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Lecture05-ApplToSpaceGroups

Lecture05-ApplToSpaceGroups - EMSE 312 DIFFRACTION...

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EMSE 312 – Diffraction Principles Lecture 5 Applications of Space Groups EMSE 312 DIFFRACTION PRINCIPLES
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EMSE 312 – Diffraction Principles 230 Space Groups • 10 Point Symmetries – 32 Point Groups • 7 Crystal Systems (defined by the point symmetry) • 5 Space Lattices (P, A (or B or C), I, F and R – 14 Bravais Lattices + 32 Point Groups gives the 73 Symmorphic Space Groups • 11 Screw Axis Symmetries – 2 1 , 3 1 , 3 2 , 4 1 , 4 2 , 4 3 , 6 1 , 6 2 , 6 3 , 6 4 and 6 5 • 5 Glide Plane Symmetries – a, b, c, n and d • 230 Unique Combinations – Space Groups
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EMSE 312 – Diffraction Principles Space Groups Symmetry About a Lattice Point – Point Symmetry (Point Group) Translation Symmetry – Lattice (Perfect Translations) Crystal System – The lattice needed for a Particular Point Symmetry Additional Translation Symmetries? – Screw Axis: Partial Translation followed by a rotation – Glide Plane: Partial Translation followed by a reflection
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EMSE 312 – Diffraction Principles Schoenflies Notation (1891) • Proper Rotation: C n (Cyclic) Groups – C 1 , C 2 , C 3 , C 4 , and C 6 (Corresponds to 1, 2, 3, 4, 6) • Improper Rotation – Inversion C i – Mirror C S (Mirror is called Spiegel in German) – Roto-Refelction (Spiegel Groups) •S 1 =C S [=m], S 2 =C i [=1], S 3 =C 3h [3/m], S 4 [=4] and S 6 = C 3i [=3] • D for Diad (2-fold axis): D n – D 2 = 22 222 - D 3 = 32 – D 4 = 42 422 - D 6 = 62 622
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EMSE 312 – Diffraction Principles Schoenflies Notation (1891) • h for Horizontal Symmetry Plane – C 2h 2/m, C 3h 3/m, C 4h 4/m, and C 6h 6/m • v for Vertical Symmetry Plane – C 2v 2mm, C 3v 3m, C 4v 4mm, and C 6v 6mm • d for Diagonal Symmetry Plane – C 3d 3m • Examples: – D 4h 4/m 2/m 2/m – D 2d 4 m 2
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EMSE 312 – Diffraction Principles Schoenflies Notation (1891) • Isometric Groups (Cubic) – O for groups having Octahedral Symmetry • Point Symmetry O for 432 and O h for 4/m 3 2/m – T for Groups having Tetrahedral Symmetry •Point Symmetry T for 23, T h for m3, and T d for 43m • Note: Schoenflies Notation will ONLY describe the Point Symmetry – the Space Groups are identified by Sequential Numbering that has no other meaning than telling different space groups apart
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EMSE 312 – Diffraction Principles
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EMSE 312 – Diffraction Principles Spherical Geometry and Group Theory
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EMSE 312 – Diffraction Principles Spherical Geometry and Group Theory Schaum’s Outline Series Mathematical Handbook of Formulas and Tables
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EMSE 312 – Diffraction Principles Azaroff Elements of X-ray Crystallography Page 12
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