Lecture 13 (11/1/2006) Crystallography Part 6: 3-D Internal Order

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Lecture 13 (11/1/2006)CrystallographyPart 6
3-D Internal Order SymmetrySpace (Bravais)
LatticesSpace Groups
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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
• c?b
• c?a
• g a?b

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14 Types of Space Lattices (Bravais Lattices)
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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
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Comparison of Symmetry Operations affecting
Motifs, Plane Lattices, and Space Lattices
External Symmetry Internal Symmetry Point
Motifs/Groups 5 Plane Lattices 14 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)
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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)
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240 Space Groups
Triclinic Monoclinic Orthorhombic Tetrag
onal 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)
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