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III Crystal Symmetry

III Crystal Symmetry. 3-3 Point group and space group. Point group. Symbols of the 32 three dimensional point groups. Rotation axis X. x. Rotation-Inversion axis . X + centre ( inversion ): Include for odd order. 2 or m even: only for even rotation symmetry .

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III Crystal Symmetry

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  1. III Crystal Symmetry 3-3 Point group and space group • Point group Symbols of the 32 three dimensional point groups Rotation axis X x Rotation-Inversion axis

  2. X + centre (inversion): Include for odd order 2 ormeven:only for even rotation symmetry X2 + centre; Xm+centre: the same result (Include mfor odd order)

  3. Ratationaxis with mirror plane normal to it X/m x x mirror Rotation axis with mirror plane (planes) parallel to it Xm x mirror x

  4. Rotation axis with diad axis (axes) normal to it X2 x x Rotation-inversion axis with diad axis (axes) normal to it 2

  5. Rotation-inversion axis with mirror plane (planes) parallel to it m mirror Rotation axis with mirror plane (planes) normal to it and mirror plane (planes) parallel to it X/mm x mirror x mirror

  6. Triclinic 1 Rotation axis X Rotation-Inversion axis X + centre Include (odd order) For odd order includes already!

  7. Monoclinic 1stsetting X 2 = m X + centre Include (odd order) 2 mirror mirror = m 2

  8. 1 Monoclinic 2st setting 2 X2 12 Xm 1m 2 ormeven 2/m X2 + centre, Xm+centre Include m (odd order) 2/m

  9. Orthorhombic 2/m Rotation axis X Already discussed Rotation-Inversion axis 2/m X + centre Include (odd order) 2/m 222 X2 Xm 2mm (2D) = mm2 2/m 2 ormeven X2 + centre, Xm+centre Include m (odd order) mmm 2/m2/m2/m

  10. Tetragonal 4 Rotation axis X Rotation-Inversion axis X + centre Include (odd order) 422 X2 Xm 4mm 2 ormeven X2 + centre, Xm+centre Include m (odd order) 4/mmm 4/m 2/m 2/m

  11. Trigonal 3 Rotation axis X Rotation-Inversion axis 2/m X + centre Include (odd order) 32 X2 Xm 3m 2 ormeven 2/m X2 + centre, Xm+centre Include m (odd order)

  12. Hexagonal 6 Rotation axis X Rotation-Inversion axis X + centre Include (odd order) 622 X2 Xm 6mm 2 ormeven X2 + centre, Xm+centre Include m (odd order) 6/mmm 6/m 2/m 2/m

  13. Cubic 23 Rotation axis X Rotation-Inversion axis 2/m m3 2/m X + centre Include (odd order) 432 X2 Xm 2/m 2 ormeven X2 + centre, Xm+centre Include m (odd order) m3m 4/m 2/m

  14. Examples of point group operation #1 Point group 222 • At a general position [x y z], the symmetry • is 1, Multiplicity = 4 The multiplicity tells us how many atoms are generated by symmetry if we place a single atom at that position. y x

  15. (2)At a special position [100], the symmetry is • 2. Multiplicity = 2 At a special position [010], the symmetry is 2. Multiplicity = 2 At a special position [001], the symmetry is 2. Multiplicity = 2

  16. #2 Point group 4 • At a general position [x y z], the symmetry • is 1. Multiplicity = 4

  17. (2) At a special position [001], the symmetry is • 4. Multiplicity = 1 #3 Point group

  18. At a general position [x y z], the symmetry • is 1. Multiplicity = 4 • (2) At a special position [001], the symmetry is • . Multiplicity = 2

  19. (3) At a special position [000], the • symmetry is. Multiplicity = 1

  20. Transformation of vector components Original vector is [, , ] i.e. When symmetry operation transform the original axesto the new axes New vector after transformation of axes becomes i.e.

  21. The angular relations between the axes may be specified by drawing up a table of direction cosines.

  22. Then i.e. In a dummy notation Similarly i.e.

  23. Moreover, by repeating the argument for the reverse transformation and we have Similarly, i.e. “old” in terms of “new”

  24. For example: #1 Point group 4 The direction cosines for the first operation is

  25. After symmetry operation, the new position is [x y z] in new axes. We can express it in old axes by i.e. [ or

  26. 14 plane lattices + 32 point groups  230 Space groups

  27. B. Space group Table for all space groups Look at the notes! Good web site to read about space group http://www.uwgb.edu/dutchs/SYMMETRY/3dSpaceGrps/3dspgrp.htm http://img.chem.ucl.ac.uk/sgp/mainmenu.htm

  28. Symmetry elements in space group • Point group • Translation symmetry + point group • Translational symmetry operations The first character: P: primitive A, B, C: A, B, C-base centered F: Face centered I: Body centered R: Romohedral

  29. Glide plane also exists for 3D space group with more possibility Symmetry planes normal to the plane of projection

  30. Symmetry planes normal to the plane of projection Projection plane

  31. Symmetry planes parallel to plane of projection 3/8 1/8 The presence of a d-glide plane automatically implies a centered lattice!

  32. Glide planes ---- translation plus reflection across the glide plane * axial glide plane (glide plane along axis) ---- translation by half lattice repeat plus reflection ---- three types of axial glide plane • i. a glide, b glide, c glide (a, b, c) • along line in plane along line parallel to projection plane

  33. e.g. b glide , b --- graphic symbol for the axial glide plane along y axis c.f. mirror (m) graphic symbol for mirror

  34. If the axial glide plane is normal to projection plane, the graphic symbol change to c glide glide plane⊥ axis If b glide plane is ⊥ axis glide plane symbol

  35. b , underneath the glide plane • c glide: along z axis or along [111] on rhombohedral axis

  36. ii. Diagonal glide (n) , , or (tetragonal, cubic system) If glide plane is perpendicular to the drawing plane (xyplane), the graphic symbol is If glide plane is parallel to the drawing plane, the graphic symbol is

  37. iii. Diamond glide (d) , (tetragonal, cubic system)

  38. Symbols of symmetry axes

  39. i. All possible screw operations screw axis --- translation τ plus rotation screw Rn along c axis = counterclockwise rotation o + translation

  40. 2 21 3 31 32 4 41 42 43

  41. 6 61 62 63 64 65

  42. 62

  43. Symmorphic space group is defined as a space group that may be specified entirely by symmetry operation acting at a common point (the operations need not involve τ) as well as the unit cell translation. (73 space groups) Nonsymmorphic space group is defined as a space group involving at least a translation τ.

  44. Cubic – The secondary symmetry symbol will always be either 3 or –3 (i.e. Ia3, Pm3m, Fd3m) • Tetragonal – The primary symmetry symbol will always be either 4, (-4), 41, 42 or 43 (i.e. P41212, I4/m, P4/mcc) • Hexagonal – The primary symmetry symbol will always be a 6, (-6), 61, 62, 63, 64 or 65 (i.e. P6mm, P63/mcm) • Trigonal – The primary symmetry symbol will always be a 3, (-3) 31 or 32 (i.e P31m, R3, R3c, P312) • Orthorhombic – All three symbols following the lattice descriptor will be either mirror planes, glide planes, 2-fold rotation or screw axes (i.e. Pnma, Cmc21, Pnc2) • Monoclinic – The lattice descriptor will be followed by either a single mirror plane, glide plane, 2-fold rotation or screw axis or an axis/plane symbol (i.e. Cc, P2, P21/n) • Triclinic – The lattice descriptor will be followed by either a 1 or a (-1).

  45. Examples Space group P1

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