Crystallography lll.

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Presentation transcript:

crystallography lll

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 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

Space group diagrams Example: P222 b a

General equipoint: 4u (x,y,z) (x,y,z) (x,y,z) (x,y,z) Space group diagrams Example: P222 b x,y,z a 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 tables International Tables for Crystallography, Vol. A

Describing crystal structures Need:. lattice parameters. space group Describing crystal structures Need: lattice parameters space group equipoints occupied positional parameters Example: LaB6 a = 4.1566 Å P 4/m 3 2/m La in 1a, B in 6f x = 0.207

Describing crystal structures LaB6 a = 4.1566 Å P 4/m 3 2/m La in 1a, B in 6f x = 0.207 In P 4/m 3 2/m 1a - (0,0,0) 6f - (x,1/2,1/2) 1/2 6f - (x,1/2,1/2) 0.207 6f - (1/2,x,1/2) 1/2 1/2 0.793 6f - (1/2,x,1/2) 1/2 6f - (1/2,1/2,x) 6f - (1/2,1/2,x)

Exercises 1. C a = 3. 500Å F 41/d 3 2/m C in 8a Exercises 1. 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]