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MIN12FAL_03_Internal Symmetry_Space Groups_Handout

The bottom row are examples of plane groups that

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Unformatted text preview: atible 17 Plane symmetry operations. The bottom row are examples of plane Groups that correspond to each lattice type. 5 unique 2D plane lattices + 10 2D point groups 17 plane groups Repeating Repeating atoms or groups of atoms can be represented by a simple array of points (lattice points). 3-D Translations and Lattices z Different ways to combine 3 non-parallel, non-coplanar axes nonnon- z Really deals with translations compatible with 32 point groups (or crystal classes) z 32 point groups fall into 6 categories, and so do the 3D crystal +c lattices lattices. 3-D Lattice Types Name axes angles +a azbzc Monoclinic azbzc Orthorhombic azbzc D E J = 90 o a 1 = a2 z c D E J = 90 o Tetragonal Hexagonal Hexagonal (4 axes) Rhombohedral Tridymite: Orthorhombic C cell E o Triclinic Isometric a 1 = a2 = a3 z c D z E z J z 90 o J = 90 E z 90 D o E = 90 J o o D E J z 90 a1 = a 2 = a3 D E J = 90 J D +b 120 a1 = a 2 = a3 o o Axial convention: “right“right-hand rule” c c c b a b b P a Triclinic Dz Ez J azbzc Primitive P c c I=C a Monoclinic BodyBody-centered D J R z E azbzc a2 a2 a1 P D 14 Bravais lattices P or C R Hexagonal Tetragonal E J R a1 = a2 z c c French mineralogist b Auguste Bravais a P (1850) C face-centered face- a1 I D Rhombohedral E R J R D E J z R a1 = a2 = a3 a1 a2 z c a3 C D F I Orthorhombic Face-centered FaceE J R a z b z c French mineralogist a Auguste Bravais 1 (1850) a2 P 14 Bravais lattices F D I Isometric E J R a1 = a2 = a3 Monoclinic Monoclinic Tetragonal Lattices Tetra Monoclinic B = P Monoclinic I = C Monoclinic Monoclinic F = C simply by a selection of appropriate crystallographic axes. Monoclinic D=JzE azbzc Monoclinic C  P No way to find any new options of crystallographic of crystallographic axes to make C = P while retaining a monoclinic lattice. Not a monoclinic cell. = 0, 1, 2... = 0.5, 1.5, 2.5 ... What’s this? = I cell in isometric or tetragona...
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