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 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 = aΛb

14 Types of Space Lattices (Bravais Lattices)

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

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

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 1 st right-handed position (circled in red)

240 Space Groups 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) Triclinic Monoclinic Orthorhombic Tetragonal Hexagonal Isometric

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