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

CONTINUED No. 200 Pm

¯

3

Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5); (13)

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates Reﬂection conditions

h,k,l cyclically permutable

General:

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

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

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

(13) ¯x, ¯y, ¯

z (14) x,y, ¯z (15) x, ¯y,z (16) ¯x, y,z

(17) ¯z, ¯x, ¯y (18) ¯z, x,y (19) z,x, ¯y (20) z, ¯x,y

(21) ¯y, ¯z, ¯x (22) y, ¯z, x (23) ¯y,z,x (24) y,z, ¯x

no conditions

Special: no extra conditions

12 km..

1

2

,y,z

1

2

, ¯y, z

1

2

,y, ¯z

1

2

, ¯y, ¯zz,

1

2

,yz,

1

2

, ¯y

¯z,

1

2

,y ¯z,

1

2

, ¯yy,z,

1

2

¯y,z,

1

2

y, ¯z,

1

2

¯y, ¯z,

1

2

12 jm.. 0,y, z 0, ¯y,z 0, y, ¯z 0, ¯y, ¯zz,0,yz,0, ¯y

¯z,0,y ¯z,0, ¯yy,z,0¯y , z,0 y, ¯z,0¯y, ¯z,0

8 i . 3 . x, x,x ¯x, ¯x,x ¯x,x, ¯xx, ¯x, ¯x

¯

x, ¯x, ¯xx,x, ¯xx, ¯x,x ¯x,x, x

6 hmm2.. x,

1

2

,

1

2

¯x,

1

2

,

1

2

1

2

,x,

1

2

1

2

, ¯x,

1

2

1

2

,

1

2

,x

1

2

,

1

2

, ¯x

6 gmm2.. x,

1

2

,0¯x,

1

2

,00, x,

1

2

0, ¯x,

1

2

1

2

,0, x

1

2

,0, ¯x

6 fmm2.. x,0,

1

2

¯x,0,

1

2

1

2

,x,0

1

2

, ¯x, 00,

1

2

,x 0,

1

2

, ¯x

6 emm2.. x, 0,0¯x,0, 00,x,00, ¯x,00,0,x 0,0, ¯x

3 dmmm..

1

2

,0, 00,

1

2

,00, 0,

1

2

3 cmmm.. 0,

1

2

,

1

2

1

2

,0,

1

2

1

2

,

1

2

,0

1 bm

¯

3 .

1

2

,

1

2

,

1

2

1 am

¯

3 . 0,0,0

Symmetry of special projections

Along [001] p2mm

a



= ab



= b

Origin at 0,0,z

Along [111] p6

a



=

1

3

(2a − b− c) b



=

1

3

(−a + 2b − c)

Origin at x,x , x

Along [110] p2mm

a



=

1

2

(−a + b) b



= c

Origin at x,x , 0

Maximal non-isomorphic subgroups

I

[2] P23 (195) 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12

[3] Pm1(Pmmm, 47) 1; 2; 3; 4; 13; 14; 15; 16

⎧

⎪

⎨

⎪

⎩

[4] P1

¯

3(R

¯

3, 148) 1; 5; 9; 13; 17; 21

[4] P1

¯

3(R

¯

3, 148) 1; 6; 12; 13; 18; 24

[4] P1

¯

3(R

¯

3, 148) 1; 7; 10; 13; 19; 22

[4] P1

¯

3(R

¯

3, 148) 1; 8; 11; 13; 20; 23

IIa none

IIb [2] Fm

¯

3(a



= 2a,b



= 2b,c



= 2c) (202); [4] Ia

¯

3(a



= 2a,b



= 2b,c



= 2c) (206); [4] Im

¯

3(a



= 2a,b



= 2b,c



= 2c) (204)

Maximal isomorphic subgroups of lowest index

IIc

[27] Pm

¯

3(a



= 3a,b



= 3b,c



= 3c) (200)

(Continued on preceding page)

615

Pn

¯

3 T

2

h

m

¯

3 Cubic

No. 201 P2/n

¯

3

Patterson symmetry Pm

¯

3

ORIGIN CHOICE 1

Origin at 23, at −

1

4

,−

1

4

,−

1

4

from centre (

¯

3)

Asymmetric unit 0 ≤ x ≤ 1; 0 ≤ y ≤

1

2

;0≤ z ≤

1

2

; y ≤ min(x,1 − x); z ≤ y

Vertices 0,0, 01,0, 0

1

2

,

1

2

,0

1

2

,

1

2

,

1

2

Symmetry operations

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

(5) 3

+

x,x, x (6) 3

+

¯x,x, ¯x (7) 3

+

x, ¯x, ¯x (8) 3

+

¯x, ¯x, x

(9) 3

−

x,x, x (10) 3

−

x, ¯x, ¯x (11) 3

−

¯x, ¯x, x (12) 3

−

¯x,x, ¯x

(13)

¯

1

4

,

1

4

,

1

4

(14) n(

1

2

,

1

2

,0) x,y,

1

4

(15) n(

1

2

,0,

1

2

) x,

1

4

,z (16) n(0,

1

2

,

1

2

)

1

4

,y,z

(17)

¯

3

+

x,x, x;

1

4

,

1

4

,

1

4

(18)

¯

3

+

¯x − 1,x + 1, ¯x; −

1

4

,

1

4

,

3

4

(19)

¯

3

+

x, ¯x + 1 , ¯x;

1

4

,

3

4

,−

1

4

(20)

¯

3

+

¯x+ 1, ¯x,x;

3

4

,−

1

4

,

1

4

(21)

¯

3

−

x,x, x;

1

4

,

1

4

,

1

4

(22)

¯

3

−

x + 1, ¯x − 1, ¯x;

1

4

,−

1

4

,

3

4

(23)

¯

3

−

¯x, ¯x + 1, x; −

1

4

,

3

4

,

1

4

(24)

¯

3

−

¯x+ 1,x, ¯x;

3

4

,

1

4

,−

1

4

616

International Tables for Crystallography (2006). Vol. A, Space group 201, pp. 616–619.

CONTINUED No. 201 Pn

¯

3

Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5); (13)

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates Reﬂection conditions

h,k,l cyclically permutable

General:

24 h 1(1)x,y, z (2) ¯x, ¯y,z (3) ¯x,y, ¯z (4) x, ¯y, ¯z

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

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

(13) ¯x +

1

2

, ¯y +

1

2

, ¯z +

1

2

(14) x +

1

2

,y +

1

2

, ¯z +

1

2

(15) x +

1

2

, ¯y +

1

2

,z +

1

2

(16) ¯x +

1

2

,y +

1

2

,z +

1

2

(17) ¯z+

1

2

, ¯x +

1

2

, ¯y +

1

2

(18) ¯z +

1

2

,x +

1

2

,y +

1

2

(19) z +

1

2

,x +

1

2

, ¯y +

1

2

(20) z +

1

2

, ¯x +

1

2

,y +

1

2

(21) ¯y +

1

2

, ¯z +

1

2

, ¯x +

1

2

(22) y +

1

2

, ¯z +

1

2

,x +

1

2

(23) ¯y +

1

2

,z +

1

2

,x +

1

2

(24) y +

1

2

,z +

1

2

, ¯x +

1

2

0kl : k + l = 2n

h00 : h = 2n

Special: as above, plus

12 g 2 .. x,

1

2

,0¯x,

1

2

,00, x,

1

2

0, ¯x,

1

2

1

2

,0, x

1

2

,0, ¯x

¯x +

1

2

,0,

1

2

x +

1

2

,0,

1

2

1

2

, ¯x +

1

2

,0

1

2

,x +

1

2

,00,

1

2

, ¯x +

1

2

0,

1

2

,x +

1

2

hkl : h + k + l = 2n

12 f 2 .. x,0,0¯x,0,00,x,00, ¯x,00,0,x 0,0, ¯x

¯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

1

2

,

1

2

, ¯x +

1

2

1

2

,

1

2

,x +

1

2

hkl : h + k + l = 2n

8 e . 3 . x,x, x ¯x, ¯x,x

¯x,x, ¯xx, ¯x , ¯x

¯x +

1

2

, ¯x +

1

2

, ¯x +

1

2

x +

1

2

,x +

1

2

, ¯x +

1

2

x +

1

2

, ¯x +

1

2

,x +

1

2

¯x +

1

2

,x +

1

2

,x +

1

2

no extra conditions

6 d 222.. 0,

1

2

,

1

2

1

2

,0,

1

2

1

2

,

1

2

,0

1

2

,0, 00,

1

2

,00,0,

1

2

hkl : h + k + l = 2n

4 c .

¯

3 .

3

4

,

3

4

,

3

4

1

4

,

1

4

,

3

4

1

4

,

3

4

,

1

4

3

4

,

1

4

,

1

4

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

4 b .

¯

3 .

1

4

,

1

4

,

1

4

3

4

,

3

4

,

1

4

3

4

,

1

4

,

3

4

1

4

,

3

4

,

3

4

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

2 a 23. 0,0,0

1

2

,

1

2

,

1

2

hkl : h + k + l = 2n

Symmetry of special projections

Along [001] c2mm

a



= ab



= b

Origin at 0,0,z

Along [111] p6

a



=

1

3

(2a − b− c) b



=

1

3

(−a + 2b − c)

Origin at x, x,x

Along [110] p2mm

a



=

1

2

(−a + b) b



= c

Origin at x,x,

1

4

Maximal non-isomorphic subgroups

I

[2] P23 (195) 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12

[3] Pn1(Pnnn, 48) 1; 2; 3; 4; 13; 14; 15; 16

⎧

⎪

⎨

⎪

⎩

[4] P1

¯

3(R

¯

3, 148) 1; 5; 9; 13; 17; 21

[4] P1

¯

3(R

¯

3, 148) 1; 6; 12; 13; 18; 24

[4] P1

¯

3(R

¯

3, 148) 1; 7; 10; 13; 19; 22

[4] P1

¯

3(R

¯

3, 148) 1; 8; 11; 13; 20; 23

IIa none

IIb [2] Fd

¯

3(a



= 2a,b



= 2b,c



= 2c) (203)

Maximal isomorphic subgroups of lowest index

IIc

[27] Pn

¯

3(a



= 3a,b



= 3b,c



= 3c) (201)

Minimal non-isomorphic supergroups

I

[2] Pn

¯

3n (222); [2] Pn

¯

3m (224)

II [2] Im

¯

3 (204); [4] Fm

¯

3 (202)

617

Pn

¯

3 T

2

h

m

¯

3 Cubic

No. 201 P2/n

¯

3

Patterson symmetry Pm

¯

3

ORIGIN CHOICE 2

Origin at centre (

¯

3),at

1

4

,

1

4

,

1

4

from 23

Asymmetric unit −

1

4

≤ x ≤

3

4

; −

1

4

≤ y ≤

1

4

; −

1

4

≤ z ≤

1

4

; y ≤ min(x,

1

2

− x); z ≤ y

Vertices −

1

4

,−

1

4

,−

1

4

3

4

,−

1

4

,−

1

4

1

4

,

1

4

,−

1

4

1

4

,

1

4

,

1

4

Symmetry operations

(1) 1 (2) 2

1

4

,

1

4

,z (3) 2

1

4

,y,

1

4

(4) 2 x,

1

4

,

1

4

(5) 3

+

x,x,x (6) 3

+

¯x,x+

1

2

, ¯x (7) 3

+

x+

1

2

, ¯x, ¯x (8) 3

+

¯x+

1

2

, ¯x+

1

2

,x

(9) 3

−

x,x,x (10) 3

−

x+

1

2

, ¯x, ¯x (11) 3

−

¯x+

1

2

, ¯x+

1

2

,x (12) 3

−

¯x,x+

1

2

, ¯x

(13)

¯

10,0,0 (14) n(

1

2

,

1

2

,0) x,y,0 (15) n(

1

2

,0,

1

2

) x,0,z (16) n(0,

1

2

,

1

2

) 0,y,z

(17)

¯

3

+

x,x,x;0,0,0 (18)

¯

3

+

¯x−1,x+

1

2

, ¯x; −

1

2

,0,

1

2

(19)

¯

3

+

x−

1

2

, ¯x+1, ¯x;0,

1

2

,−

1

2

(20)

¯

3

+

¯x+

1

2

, ¯x−

1

2

,x;

1

2

,−

1

2

,0

(21)

¯

3

−

x,x,x;0,0,0 (22)

¯

3

−

x+

1

2

, ¯x−1, ¯x;0,−

1

2

,

1

2

(23)

¯

3

−

¯x−

1

2

, ¯x+

1

2

,x; −

1

2

,

1

2

,0 (24)

¯

3

−

¯x+1,x−

1

2

, ¯x;

1

2

,0,−

1

2

618

CONTINUED No. 201 Pn

¯

3

Generators selected (1); t(1, 0,0); t(0,1,0); t(0,0,1); (2); (3); (5); (13)

Positions

Multiplicity,

Wyckoff letter,

Site symmetry

Coordinates Reﬂection conditions

h,k,l cyclically permutable

General:

24 h 1(1)x,y, z (2) ¯x +

1

2

, ¯y +

1

2

,z (3) ¯x +

1

2

,y, ¯z +

1

2

(4) x, ¯y +

1

2

, ¯z +

1

2

(5) z,x,y (6) z, ¯x +

1

2

, ¯y +

1

2

(7) ¯z +

1

2

, ¯x +

1

2

,y (8) ¯z +

1

2

,x, ¯y +

1

2

(9) y,z,x (10) ¯y +

1

2

,z, ¯x +

1

2

(11) y, ¯z +

1

2

, ¯x +

1

2

(12) ¯y +

1

2

, ¯z +

1

2

,x

(13) ¯x, ¯y, ¯z (14) x +

1

2

,y +

1

2

, ¯z (15) x +

1

2

, ¯y, z +

1

2

(16) ¯x,y +

1

2

,z +

1

2

(17) ¯z, ¯x, ¯y (18) ¯z, x +

1

2

,y +

1

2

(19) z +

1

2

,x +

1

2

, ¯y (20) z+

1

2

, ¯x,y +

1

2

(21) ¯y, ¯z, ¯x (22) y +

1

2

, ¯z, x +

1

2

(23) ¯y,z +

1

2

,x +

1

2

(24) y +

1

2

,z +

1

2

, ¯x

0kl : k + l = 2 n

h00 : h = 2n

Special: as above, plus

12 g 2 .. x,

3

4

,

1

4

¯x +

1

2

,

3

4

,

1

4

1

4

,x,

3

4

1

4

, ¯x +

1

2

,

3

4

3

4

,

1

4

,x

3

4

,

1

4

, ¯x +

1

2

¯x,

1

4

,

3

4

x +

1

2

,

1

4

,

3

4

3

4

, ¯x,

1

4

3

4

,x +

1

2

,

1

4

1

4

,

3

4

, ¯x

1

4

,

3

4

,x +

1

2

hkl : h + k + l = 2n

12 f 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

1

4

,

1

4

,x

1

4

,

1

4

, ¯x +

1

2

¯x,

3

4

,

3

4

x +

1

2

,

3

4

,

3

4

3

4

, ¯x,

3

4

3

4

,x +

1

2

,

3

4

3

4

,

3

4

, ¯x

3

4

,

3

4

,x +

1

2

hkl : h + k + l = 2n

8 e . 3 . x,x,x ¯x +

1

2

, ¯x +

1

2

,x ¯x +

1

2

,x, ¯x +

1

2

x, ¯x +

1

2

, ¯x +

1

2

¯x, ¯x, ¯xx+

1

2

,x +

1

2

, ¯xx+

1

2

, ¯x, x +

1

2

¯x,x +

1

2

,x +

1

2

no extra conditions

6 d 222..

1

4

,

3

4

,

3

4

3

4

,

1

4

,

3

4

3

4

,

3

4

,

1

4

3

4

,

1

4

,

1

4

1

4

,

3

4

,

1

4

1

4

,

1

4

,

3

4

hkl : h + k + l = 2n

4 c .

¯

3 .

1

2

,

1

2

,

1

2

0,0,

1

2

0,

1

2

,0

1

2

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

4 b .

¯

3 . 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

2 a 23.

1

4

,

1

4

,

1

4

3

4

,

3

4

,

3

4

hkl : h + k + l = 2n

Symmetry of special projections

Along [001] c2mm

a



= ab



= b

Origin at

1

4

,

1

4

,z

Along [111] p6

a



=

1

3

(2a − b− c) b



=

1

3

(−a + 2b − c)

Origin at x, x,x

Along [110] p2mm

a



=

1

2

(−a + b) b



= c

Origin at x,x, 0

Maximal non-isomorphic subgroups

I

[2] P23 (195) 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12

[3] Pn1(Pnnn, 48) 1; 2; 3; 4; 13; 14; 15; 16

⎧

⎪

⎨

⎪

⎩

[4] P1

¯

3(R

¯

3, 148) 1; 5; 9; 13; 17; 21

[4] P1

¯

3(R

¯

3, 148) 1; 6; 12; 13; 18; 24

[4] P1

¯

3(R

¯

3, 148) 1; 7; 10; 13; 19; 22

[4] P1

¯

3(R

¯

3, 148) 1; 8; 11; 13; 20; 23

IIa none

IIb [2] Fd

¯

3(a



= 2a,b



= 2b,c



= 2c) (203)

Maximal isomorphic subgroups of lowest index

IIc

[27] Pn

¯

3(a



= 3a,b



= 3b,c



= 3c) (201)

Minimal non-isomorphic supergroups

I

[2] Pn

¯

3n (222); [2] Pn

¯

3m (224)

II [2] Im

¯

3 (204); [4] Fm

¯

3 (202)

619

Fm

¯

3 T

3

h

m

¯

3 Cubic

No. 202 F2/m

¯

3

Patterson symmetry Fm

¯

3

Origin at centre (m

¯

3)

Asymmetric unit 0 ≤ x ≤

1

2

;0≤ y ≤

1

2

;0≤ z ≤

1

4

; y ≤ x; z ≤ min(

1

2

− x,y)

Vertices 0,0, 0

1

2

,0, 0

1

2

,

1

2

,0

1

4

,

1

4

,

1

4

Symmetry operations

For (0,0,0)+ set

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

(5) 3

+

x,x,x (6) 3

+

¯x,x, ¯x (7) 3

+

x, ¯x, ¯x (8) 3

+

¯x, ¯x,x

(9) 3

−

x,x,x (10) 3

−

x, ¯x, ¯x (11) 3

−

¯x, ¯x,x (12) 3

−

¯x,x, ¯x

(13)

¯

10,0,0 (14) mx,y,0 (15) mx,0,z (16) m 0,y,z

(17)

¯

3

+

x,x,x;0,0,0 (18)

¯

3

+

¯x,x, ¯x;0,0,0 (19)

¯

3

+

x, ¯x, ¯x;0,0,0 (20)

¯

3

+

¯x, ¯x,x ;0,0,0

(21)

¯

3

−

x,x,x;0,0,0 (22)

¯

3

−

x, ¯x, ¯x;0,0,0 (23)

¯

3

−

¯x, ¯x,x;0,0,0 (24)

¯

3

−

¯x,x, ¯x;0,0,0

620

International Tables for Crystallography (2006). Vol. A, Space group 202, pp. 620–622.

CONTINUED No. 202 Fm

¯

3

Symmetry operations (continued)

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) 3

+

(

1

3

,

1

3

,

1

3

) x−

1

3

,x−

1

6

,x (6) 3

+

¯x,x+

1

2

, ¯x (7) 3

+

(−

1

3

,

1

3

,

1

3

) x+

1

3

, ¯x−

1

6

, ¯x (8) 3

+

¯x, ¯x+

1

2

,x

(9) 3

−

(

1

3

,

1

3

,

1

3

) x−

1

6

,x+

1

6

,x (10) 3

−

(−

1

3

,

1

3

,

1

3

) x+

1

6

, ¯x+

1

6

, ¯x (11) 3

−

¯x+

1

2

, ¯x+

1

2

,x (12) 3

−

¯x−

1

2

,x+

1

2

, ¯x

(13)

¯

10,

1

4

,

1

4

(14) bx,y,

1

4

(15) cx,

1

4

,z (16) n(0,

1

2

,

1

2

) 0,y,z

(17)

¯

3

+

x,x+

1

2

,x;0,

1

2

,0 (18)

¯

3

+

¯x−1,x+

1

2

, ¯x; −

1

2

,0,

1

2

(19)

¯

3

+

x, ¯x+

1

2

, ¯x;0,

1

2

,0 (20)

¯

3

+

¯x+1, ¯x+

1

2

,x;

1

2

,0,

1

2

(21)

¯

3

−

x−

1

2

,x−

1

2

,x;0,0,

1

2

(22)

¯

3

−

x+

1

2

, ¯x−

1

2

, ¯x;0,0,

1

2

(23)

¯

3

−

¯x−

1

2

, ¯x+

1

2

,x; −

1

2

,

1

2

,0 (24)

¯

3

−

¯x+

1

2

,x+

1

2

, ¯x;

1

2

,

1

2

,0

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) 3

+

(

1

3

,

1

3

,

1

3

) x+

1

6

,x−

1

6

,x (6) 3

+

(

1

3

,−

1

3

,

1

3

) ¯x+

1

6

,x+

1

6

, ¯x (7) 3

+

x+

1

2

, ¯x−

1

2

, ¯x (8) 3

+

¯x+

1

2

, ¯x+

1

2

,x

(9) 3

−

(

1

3

,

1

3

,

1

3

) x−

1

6

,x−

1

3

,x (10) 3

−

x+

1

2

, ¯x, ¯x (11) 3

−

¯x+

1

2

, ¯x,x (12) 3

−

(

1

3

,−

1

3

,

1

3

) ¯x−

1

6

,x+

1

3

, ¯x

(13)

¯

1

4

,0,

1

4

(14) ax,y,

1

4

(15) n(

1

2

,0,

1

2

) x,0,z (16) c

1

4

,y,z

(17)

¯

3

+

x−

1

2

,x−

1

2

,x;0,0,

1

2

(18)

¯

3

+

¯x−

1

2

,x+

1

2

, ¯x;0,0,

1

2

(19)

¯

3

+

x+

1

2

, ¯x+

1

2

, ¯x;

1

2

,

1

2

,0 (20)

¯

3

+

¯x+

1

2

, ¯x−

1

2

,x;

1

2

,−

1

2

,0

(21)

¯

3

−

x+

1

2

,x,x;

1

2

,0,0 (22)

¯

3

−

x+

1

2

, ¯x−1, ¯x;0,−

1

2

,

1

2

(23)

¯

3

−

¯x+

1

2

, ¯x+1,x ;0,

1

2

,

1

2

(24)

¯

3

−

¯x+

1

2

,x, ¯x;

1

2

,0,0

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) 3

+

(

1

3

,

1

3

,

1

3

) x+

1

6

,x+

1

3

,x (6) 3

+

¯x+

1

2

,x, ¯x (7) 3

+

x+

1

2

, ¯x, ¯x (8) 3

+

(

1

3

,

1

3

,−

1

3

) ¯x+

1

6

, ¯x+

1

3

,x

(9) 3

−

(

1

3

,

1

3

,

1

3

) x+

1

3

,x+

1

6

,x (10) 3

−

x, ¯x+

1

2

, ¯x (11) 3

−

(

1

3

,

1

3

,−

1

3

) ¯x+

1

3

, ¯x+

1

6

,x (12) 3

−

¯x,x+

1

2

, ¯x

(13)

¯

1

4

,

1

4

,0 (14) n(

1

2

,

1

2

,0) x,y,0 (15) ax,

1

4

,z (16) b

1

4

,y,z

(17)

¯

3

+

x+

1

2

,x,x;

1

2

,0,0 (18)

¯

3

+

¯x−

1

2

,x+1, ¯x;0,

1

2

,

1

2

(19)

¯

3

+

x−

1

2

, ¯x+1, ¯x;0,

1

2

,−

1

2

(20)

¯

3

+

¯x+

1

2

, ¯x,x;

1

2

,0,0

(21)

¯

3

−

x,x+

1

2

,x;0,

1

2

,0 (22)

¯

3

−

x+1, ¯x−

1

2

, ¯x;

1

2

,0,

1

2

(23)

¯

3

−

¯x, ¯x+

1

2

,x;0,

1

2

,0 (24)

¯

3

−

¯x+1,x−

1

2

, ¯x;

1

2

,0,−

1

2

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); (13)

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

h,k,l cyclically permutable

General:

96 i 1(1)x, y,z (2) ¯x, ¯y,z (3) ¯x,y, ¯z (4) x, ¯y, ¯z

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

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

(13) ¯x,

¯y, ¯z (14) x,y, ¯z (15) x, ¯y,z (16) ¯x,y, z

(17) ¯z, ¯x, ¯y (18) ¯z, x,y (19) z,x, ¯y (20) z, ¯x,y

(21) ¯y, ¯z, ¯x (22) y, ¯z, x (23) ¯y,z,x (24) y,z, ¯x

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

0kl : k

,l = 2n

hhl : h + l = 2n

h00 : h = 2n

Special: as above, plus

48 hm.. 0,y, z 0, ¯y, z 0,y, ¯z 0, ¯y, ¯zz,0, yz,0, ¯y

¯z,0,y ¯z,0, ¯yy,z,0¯y,z,0 y, ¯z,0¯y, ¯z,0

no extra conditions

48 g 2 .. x,

1

4

,

1

4

¯x,

3

4

,

1

4

1

4

,x,

1

4

1

4

, ¯x,

3

4

1

4

,

1

4

,x

3

4

,

1

4

, ¯x

¯x,

3

4

,

3

4

x,

1

4

,

3

4

3

4

, ¯x,

3

4

3

4

,x,

1

4

3

4

,

3

4

, ¯x

1

4

,

3

4

,x

hkl : h = 2n

32 f . 3 . x,x,x ¯x, ¯x,x ¯x,x, ¯xx, ¯x, ¯x

¯x, ¯x, ¯xx, x, ¯xx, ¯x,x ¯x,x, x

no extra conditions

24 emm2.. x,0,0¯x,0,00,x,00, ¯x,00,0,x 0,0, ¯x no extra conditions

24 d 2/

m .. 0,

1

4

,

1

4

0,

3

4

,

1

4

1

4

,0,

1

4

1

4

,0,

3

4

1

4

,

1

4

,0

3

4

,

1

4

,0 hk l : h = 2n

8 c 23.

1

4

,

1

4

,

1

4

3

4

,

3

4

,

3

4

hkl : h = 2n

4 bm

¯

3 .

1

2

,

1

2

,

1

2

no extra conditions

4 am

¯

3 . 0,0,0 no extra conditions

621

Fm

¯

3 No. 202 CONTINUED

Symmetry of special projections

Along [001] p2mm

a



=

1

2

ab



=

1

2

b

Origin at 0,0,z

Along [111] p6

a



=

1

6

(2a − b− c) b



=

1

6

(−a + 2b − c)

Origin at x,x , x

Along [110] c2mm

a



=

1

2

(−a + b) b



= c

Origin at x,x , 0

Maximal non-isomorphic subgroups

I

[2] F 23 (196) (1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12)+

[3] Fm1(F mmm, 69) (1; 2; 3; 4; 13; 14; 15; 16)+

⎧

⎪

⎨

⎪

⎩

[4] F 1

¯

3(R

¯

3, 148) (1; 5; 9; 13; 17; 21)+

[4] F 1

¯

3(R

¯

3, 148) (1; 6; 12; 13; 18; 24)+

[4] F 1

¯

3(R

¯

3, 148) (1; 7; 10; 13; 19; 22)+

[4] F 1

¯

3(R

¯

3, 148) (1; 8; 11; 13; 20; 23)+

IIa

⎧

⎪

⎨

⎪

⎩

[4] Pa

¯

3 (205) 1; 5; 9; 13; 17; 21; (2; 7; 12; 14; 19; 24)+(0,

1

2

,

1

2

); (4; 6; 11; 16; 18; 23)+(

1

2

,0,

1

2

); (3; 8; 10;

15; 20; 22)+(

1

2

,

1

2

,0)

[4] Pa

¯

3 (205) 1; 7; 10; 13; 19; 22; (2; 5; 11; 14; 17; 23)+(0,

1

2

,

1

2

); (4; 8; 12; 16; 20; 24)+(

1

2

,0,

1

2

); (3; 6; 9;

15; 18; 21)+(

1

2

,

1

2

,0)

[4] Pa

¯

3 (205) 1; 8; 11; 13; 20; 23; (2; 6; 10; 14; 18; 22)+(0,

1

2

,

1

2

); (4; 7; 9; 16; 19; 21)+(

1

2

,0,

1

2

); (3; 5; 12;

15; 17; 24)+(

1

2

,

1

2

,0)

[4] Pa

¯

3 (205) 1; 6; 12; 13; 18; 24; (2; 8; 9; 14; 20; 21)+(0,

1

2

,

1

2

); (4; 5; 10; 16; 17; 22)+(

1

2

,0,

1

2

); (3; 7; 11;

15; 19; 23)+(

1

2

,

1

2

,0)

⎧

⎪

⎨

⎪

⎩

[4] Pa

¯

3 (205) 1; 5; 9; 13; 17; 21; (3; 8; 10; 15; 20; 22)+(0,

1

2

,

1

2

); (2; 7; 12; 14; 19; 24)+(

1

2

,0,

1

2

); (4; 6; 11;

16; 18; 23)+(

1

2

,

1

2

,0)

[4] Pa

¯

3 (205) 1; 7; 10; 13; 19; 22; (3; 6; 9; 15; 18; 21)+(0,

1

2

,

1

2

); (2; 5; 11; 14; 17; 23)+(

1

2

,0,

1

2

); (4; 8; 12;

16; 20; 24)+(

1

2

,

1

2

,0)

[4] Pa

¯

3 (205) 1; 8; 11; 13; 20; 23; (3; 5; 12; 15; 17; 24)+(0,

1

2

,

1

2

); (2; 6; 10; 14; 18; 22)+(

1

2

,0,

1

2

); (4; 7; 9;

16; 19; 21)+(

1

2

,

1

2

,0)

[4] Pa

¯

3 (205) 1; 6; 12; 13; 18; 24; (3; 7; 11; 15; 19; 23)+(0,

1

2

,

1

2

); (2; 8; 9; 14; 20; 21)+(

1

2

,0,

1

2

); (4; 5; 10;

16; 17; 22)+(

1

2

,

1

2

,0)

⎧

⎪

⎨

⎪

⎩

[4] Pn

¯

3 (201) 1; 5; 9; 13; 17; 21; (4; 6; 11; 16; 18; 23)+(0,

1

2

,

1

2

); (3; 8; 10; 15; 20; 22)+(

1

2

,0,

1

2

); (2; 7; 12;

14; 19; 24)+(

1

2

,

1

2

,0)

[4] Pn

¯

3 (201) 1; 7; 10; 13; 19; 22; (4; 8; 12; 16; 20; 24)+(0,

1

2

,

1

2

); (3; 6; 9; 15; 18; 21)+(

1

2

,0,

1

2

); (2; 5; 11;

14; 17; 23)+(

1

2

,

1

2

,0)

[4] Pn

¯

3 (201) 1; 8; 11; 13; 20; 23; (4; 7; 9; 16; 19; 21)+(0,

1

2

,

1

2

); (3; 5; 12; 15; 17; 24)+(

1

2

,0,

1

2

); (2; 6; 10;

14; 18; 22)+(

1

2

,

1

2

,0)

[4] Pn

¯

3 (201) 1; 6; 12; 13; 18; 24; (4; 5; 10; 16; 17; 22)+(0,

1

2

,

1

2

); (3; 7; 11; 15; 19; 23)+(

1

2

,0,

1

2

); (2; 8; 9;

14; 20; 21)+(

1

2

,

1

2

,0)

⎧

⎪

⎨

⎪

⎩

[4] Pm

¯

3 (200) 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24

[4] Pm

¯

3 (200) 1; 2; 3; 4; 13; 14; 15; 16; (5; 6; 7; 8; 17; 18; 19; 20)+(0,

1

2

,

1

2

); (9; 10; 11; 12; 21; 22; 23;

24)+(

1

2

,

1

2

,0)

[4] Pm

¯

3 (200) 1; 2; 3; 4; 13; 14; 15; 16; (5; 6; 7; 8; 17; 18; 19; 20)+(

1

2

,0,

1

2

); (9; 10; 11; 12; 21; 22; 23;

24)+(0,

1

2

,

1

2

)

[4] Pm

¯

3 (200) 1; 2; 3; 4; 13; 14; 15; 16; (5; 6; 7; 8; 17; 18; 19; 20)+(

1

2

,

1

2

,0); (9; 10; 11; 12; 21; 22; 23;

24)+(

1

2

,0,

1

2

)

IIb none

Maximal isomorphic subgroups of lowest index

IIc

[27] Fm

¯

3(a



= 3a,b



= 3b,c



= 3c) (202)

Minimal non-isomorphic supergroups

I

[2] Fm

¯

3m (225); [2] Fm

¯

3c (226)

II [2] Pm

¯

3(a



=

1

2

a,b



=

1

2

b,c



=

1

2

c) (200)

622

Fd

¯

3 T

4

h

m

¯

3 Cubic

No. 203 F 2/d

¯

3

Patterson symmetry Fm

¯

3

ORIGIN CHOICE 1

Origin at 23, at −

1

8

,−

1

8

,−

1

8

from centre (

¯

3)

Asymmetric unit 0 ≤ x ≤

1

2

;0≤ y ≤

1

4

; −

1

4

≤ z ≤

1

4

; y ≤ min(x,

1

2

− x); −y ≤ z ≤ y

Vertices 0,0, 0

1

2

,0, 0

1

4

,

1

4

,

1

4

1

4

,

1

4

,−

1

4

624

International Tables for Crystallography (2006). Vol. A, Space group 203, pp. 623–627.

CONTINUED No. 203 Fd

¯

3

Symmetry operations

For (0,0,0)+ set

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

(5) 3

+

x,x,x (6) 3

+

¯x,x, ¯x (7) 3

+

x, ¯x, ¯x (8) 3

+

¯x, ¯x,x

(9) 3

−

x,x,x (10) 3

−

x, ¯x, ¯x (11) 3

−

¯x, ¯x,x (12) 3

−

¯x,x, ¯x

(13)

¯

1

8

,

1

8

,

1

8

(14) d(

1

4

,

1

4

,0) x,y,

1

8

(15) d(

1

4

,0,

1

4

) x,

1

8

,z (16) d(0,

1

4

,

1

4

)

1

8

,y,z

(17)

¯

3

+

x,x,x;

1

8

,

1

8

,

1

8

(18)

¯

3

+

¯x−

1

2

,x+

1

2

, ¯x; −

1

8

,

1

8

,

3

8

(19)

¯

3

+

x, ¯x+

1

2

, ¯x;

1

8

,

3

8

,−

1

8

(20)

¯

3

+

¯x+

1

2

, ¯x,x;

3

8

,−

1

8

,

1

8

(21)

¯

3

−

x,x,x;

1

8

,

1

8

,

1

8

(22)

¯

3

−

x+

1

2

, ¯x−

1

2

, ¯x;

1

8

,−

1

8

,

3

8

(23)

¯

3

−

¯x, ¯x+

1

2

,x; −

1

8

,

3

8

,

1

8

(24)

¯

3

−

¯x+

1

2

,x, ¯x;

3

8

,

1

8

,−

1

8

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) 3

+

(

1

3

,

1

3

,

1

3

) x−

1

3

,x−

1

6

,x (6) 3

+

¯x,x+

1

2

, ¯x (7) 3

+

(−

1

3

,

1

3

,

1

3

) x+

1

3

, ¯x−

1

6

, ¯x (8) 3

+

¯x, ¯x+

1

2

,x

(9) 3

−

(

1

3

,

1

3

,

1

3

) x−

1

6

,x+

1

6

,x (10) 3

−

(−

1

3

,

1

3

,

1

3

) x+

1

6

, ¯x+

1

6

, ¯x (11) 3

−

¯x+

1

2

, ¯x+

1

2

,x (12) 3

−

¯x−

1

2

,x+

1

2

, ¯x

(13)

¯

1

8

,

3

8

,

3

8

(14) d(

1

4

,

3

4

,0) x,y,

3

8

(15) d(

1

4

,0,

3

4

) x,

3

8

,z (16) d(0,

3

4

,

3

4

)

1

8

,y,z

(17)

¯

3

+

x,x+

1

2

,x;

1

8

,

5

8

,

1

8

(18)

¯

3

+

¯x−

3

2

,x+1, ¯x; −

5

8

,

1

8

,

7

8

(19)

¯

3

+

x, ¯x+1, ¯x;

1

8

,

7

8

,−

1

8

(20)

¯

3

+

¯x+

3

2

, ¯x+

1

2

,x;

7

8

,−

1

8

,

5

8

(21)

¯

3

−

x−

1

2

,x−

1

2

,x;

1

8

,

1

8

,

5

8

(22)

¯

3

−

x+1, ¯x−1, ¯x;

1

8

,−

1

8

,

7

8

(23)

¯

3

−

¯x−

1

2

, ¯x+1,x ; −

5

8

,

7

8

,

1

8

(24)

¯

3

−

¯x+1,x+

1

2

, ¯x;

7

8

,

5

8

,−

1

8

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) 3

+

(

1

3

,

1

3

,

1

3

) x+

1

6

,x−

1

6

,x (6) 3

+

(

1

3

,−

1

3

,

1

3

) ¯x+

1

6

,x+

1

6

, ¯x (7) 3

+

x+

1

2

, ¯x−

1

2

, ¯x (8) 3

+

¯x+

1

2

, ¯x+

1

2

,x

(9) 3

−

(

1

3

,

1

3

,

1

3

) x−

1

6

,x−

1

3

,x (10) 3

−

x+

1

2

, ¯x, ¯x (11) 3

−

¯x+

1

2

, ¯x,x (12) 3

−

(

1

3

,−

1

3

,

1

3

) ¯x−

1

6

,x+

1

3

, ¯x

(13)

¯

1

3

8

,

1

8

,

3

8

(14) d(

3

4

,

1

4

,0) x,y,

3

8

(15) d(

3

4

,0,

3

4

) x,

1

8

,z (16) d(0,

1

4

,

3

4

)

3

8

,y,z

(17)

¯

3

+

x−

1

2

,x−

1

2

,x;

1

8

,

1

8

,

5

8

(18)

¯

3

+

¯x−1,x+1, ¯x; −

1

8

,

1

8

,

7

8

(19)

¯

3

+

x+

1

2

, ¯x+1, ¯x;

5

8

,

7

8

,−

1

8

(20)

¯

3

+

¯x+1, ¯x−

1

2

,x;

7

8

,−

5

8

,

1

8

(21)

¯

3

−

x+

1

2

,x,x;

5

8

,

1

8

,

1

8

(22)

¯

3

−

x+1, ¯x−

3

2

, ¯x;

1

8

,−

5

8

,

7

8

(23)

¯

3

−

¯x+

1

2

, ¯x+

3

2

,x; −

1

8

,

7

8

,

5

8

(24)

¯

3

−

¯x+1,x, ¯x;

7

8

,

1

8

,−

1

8

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) 3

+

(

1

3

,

1

3

,

1

3

) x+

1

6

,x+

1

3

,x (6) 3

+

¯x+

1

2

,x, ¯x (7) 3

+

x+

1

2

, ¯x, ¯x (8) 3

+

(

1

3

,

1

3

,−

1

3

) ¯x+

1

6

, ¯x+

1

3

,x

(9) 3

−

(

1

3

,

1

3

,

1

3

) x+

1

3

,x+

1

6

,x (10) 3

−

x, ¯x+

1

2

, ¯x (11) 3

−

(

1

3

,

1

3

,−

1

3

) ¯x+

1

3

, ¯x+

1

6

,x (12) 3

−

¯x,x+

1

2

, ¯x

(13)

¯

1

3

8

,

3

8

,

1

8

(14) d(

3

4

,

3

4

,0) x,y,

1

8

(15) d(

3

4

,0,

1

4

) x,

3

8

,z (16) d(0,

3

4

,

1

4

)

3

8

,y,z

(17)

¯

3

+

x+

1

2

,x,x;

5

8

,

1

8

,

1

8

(18)

¯

3

+

¯x−1,x+

3

2

, ¯x; −

1

8

,

5

8

,

7

8

(19)

¯

3

+

x−

1

2

, ¯x+

3

2

, ¯x;

1

8

,

7

8

,−

5

8

(20)

¯

3

+

¯x+1, ¯x,x;

7

8

,−

1

8

,

1

8

(21)

¯

3

−

x,x+

1

2

,x;

1

8

,

5

8

,

1

8

(22)

¯

3

−

x+

3

2

, ¯x−1, ¯x;

5

8

,−

1

8

,

7

8

(23)

¯

3

−

¯x, ¯x+1,x; −

1

8

,

7

8

,

1

8

(24)

¯

3

−

¯x+

3

2

,x−

1

2

, ¯x;

7

8

,

1

8

,−

5

8

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); (13)

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

h,k,l cyclically permutable

General:

96 g 1(1)x,y, z (2) ¯x, ¯y,z (3) ¯x,y, ¯z (4) x, ¯y, ¯z

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

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

(13) ¯x +

1

4

, ¯y +

1

4

, ¯z +

1

4

(14) x +

1

4

,y +

1

4

, ¯z +

1

4

(15) x +

1

4

, ¯y +

1

4

,z +

1

4

(16) ¯x +

1

4

,y +

1

4

,z +

1

4

(17) ¯z+

1

4

, ¯x +

1

4

, ¯y +

1

4

(18) ¯z +

1

4

,x +

1

4

,y +

1

4

(19) z +

1

4

,x +

1

4

, ¯y +

1

4

(20) z +

1

4

, ¯x +

1

4

,y +

1

4

(21) ¯y +

1

4

, ¯z +

1

4

, ¯x +

1

4

(22) y +

1

4

, ¯z +

1

4

,x +

1

4

(23) ¯y +

1

4

,z +

1

4

,x +

1

4

(24) y +

1

4

,z +

1

4

, ¯x +

1

4

hkl : h + k = 2n and

h + l,k + l = 2n

0kl : k + l = 4n and

k,l = 2n

hhl : h + l = 2n

h00 : h = 4n

Special: as above, plus

48 f 2 .. x,0,0¯x , 0,00,x, 00, ¯x,00, 0,x 0 , 0, ¯x

¯x +

1

4

,

1

4

,

1

4

x +

1

4

,

1

4

,

1

4

1

4

, ¯x +

1

4

,

1

4

1

4

,x +

1

4

,

1

4

1

4

,

1

4

, ¯x +

1

4

1

4

,

1

4

,x +

1

4

hkl : h = 2n + 1

or h + k + l = 4n

32 e . 3 . x,x, x ¯x, ¯x,x ¯x,x, ¯xx, ¯x, ¯x

¯x +

1

4

, ¯x +

1

4

, ¯x +

1

4

x +

1

4

,x +

1

4

, ¯x +

1

4

x +

1

4

, ¯x +

1

4

,x +

1

4

¯x +

1

4

,x +

1

4

,x +

1

4

no extra conditions

16 d .

¯

3 .

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 .

¯

3 .

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 23.

1

2

,

1

2

,

1

2

3

4

,

3

4

,

3

4

8 a 23. 0,0,0

1

4

,

1

4

,

1

4



hkl : h = 2n + 1

or h + k + l = 4n

(Continued on page 623)

625

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

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