Crystallography lll. 230 space groups Combine 32 point groups (rotational symmetry) with a. 14 Bravais lattices (translational symmetry) b.glide planes.

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Presentation transcript:

crystallography lll

230 space groups Combine 32 point groups (rotational symmetry) with a. 14 Bravais lattices (translational symmetry) b.glide planes (rotational + translational symmetry) - a, b, c, n, d c.screw axes (rotational + translational symmetry) - 2 1, 3 1, 3 2, 4 1, 4 2,4 3, 6 1, 6 2, 6 3, 6 4, space groups Combine 32 point groups (rotational symmetry) with a. 14 Bravais lattices (translational symmetry) b.glide planes (rotational + translational symmetry) - a, b, c, n, d c.screw axes (rotational + translational symmetry) - 2 1, 3 1, 3 2, 4 1, 4 2,4 3, 6 1, 6 2, 6 3, 6 4, 6 5 lattice centering screw axis glide plane 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 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

Space group diagrams Example: P222 Space group diagrams Example: P222 a b

Space group diagrams Example: P222 Space group diagrams Example: P222 a b 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 tables International Tables for Crystallography, Vol. A

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

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

Exercises 1. C a = 3.500Å F 4 1 /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. CaCu 5 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]