(International Tables) Space group 70

Fddd	D_2h²⁴	mmm	Orthorhombic
No. 70	F2/d2/d2/d	Patterson symmetry Fmmm
ORIGIN CHOICE 1

symmetry group diagram

Origin at 2 2 2, at -1/8, -1/8, -1/8 from -1

Asymmetric unit

0 ≤ x ≤ 1/8; 0 ≤ y ≤ 1/4; 0 ≤ z ≤ 1

Symmetry operations

For (0, 0, 0)+ set

(1) 1	(2) 2 0, 0, z	(3) 2 0, y, 0	(4) 2 x, 0, 0
(5) -1 1/8, 1/8, 1/8	(6) d(1/4, 1/4, 0) x, y, 1/8	(7) d(1/4, 0, 1/4) x, 1/8, z	(8) d(0, 1/4, 1/4) 1/8, y, z

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) -1 1/8, 3/8, 3/8	(6) d(1/4, 3/4, 0) x, y, 3/8	(7) d(1/4, 0, 3/4) x, 3/8, z	(8) d(0, 3/4, 3/4) 1/8, y, z

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) -1 3/8, 1/8, 3/8	(6) d(3/4, 1/4, 0) x, y, 3/8	(7) d(3/4, 0, 3/4) x, 1/8, z	(8) d(0, 1/4, 3/4) 3/8, y, z

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) -1 3/8, 3/8, 1/8	(6) d(3/4, 3/4, 0) x, y, 1/8	(7) d(3/4, 0, 1/4) x, 3/8, z	(8) d(0, 3/4, 1/4) 3/8, y, z

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)

Positions

Multiplicity, Wyckoff letter,
Site symmetry

Coordinates

Reflection conditions

(0, 0, 0)+ (0, 1/2, 1/2)+ (1/2, 0, 1/2)+ (1/2, 1/2, 0)+

General:

(1) x, y, z	(2) -x, -y, z	(3) -x, y, -z	(4) x, -y, -z
(5) -x + 1/4, -y + 1/4, -z + 1/4	(6) x + 1/4, y + 1/4, -z + 1/4	(7) x + 1/4, -y + 1/4, z + 1/4	(8) -x + 1/4, y + 1/4, z + 1/4

hkl : h + k = 2n
and h + l, k + l = 2n
0kl : k + l = 4n
and k, l = 2n
h0l : h + l = 4n
and h, l = 2n
hk0 : h + k = 4n
and h, k = 2n
h00 : h = 4n
0k0 : k = 4n
00l : l = 4n

Special: as above, plus

. . 2

0, 0, z

0, 0, -z

1/4, 1/4, -z + 1/4

1/4, 1/4, z + 1/4

hkl : h = 2n + 1 or h + k + l = 4n

. 2 .

0, y, 0

0, -y, 0

1/4, -y + 1/4, 1/4

1/4, y + 1/4, 1/4

2 . .

x, 0, 0

-x, 0, 0

-x + 1/4, 1/4, 1/4

x + 1/4, 1/4, 1/4

-1

5/8, 5/8, 5/8

3/8, 3/8, 5/8

3/8, 5/8, 3/8

5/8, 3/8, 3/8

hkl : h = 2n + 1 or h, k, l = 4n + 2 or h, k, l = 4n

-1

1/8, 1/8, 1/8

7/8, 7/8, 1/8

7/8, 1/8, 7/8

1/8, 7/8, 7/8

2 2 2

0, 0, 1/2

1/4, 1/4, 3/4

hkl : h = 2n + 1 or h + k + l = 4n

2 2 2

0, 0, 0

1/4, 1/4, 1/4

Symmetry of special projections

Along [001] c2mm
a' = 1/2a b' = 1/2b
Origin at 0, 0, z

Along [100] c2mm
a' = 1/2b b' = 1/2c
Origin at x, 0, 0

Along [010] c2mm
a' = 1/2c b' = 1/2a
Origin at 0, y, 0

Maximal non-isomorphic subgroups

I	[2] Fdd2 (43)	(1; 2; 7; 8)+
	[2] Fd2d (Fdd2, 43)	(1; 3; 6; 8)+
	[2] F2dd (Fdd2, 43)	(1; 4; 6; 7)+
	[2] F222 (22)	(1; 2; 3; 4)+
	[2] F112/d (C2/c, 15)	(1; 2; 5; 6)+
	[2] F12/d1 (C2/c, 15)	(1; 3; 5; 7)+
	[2] F2/d11 (C2/c, 15)	(1; 4; 5; 8)+

IIa

none

IIb

none

Maximal isomorphic subgroups of lowest index

IIc

[3] Fddd (a' = 3a or b' = 3b or c' = 3c) (70)

Minimal non-isomorphic supergroups

I	[2] I4₁/amd (141); [2] I4₁/acd (142); [3] Fd-3 (203)

II	[2] Pnnn (a' = 1/2a, b' = 1/2b, c' = 1/2c) (48)

Fddd	D_2h²⁴	mmm	Orthorhombic
No. 70	F2/d2/d2/d	Patterson symmetry Fmmm
ORIGIN CHOICE 2

symmetry group diagram

Origin at -1 at d d d, at 1/8, 1/8, 1/8 from 2 2 2

Asymmetric unit

0 ≤ x ≤ 1/8; -1/8 ≤ y ≤ 1/8; 0 ≤ z ≤ 1

Symmetry operations

For (0, 0, 0)+ set

(1) 1	(2) 2 3/8, 3/8, z	(3) 2 3/8, y, 3/8	(4) 2 x, 3/8, 3/8
(5) -1 0, 0, 0	(6) d(1/4, 1/4, 0) x, y, 0	(7) d(1/4, 0, 1/4) x, 0, z	(8) d(0, 1/4, 1/4) 0, y, z

For (0, 1/2, 1/2)+ set

(1) t(0, 1/2, 1/2)	(2) 2(0, 0, 1/2) 3/8, 1/8, z	(3) 2(0, 1/2, 0) 3/8, y, 1/8	(4) 2 x, 1/8, 1/8
(5) -1 0, 1/4, 1/4	(6) d(1/4, 3/4, 0) x, y, 1/4	(7) d(1/4, 0, 3/4) x, 1/4, z	(8) d(0, 3/4, 3/4) 0, y, z

For (1/2, 0, 1/2)+ set

(1) t(1/2, 0, 1/2)	(2) 2(0, 0, 1/2) 1/8, 3/8, z	(3) 2 1/8, y, 1/8	(4) 2(1/2, 0, 0) x, 3/8, 1/8
(5) -1 1/4, 0, 1/4	(6) d(3/4, 1/4, 0) x, y, 1/4	(7) d(3/4, 0, 3/4) x, 0, z	(8) d(0, 1/4, 3/4) 1/4, y, z

For (1/2, 1/2, 0)+ set

(1) t(1/2, 1/2, 0)	(2) 2 1/8, 1/8, z	(3) 2(0, 1/2, 0) 1/8, y, 3/8	(4) 2(1/2, 0, 0) x, 1/8, 3/8
(5) -1 1/4, 1/4, 0	(6) d(3/4, 3/4, 0) x, y, 0	(7) d(3/4, 0, 1/4) x, 1/4, z	(8) d(0, 3/4, 1/4) 1/4, y, z

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)

Positions

Multiplicity, Wyckoff letter,
Site symmetry

Coordinates

Reflection conditions

(0, 0, 0)+ (0, 1/2, 1/2)+ (1/2, 0, 1/2)+ (1/2, 1/2, 0)+

General:

(1) x, y, z	(2) -x + 3/4, -y + 3/4, z	(3) -x + 3/4, y, -z + 3/4	(4) x, -y + 3/4, -z + 3/4
(5) -x, -y, -z	(6) x + 1/4, y + 1/4, -z	(7) x + 1/4, -y, z + 1/4	(8) -x, y + 1/4, z + 1/4

hkl : h + k, h + l, k + l = 2n
0kl : k + l = 4n, k, l = 2n
h0l : h + l = 4n, h, l = 2n
hk0 : h + k = 4n, h, k = 2n
h00 : h = 4n
0k0 : k = 4n
00l : l = 4n

Special: as above, plus

. . 2

1/8, 1/8, z

5/8, 1/8, -z + 3/4

7/8, 7/8, -z

3/8, 7/8, z + 1/4

hkl : h = 2n + 1 or h + k + l = 4n

. 2 .

1/8, y, 1/8

5/8, -y + 3/4, 1/8

7/8, -y, 7/8

3/8, y + 1/4, 7/8

2 . .

x, 1/8, 1/8

-x + 3/4, 5/8, 1/8

-x, 7/8, 7/8

x + 1/4, 3/8, 7/8

-1

1/2, 1/2, 1/2

1/4, 1/4, 1/2

1/4, 1/2, 1/4

1/2, 1/4, 1/4

hkl : h = 2n + 1 or h, k, l = 4n + 2 or h, k, l = 4n

-1

0, 0, 0

3/4, 3/4, 0

3/4, 0, 3/4

0, 3/4, 3/4

2 2 2

1/8, 1/8, 5/8

7/8, 7/8, 3/8

hkl : h = 2n + 1 or h + k + l = 4n

2 2 2

1/8, 1/8, 1/8

7/8, 7/8, 7/8

Symmetry of special projections

Along [001] c2mm
a' = 1/2a b' = 1/2b
Origin at 1/8, 1/8, z

Along [100] c2mm
a' = 1/2b b' = 1/2c
Origin at x, 1/8, 1/8

Along [010] c2mm
a' = 1/2c b' = 1/2a
Origin at 1/8, y, 1/8

Maximal non-isomorphic subgroups

I	[2] Fdd2 (43)	(1; 2; 7; 8)+
	[2] Fd2d (Fdd2, 43)	(1; 3; 6; 8)+
	[2] F2dd (Fdd2, 43)	(1; 4; 6; 7)+
	[2] F222 (22)	(1; 2; 3; 4)+
	[2] F112/d (C2/c, 15)	(1; 2; 5; 6)+
	[2] F12/d1 (C2/c, 15)	(1; 3; 5; 7)+
	[2] F2/d11 (C2/c, 15)	(1; 4; 5; 8)+

IIa

none

IIb

none

Maximal isomorphic subgroups of lowest index

IIc

[3] Fddd (a' = 3a or b' = 3b or c' = 3c) (70)

Minimal non-isomorphic supergroups

I	[2] I4₁/amd (141); [2] I4₁/acd (142); [3] Fd-3 (203)

II	[2] Pnnn (a' = 1/2a, b' = 1/2b, c' = 1/2c) (48)

International Tables for Crystallography (2006). Vol. A, ch. 7.1, Space group 70.