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
g = 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
Point Motifs/Groups
Internal Symmetry
5 Plane Lattices
14 Space Lattices
No Translation
Translation in 2D
Translation in 3D
Rotation Pts/Axes
Rotation Points
Rotation Axes
Mirror Lines/Planes
Mirror Lines
Mirror Planes
Glide Lines
Glide Planes
Center of Symmetry (3D)
Roto-inversion (3D)
10 2D Point Motifs
(Fig. 5.55)
32 3D Point Groups
(Fig. 5.20)
Screw Axes
17 Plane Groups
(Fig. 5.59)
240 Space Groups
(Table 5.10)
Screw Axis
Operations
Right-handed – motif
moves clockwise when
screwed downward
Left-handed – motif
moves counterclockwise 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)
Triclinic
Monoclinic
Orthorhombic
Tetragonal
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
Hexagonal
Isometric
Glide Plane notation
indicates the direction of
glide – a, b, c, n
(diagonal) or d (diamond)
Next Lecture
Crystallography
Jeopardy
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