1. Lecture 8 (10/09/2006) Crystallography Part 1: Symmetry Operations

2. Crystallography
The study of crystalline solids and the principles that govern their growth, external shape, and internal structure
Crystal – a solid bounded by geometrically-arranged smooth faces
Perfection of crystal face development:
Euhedral – perfect crystal faces
Subhedral – imperfectly developed crystal faces
Anhedral – no crystal faces

3. Recognition of Mineral Symmetry
External Crystal Form (Morphology)
- macroscopic observations/measurements of crystal faces
- 32 crystal classes (point groups) possible symmetry elements or combinations of elements
- 1st Half of Chapter 5 (p )
Internal Atomic Arrangement
- determined from X-ray diffraction
- takes into account 3D translation
- 14 lattice types (Bravais lattices)
- Combination of 32 point groups and 14 lattice types yields 230 space groups
- 2nd Half of Chapter 5 (p )

4. External Symmetry Elements Crystal Morphology
Rotation Reflection Center of Symmetry Rotation with
Inversion

5. External Symmetry Elements Motif Patterns
Rotation
Reflection
Center of
Symmetry
Rotation with
Inversion

6. Rotational Axes
As an external symmetry element, rotation increments (n) range from 1 to ∞
When considering the
limits of translation,
rotation increments are
limited to 1, 2, 3, 4,
and 6-fold

7. Rotation with Inversion (Rotoinversion) Equivalent to other symmetry operations
Equivalent to Center
of Symmetry

8. Combination of Symmetry Elements – Multiple Rotational Axes
Axes at 90º (except 3-fold axes in cubic symmetry at 54º44’)
Axes intersect at point
Possible symmetry combinations:
422, 622, 222, 32, 23, 432
(View 422 Symmetry.ai)

9. Combination of Symmetry Elements – Multiple Rotational Axes with 32 Symmetry
motif projections
do not require a
second set of 2-fold
axes

10. Combination of Symmetry Elements – Multiple Rotational Axes in the Cubic System
432 Point Group

11. Next Lecture External Symmetry (Continued)
Combination of Rotation Axes and Mirror Planes
32 Crystal Classes/Point Groups
6 Crystal Systems
Crystal Morphology
Read: Chapter 5, p