Internal Symmetry of Crystals

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Presentation transcript:

Internal Symmetry of Crystals Space Groups Internal Symmetry of Crystals

Space Groups If translation operations are included with rotation and inversion, We have 230 three-dimensional space groups. Translation operations Unit cell translations Centering operations (Lattices) (A, B, C, I, F, R) Glide planes (reflection + translation) (a, b, c, n, d) Screw axes (rotation + translation) (21, 31, 32)

Space Groups 230 three-dimensional space groups Hermann-Mauguin symbols. (4 positions) First position is Lattice type (P, A, B, C, I, F or R) Second, third and fourth positions as with point groups

Lattices (centering operations) P is for primitive

Lattices A, B, and C are end-centered

Lattices I is body-centered F is face-centered point @ (0.5, 0.5, 0.5) multiplicity = 2 F is face-centered 0.5, 0.5, 0 0.5, 0, 0.5 0, 0.5, 0.5 multiplicity = 4

Lattices R is rhombohedral (2/3, 1/3, 1/3) (1/3, 2/3, 2/3) multiplicity = 3 Trigonal system

Rotation + Translation: Screw Diads 21 is a 180º rotation plus 1/2 cell translation.

Screw Triads 31 is a 120º rotation plus a 1/3 cell translation.

Screw Tetrads 41 is a 90º rotation plus a 1/4 cell translation (right-handed). 42 is a 180º rotation plus a 1/2 cell translation (no handedness). 43 is a 90º rotation plus a 3/4 cell translation (left-handed).

Screw Hexads 61 is a 60º rotation plus a 1/6 cell translation (right-handed). 62 is a 120º rotation plus a 1/3 cell translation (right-handed). 63 is a 180º rotation plus a 1/2 cell translation (no handedness). 64 is a 240º rotation plus a 2/3 cell translation (left-handed). 65 is a 300º rotation plus a 5/6 cell translation (left-handed).

Screw Hexads

Glide Planes Reflection plus 1/2 cell translation a-glide: a/2 translation b-glide: b/2 translation c-glide: c/2 translation n-glide(normal to a): b/2+c/2 translation n-glide(normal to b): a/2+c/2 translation n-glide(normal to c): a/2+b/2 translation d-glide (tetragonal + cubic systems)

Space Group Symmetry Diagrams a-vertical b-horizontal c-normal to page

Space Group Symmetry Diagrams Axis Plane a 21 b b 2 c c 21 n P 21/b 2/c 21/n = Pbcn

Space Group Symmetry Diagrams Axis Plane a 2 b b 2 a c 2 n P 2/b 2/a 2/n = Pban

Pbcn (x, y, z) (-x, -y, -z) (x, -y 1/2+z) (-x, y 1/2-z)

Pbcn General (0.1, 0.2, 0.3) (x, y, z) (0.9, 0.8, 0.7) (-x, -y, -z) (0.1, 0.8, 0.8) (0.9, 0.2, 0.2) (0.4, 0.7, 0.3) (0.6, 0.3, 0.7) (0.4, 0.3, 0.8) (0.6, 0.7, 0.2) (x, y, z) (-x, -y, -z) (x, -y 1/2+z) (-x, y 1/2-z) (1/2 - x, 1/2 + y, z) (1/2 + x, 1/2 - y, -z) (1/2 - x, 1/2 - y, 1/2 + z) (1/2 + x, 1/2 + y, 1/2 - z)

Pbcn Special (x, y, z) (-x, -y, -z) (x, -y 1/2+z) (-x, y 1/2-z) (0, 0, 0) (0, 0, 0.5) (0.5, 0.5, 0) (0.5, 0.5, 0.5)

Enstatite Pbcn Cell a=9.306; b=8.892 c= 5.349 Å Mg1 (0, 0.1006, 0.25) Si (0.2928, 0.0897,0.0739) O1 (0.1200, 0.0942, 0.0770) O2 (0.3773, 0.2463, 0.0677) O3 (0.3481, 0.9836, 0.3079)

Enstatite Pbcn

Quartz P3121 Cell a=4.9137; b=4.9137 c= 5.4047Å Si (0.4697, 0.0, 0.0) O (0.4133, 0.2672, 0.1188)

Quartz P3121