1

4

,

3

4

,

1

2

3

4

,

1

4

,

1

2

hkl : h + k,l = 2n

4 c 222.

1

4

,

3

4

,

3

4

3

4

,

1

4

,

3

4

3

4

,

1

4

,

1

4

1

4

,

3

4

,

1

4

hkl : h + k,l = 2n

2 b 422

1

4

,

1

4

,

3

4

3

4

,

3

4

,

1

4

hkl : h + k + l = 2n

2 a 422

1

4

,

1

4

,

1

4

3

4

,

3

4

,

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

1

4

,

1

4

,z

Along [100] c2mm

a



= bb



= c

Origin at x,

1

4

,

1

4

Along [110] p2mm

a



=

1

2

(−a + b) b



=

1

2

c

Origin at x,x , 0

Maximal non-isomorphic subgroups

I

[2] P

¯

4n2 (118) 1; 2; 7; 8; 11; 12; 13; 14

[2] P

¯

42c (112) 1; 2; 5; 6; 11; 12; 15; 16

[2] P4nc (104) 1; 2; 3; 4; 13; 14; 15; 16

[2] P422 (89) 1; 2; 3; 4; 5; 6; 7; 8

[2] P4/n11(P4/n, 85) 1; 2; 3; 4; 9; 10; 11; 12

[2] P2/n12/c (Ccce, 68) 1; 2; 7; 8; 9; 10; 15; 16

[2] P2/n2/n1(Pnnn, 48) 1; 2; 5; 6; 9; 10; 13; 14

IIa none

IIb none

Maximal isomorphic subgroups of lowest index

IIc

[3] P4/nnc(c



= 3c) (126); [9] P4/nnc(a



= 3a,b



= 3b) (126)

Minimal non-isomorphic supergroups

I

[3] Pn

¯

3n (222)

II [2] I 4/mmm(139); [2] C 4/mcc (P4/mcc, 124); [2] P4/nbm(c



=

1

2

c) (125)

441

P4/mbm D

5

4h

4/mmm Tetragonal

No. 127 P 4/m 2

1

/b 2/m Patterson symmetry P4/mmm

Origin at centre (4/m) at 4/m12

1

/g

Asymmetric unit 0 ≤ x ≤

1

2

;0≤ y ≤

1

2

;0≤ z ≤

1

2

; y ≤

1

2

− x

Symmetry operations

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

+

0,0,z (4) 4

−

0,0,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)

¯

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

¯

4

+

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

¯

4

−

0,0,z;0,0, 0

(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

442

International Tables for Crystallography (2006). Vol. A, Space group 127, pp. 442–443.

CONTINUED No. 127 P4/mbm

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 l 1(1)x,y,z (2) ¯x, ¯y, z (3) ¯y,x,z (4) y, ¯x,z

(5) ¯x +

1

2

,y +

1

2

, ¯z (6) x +

1

2

, ¯y +

1

2

, ¯z (7) y +

1

2

,x +

1

2

, ¯z (8) ¯y +

1

2

, ¯x +

1

2

, ¯z

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

(13) x +

1

2

, ¯y +

1

2

,z (14) ¯x+

1

2

,y +

1

2

,z (15) ¯y +

1

2

, ¯x +

1

2

,z (16) y +

1

2

,x +

1

2

,z

0kl : k = 2n

h00 : h = 2n

Special: as above, plus

8 k ..mx, x +

1

2

,z ¯x, ¯x +

1

2

,z ¯x +

1

2

,x,zx+

1

2

, ¯x, z

¯x +

1

2

,x, ¯zx+

1

2

, ¯x, ¯zx,x +

1

2

, ¯z ¯x, ¯x +

1

2

, ¯z

no extra conditions

8 jm.. x,y,

1

2

¯x, ¯y,

1

2

¯y,x,

1

2

y, ¯x,

1

2

¯x +

1

2

,y +

1

2

,

1

2

x +

1

2

, ¯y +

1

2

,

1

2

y +

1

2

,x +

1

2

,

1

2

¯y +

1

2

, ¯x +

1

2

,

1

2

no extra conditions

8 im.. x,y,0¯x, ¯y,0¯y, x,0 y, ¯x, 0

¯x +

1

2

,y +

1

2

,0 x +

1

2

, ¯y +

1

2

,0 y+

1

2

,x +

1

2

,0¯y +

1

2

, ¯x +

1

2

,0

no extra conditions

4 hm. 2mx,x +

1

2

,

1

2

¯x, ¯x +

1

2

,

1

2

¯x+

1

2

,x,

1

2

x +

1

2

, ¯x,

1

2

no extra conditions

4 gm. 2mx,x +

1

2

,0¯x, ¯x +

1

2

,0¯x +

1

2

,x,0 x+

1

2

, ¯x,0 no extra conditions

4 f 2 . mm 0,

1

2

,z

1

2

,0, z

1

2

,0, ¯z 0,

1

2

, ¯zhkl: h + k = 2n

4 e 4 .. 0,0,z

1

2

,

1

2

, ¯z 0,0, ¯z

1

2

,

1

2

,zhkl: h + k = 2n

2 dm. mm 0,

1

2

,0

1

2

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

2 cm. mm 0,

1

2

,

1

2

1

2

,0,

1

2

hkl : h + k = 2n

2 b 4/ m .. 0, 0,

1

2

1

2

,

1

2

,

1

2

hkl : h + k = 2n

2 a 4/ m .. 0, 0,0

1

2

,

1

2

,0 hkl : h + k = 2n

Symmetry of special projections

Along [001] p4gm

a



= ab



= b

Origin at 0,0,z

Along [100] p2mm

a



=

1

2

bb



= 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

¯

4b2 (117) 1; 2; 7; 8; 11; 12; 13; 14

[2] P

¯

42

1

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

[2] P4bm (100) 1; 2; 3; 4; 13; 14; 15; 16

[2] P42

1

2 (90) 1; 2; 3; 4; 5; 6; 7; 8

[2] P4/m11(P4/m, 83) 1; 2; 3; 4; 9; 10; 11; 12

[2] P2/m12/m (Cmmm, 65) 1; 2; 7; 8; 9; 10; 15; 16

[2] P2/m2

1

/b1(Pbam, 55) 1; 2; 5; 6; 9; 10; 13; 14

IIa none

IIb [2] P4

2

/mnm (c



= 2c) (136); [2] P4

2

/mbc (c



= 2c) (135); [2] P4/mnc (c



= 2c) (128)

Maximal isomorphic subgroups of lowest index

IIc

[2] P4/mbm(c



= 2c) (127); [9] P4/mbm(a



= 3a,b



= 3b) (127)

Minimal non-isomorphic supergroups

I

none

II [2] C 4/mmm (P4/mmm, 123); [2] I 4/mcm(140)

443

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

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