1. Lecture 9 (10/11/2006) Crystallography Part 2: Multiple Symmetry Operations Crystal Morphology

2. Rotation with Inversion (Rotoinversion) Equivalency to other symmetry operations

3. 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)

4. Combination of Symmetry Elements – Multiple Rotational Axes and Mirrors
mirror plane
perpendicular
to rotational
axis

5. Hermann-Maugin notation for Crystal Classes (Point Groups)

6. Relationship of Mirrors and Rotational Axes
Line traced by intersecting of X mirrors corresponds to X-fold rotation axis

7. 32 Point Groups (Crystal Classes)

8. 32 Crystal Classes grouped by Crystal System
Least
Symmetry
Greatest
Symmetry

9. Graphical Representation of Point Groups

10. 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

11. Steno’s Law of Interfacial Angles
Angles between adjacent crystal faces will be constant, regardless of crystal shape and size.

12. 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).

13. Next Lecture Crystal Symmetry (Continued) Crystallographic Axes
Numerical Notation of Crystal Faces and Atomic Planes – Miller Indicies
Read: Chapter 5, p