Rotation with Inversion (Rotoinversion) Equivalency to other symmetry operations

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Presentation transcript:

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 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