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crystallography lll. Examples: Ccc2, P1, I 4 1 /a 2/c 2/d, F 4/m 3 2/m, P 6 3 /m 2/m 2/c. lattice centering screw axis glide plane. 230 space groups Combine 32 point groups (rotational symmetry) with a. 14 Bravais lattices (translational symmetry)

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slide2

Examples:

Ccc2, P1, I 41/a 2/c 2/d, F 4/m 3 2/m, P 63/m 2/m 2/c

lattice centering screw axis glide plane

230 space groups

Combine 32 point groups (rotational symmetry) with

a. 14 Bravais lattices (translational symmetry)

glide planes (rotational + translational symmetry) -

a, b, c, n, d

screw axes (rotational + translational symmetry) -

21, 31, 32, 41, 42,43, 61, 62, 63, 64, 65

slide3

Space group diagrams

Example: P222

b

a

slide4

x,y,z

x,y,z

x,y,z

x,y,z

General equipoint: 4u (x,y,z) (x,y,z) (x,y,z) (x,y,z)

Special equipoints: 1a (0,0,0); 1b (1/2,0,0); … 2i (x,0,0), (x,0,0); 2j (x,0,1/2), (x,0,1/2); … 2m (0,y,z) (0,y,0); …

Space group diagrams

Example: P222

b

a

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Example:

LaB6 a = 4.1566 Å

P 4/m 3 2/m

La in 1a, B in 6f

x = 0.207

Describing crystal structuresNeed: lattice parameters space group equipoints occupied positional parameters
describing crystal structures

In P 4/m 3 2/m

1a - (0,0,0)

6f - (x,1/2,1/2)

6f - (1/2,x,1/2)

6f - (1/2,1/2,x)

Describing crystal structures

LaB6 a = 4.1566 Å

P 4/m 3 2/m

La in 1a, B in 6f

x = 0.207

6f - (x,1/2,1/2)

1/2

0.207

1/2

1/2

6f - (1/2,x,1/2)

0.793

1/2

6f - (1/2,1/2,x)

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Exercises1. C a = 3.500Å F 41/d 3 2/m C in 8a F (0,0,0)+ (1/2,1/2,0)+ (1/2 ,0,1/2)+ (0,1/2,1/2)+ 8a: (0,0,0) (3/4,1/4,3/4) projection down [001]2. CaCu5 a = 5.082, c = 4.078Å P6/mmm Ca in 1a, Cu in 2c, 3g 2c: (1/3,2/3,0) (2/3,1/3,0) 3g: (1/2,0,1/2) (0,1/2,1/2) (1/2,1/2,1/2) projection down [001]