1 Watkins/Fronczek - Space Groups Translational + Rotational Symmetry Introduction to Space Groups

2 Watkins/Fronczek - Space Groups Rotational Symmetry 1on1on1on1on 2 o n’ Angle n-n’ Crystal Systems (7) Crystal Classes (32) 11-TriclinicC 1, C i 21-MonoclinicC 2, C s, C 2h 2290OrthorhombicD 2, C 2v, D 2h 31 or 290TrigonalC 3, S 6, D 3, C 3v, D 3d 41 or 290TetragonalC 4, S 4, C 4h, D 4, C 4v, D 2d, D 4h 61 or 290HexagonalC 6, C 3h, C 6h, D 6, C 6v, D 3h, D 6h 2; CubicT, T h, T d ; O, O h

3 Watkins/Fronczek - Space Groups Rotational Symmetry The external form (faces, edges and vertices) of any given crystal is a symmetric pattern which corresponds to one of the 32 crystallographic point groups.

4 Watkins/Fronczek - Space Groups Translational Symmetry The internal form of a crystal is also a symmetric pattern of atoms, ions and molecules - the crystal structure. rotational symmetry translational symmetry The internal pattern combines the rotational symmetry of the external form with the translational symmetry of the lattice.

5 Watkins/Fronczek - Space Groups Translation reproduces the motif at equally spaced intervals called the repeat distance along a specific translation direction. Translational Symmetry aaaa a'

6 Watkins/Fronczek - Space Groups Translational Symmetry Translation can occur in 7 border groups1-dimension (7 border groups) 17 plane groups2-dimensions (17 plane groups) 230 space groups3-dimensions (230 space groups)

7 Watkins/Fronczek - Space Groups Definitions Symmetry GroupSymmetry Group - a set of symmetry operations which conforms to the four group postulates: - associativity - closure - identity - inverse Point GroupPoint Group - a repetitive pattern produced by a group of 2-D and 3-D rotational symmetry operations, all axes of which pass through a single point in space

8 Watkins/Fronczek - Space Groups Definitions Border GroupBorder Group - a repetitive pattern in two dimensions, with 2-D rotational symmetry and 1-D translational symmetry. Plane GroupPlane Group - a repetitive pattern in two dimensions, with 2-D rotational symmetry and 2-D translational symmetry. Space GroupSpace Group - a repetitive pattern in three dimensions, with 2-D and 3-D rotational symmetry and 3-D translational symmetry.

9 Watkins/Fronczek - Space Groups Border Groups MotifMotif - the smallest part of the pattern which contains no rotational or translational symmetry. For the border groups which follow, the motif shown below, of inherent dimension l, is chosen to illustrate all of the patterns: l

10 Watkins/Fronczek - Space Groups Simplest Border Group Translate Translate the motif in the "x" direction by distance a, and repeat to generate a pattern: This pattern has repeat distance a equal to the motif dimension l (but this is not a necessary condition). x

11 Watkins/Fronczek - Space Groups Simplest Border Group The pattern lies in the "xy" plane, and extends along the "x" axis; axis "z" is perpendicular to the plane of the pattern. 111 The rotational symmetry along each of the three directions is "1" (unit rotation), so this border group is named 111.

12 Watkins/Fronczek - Space Groups Simplest Border Group Choose a set of imaginary points on the pattern with identical surroundings (the first point is arbitrary). a These points are called lattice points. The array of points is called a lattice row and has the same repeat distance as the pattern. 111

13 Watkins/Fronczek - Space Groups Simplest Border Group A general representation of any border groups with 111 symmetry replaces the specific motif with a general equipoint. The unit cell pattern consists of one equipoint (Z = 1) a111 Z = 1

14 Watkins/Fronczek - Space Groups Border Group 2 With the chosen motif, perform a two-fold rotation somewhere in the "xy" plane about the "z" axis at a lattice point, then translate.

15 Watkins/Fronczek - Space Groups Border Group 2 The pattern is named according to the symmetry elements:112 With the chosen motif, perform a two-fold rotation somewhere in the "xy" plane about the "z" axis at a lattice point, then translate.

16 Watkins/Fronczek - Space Groups Border Group 2 The relative positions of the motif and the 2- fold symmetry element determines the unit cell content and the total pattern. 112

17 Watkins/Fronczek - Space Groups Border Group 2 Note the position of another 2-fold axis at ½a, dictated by the "closure" group postulate. a112 Z = 2

18 Watkins/Fronczek - Space Groups Border Group 3 With the chosen motif, perform a mirror operation perpendicular to the "y" axis (“mirror in y”). An improper rotation creates a motif with the opposite hand. 1m1 Z = 2

19 Watkins/Fronczek - Space Groups Border Group 4 glide plane With the chosen motif, perform a mirror operation perpendicular to "y" followed by translation of ½a along "x". This compound symmetry element is called a glide plane.

20 Watkins/Fronczek - Space Groups Border Group 4 1g1 1g1 The pattern is named 1g1 glide plane With the chosen motif, perform a mirror operation perpendicular to "y" followed by translation of ½a along "x". This compound symmetry element is called a glide plane.

21 Watkins/Fronczek - Space Groups Border Group 4 a 1g1 Z = 2

22 Watkins/Fronczek - Space Groups Border Group 5 With the chosen motif, perform a mirror operation perpendicular to the "x" axis (“mirror in x”). m11

23 Watkins/Fronczek - Space Groups Border Group 5 The lattice point spacing is a, but the spacing between mirrors is ½a a m11 Z = 2

24 Watkins/Fronczek - Space Groups Border Group 6 With the chosen motif, perform two mirror operations, one perpendicular to “x” and one perpendicular to “y” and sharing a lattice point. These two patterns have the same symmetry and the same motif. What is the difference (besides how they look)?

25 Watkins/Fronczek - Space Groups Border Group 6 Note that the unit cell (repeat) length is determined by the pattern (unit cell content), not the dimensions of the motif. a a

26 Watkins/Fronczek - Space Groups Border Group 6 Another "x" mirror is created at ½a, and 2-fold axes are generate along "z" at the intersections of perpendicular mirrors.,,,, Z = 4 mm2

27 Watkins/Fronczek - Space Groups Border Group 7 With the chosen motif, perform a mirror operation perpendicular to "x" and a glide operation perpendicular to "y".

28 Watkins/Fronczek - Space Groups Border Group 7 mg2 Another "x" mirror is created at ½a, and a 2-fold axis along "z" is generated at ¼a. The pattern is named mg2 mg2 Z = 4 a

29 Watkins/Fronczek - Space Groups Seven Border Groups Border groups 111 and 112 contain equipoints of the same hand. The remaining five border groups (next slide) contain equipoints of both hands generated by reflection. Each pattern displays a unique unit cell content and repeat (unit cell) length.

30 Watkins/Fronczek - Space Groups Seven Border Groups mg2 mm2 m11 1g1 1m1,,,,

31 Watkins/Fronczek - Space Groups More Examples of the Seven Border groups:

32 Watkins/Fronczek - Space Groups Seven Border Groups mm2 All seven patterns reside on a single array of lattice points. The lattice row itself has symmetry mm2.,,,,

33 Watkins/Fronczek - Space Groups Quiz Question: Determine the border group and lattice constant (arbitrary units) of this border pattern. Border Groups