Lecture 9 (10/11/2006)
Crystallography
Part 2:
Multiple Symmetry Operations
Crystal Morphology
Rotation with Inversion
(Rotoinversion)
Equivalency to other symmetry
operations
Combination of Symmetry Elements –
Multiple Rotational Axes

Axes at 90º (except
3-fold axes in cubic
symmetry at 54º44’)
Axes intersect at a
point
 Possible symmetry
combinations:
422, 622, 222, 32,
23, 432

(View 422 Symmetry.ai)
Combination of Symmetry Elements –
Multiple Rotational Axes and Mirrors
A#
m
-mirror plane
perpendicular
to rotational
axis
Hermann-Maugin notation
for Crystal Classes (Point Groups)
Relationship of Mirrors and
Rotational Axes
Line traced by intersecting of X
mirrors corresponds to X-fold
rotation axis
32 Point Groups (Crystal Classes)
32 Crystal Classes grouped by
Crystal System
Least
Symmetry
Greatest
Symmetry
Graphical Representation of Point Groups
Crystal Morphology
The angular relationships, size and shape
of faces on a crystal
 Bravais Law – crystal faces will most
commonly occur on lattice planes with the
highest density of atoms

Planes AB and AC will be the most
common crystal faces in this cubic
lattice array
Steno’s Law of Interfacial Angles

Angles between adjacent crystal faces will
be constant, regardless of crystal shape
and size.
Paradox of the
Growth of
Crystal Faces
Lattice planes with the highest
density are the most stable, but
experience slow growth due to
the abundance of atoms needed
to construct them.
These stable faces will appear at
the nucleation stages of growth
(1), but then will diminish due to
fast growth in these directions
(2-4).
Next Lecture
Crystal Symmetry (Continued)
 Crystallographic Axes
 Numerical Notation of Crystal Faces
and Atomic Planes – Miller Indicies
Read: Chapter 5, p. 192-201