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Lecture 13 (11/1/2006) Crystallography Part 6: 3-D Internal Order & Symmetry Space (Bravais) Lattices Space Groups

Lecture 13 (11/1/2006) Crystallography Part 6: 3-D Internal Order & Symmetry Space (Bravais) Lattices Space Groups. Three-Dimensional Lattices. Translation in three directions: x , y & z axes Translation distance: a along x b along y c along z

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Lecture 13 (11/1/2006) Crystallography Part 6: 3-D Internal Order & Symmetry Space (Bravais) Lattices Space Groups

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  1. Lecture 13 (11/1/2006)CrystallographyPart 6:3-D Internal Order & SymmetrySpace (Bravais) LatticesSpace Groups

  2. Three-Dimensional Lattices • Translation in three directions: x, y & z axes • Translation distance: • a along x • b along y • c along z • A lattice point in 3D space corresponds to a vector (r), which is defined by three axial vector components: a, b, and c • Angles between axes: • = cΛb • = cΛa g = aΛb

  3. 14 Types of Space Lattices (Bravais Lattices)

  4. Unit Cell Types in Bravais Lattices P – Primitive; nodes at corners only C – Side-centered; nodes at corners and in center of one set of faces (usually C) F – Face-centered; nodes at corners and in center of all faces I – Body-centered; nodes at corners and in center of cell

  5. Comparison of Symmetry Operations affecting Motifs, Plane Lattices, and Space Lattices External Symmetry Internal Symmetry Point Motifs/Groups5 Plane Lattices14 Space Lattices No Translation Translation in 2D Translation in 3D Center of Symmetry (3D) Rotation Pts/Axes Rotation Points Rotation Axes Mirror Lines/Planes Mirror Lines Mirror Planes Roto-inversion (3D) Glide Lines Glide Planes 10 2D Point Motifs Screw Axes (Fig. 5.55) 32 3D Point Groups 17 Plane Groups 240 Space Groups (Fig. 5.20) (Fig. 5.59) (Table 5.10)

  6. Screw Axis Operations Right-handed – motif moves clockwise when screwed downward Left-handed – motif moves counter-clockwise when screwed downward Notation lists rotation axis type (#) and subscript which indicates number of 1/# turns to reach the 1st right-handed position (circled in red)

  7. 240 Space Groups Triclinic Monoclinic Orthorhombic Tetragonal Hexagonal Isometric Notation indicates lattice type (P,I,F,C) and Hermann-Maugin notation for basic symmetry operations (rotation and mirrors) Screw Axis notation as previously noted Glide Plane notation indicates the direction of glide – a, b, c, n (diagonal) or d (diamond)

  8. Next Lecture Crystallography Jeopardy Bring your textbook!!

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