Lecture05-ApplToSpaceGroups

Lecture05-ApplToSpaceGroups - EMSE 312 DIFFRACTION...

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Lecture 5 Applications of Space Groups EMSE 312 DIFFRACTION PRINCIPLES
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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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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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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 4 = 42 422 - D 6 = 62 622
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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 2v 2mm, C 3v 3m, C 4v 4mm, and C 6v 6mm • d for Diagonal Symmetry Plane 3d 3m •Examples: –D 4h 4/m 2/m 2/m 2d 4 m 2
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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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Spherical Geometry and Group Theory
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Spherical Geometry and Group Theory Schaum’s Outline Series Mathematical Handbook of Formulas and Tables
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Azaroff Elements of X-ray Crystallography Page 12
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Spherical Geometry and Group Theory () ( ) ( ) ()() cos / 2 cos / 2 cos / 2 cos sin / 2 sin / 2 A β⋅ γ + α = γ Schaum’s Outline Series sin sin sin sin sin sin abc A BC == cos cos cos sin sin cos cos cos cos sin sin cos ab c b c A A B Ca =⋅ + =− + Azaroff A B C a b c α β γ A B C ( ) ( ) ( ) cos / 2 cos / 2 cos / 2 cos sin / 2 sin / 2 B γ⋅ α+ β = α ( ) ( ) ( ) cos / 2 cos / 2 cos / 2 cos sin / 2 sin / 2 C α⋅ β+ γ = β
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Spherical Geometry and Group Theory Azaroff α β γ A B C Example 1: Two 2-fold axes at 90 o A BC ⋅= 22 ? 18 90 180 0 ,, o o o C α= = β= () ( ) ( ) ( ) cos / 2 cos / 2 cos / 2 cos sin / 2 sin / 2 cos 90 cos 90 cos / 2 00 co s /2 cos90 0 11 sin 90 sin 90 cos / 2 0; / 2 90 180 oo o C and α⋅ β+ γ = β ⋅+ γ γ == = γ = γ = γ = a) What is the “throw” of the third axis – i.e., what is the angle γ ?
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This note was uploaded on 12/01/2011 for the course EMSE 312 taught by Professor Lagerlof,p during the Spring '08 term at Case Western.

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Lecture05-ApplToSpaceGroups - EMSE 312 DIFFRACTION...

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