# Space Groups

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 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 for rhombohedral

(2/3, 1/3, 1/3) (1/3, 2/3, 2/3) multiplicity = 3 Trigonal system Screw Diads

21 is a 180 rotation plus 1/2 cell translation. Screw Triads 31 is a 120 rotation plus a 1/3 cell translation. 32 is a 120 rotation plus a 2/3 cell translation

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

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

65 is a 300 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 a b c Axis

Plane 21 2 21 b c

n P 21/b 2/c 21/n = Pbcn Pbcn (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)

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