Lecture 8 (10/09/2006) Crystallography Part 1: Symmetry OperationsCrystallographyRecognition of Mineral SymmetryExternal Symmetry Elements Crystal MorphologyExternal Symmetry Elements Motif PatternsRotational AxesRotation with Inversion (Rotoinversion) Equivalent to other symmetry operationsCombination of Symmetry Elements – Multiple Rotational AxesCombination of Symmetry Elements – Multiple Rotational Axes with 32 SymmetryCombination of Symmetry Elements – Multiple Rotational Axes in the Cubic SystemNext LectureLecture 8 (10/09/2006)Lecture 8 (10/09/2006)CrystallographyCrystallographyPart 1: Symmetry Part 1: Symmetry OperationsOperationsCrystallographyCrystallographyThe study of crystalline solids and the The study of crystalline solids and the principles that govern their growth, principles that govern their growth, external shape, and internal structureexternal shape, and internal structureCrystalCrystal – a solid bounded by geometrically-arranged – a solid bounded by geometrically-arranged smooth facessmooth facesPerfection of crystal face development:Perfection of crystal face development:Euhedral – perfect crystal facesEuhedral – perfect crystal facesSubhedral – imperfectly developed crystal facesSubhedral – imperfectly developed crystal facesAnhedral – no crystal facesAnhedral – no crystal facesRecognition of Mineral SymmetryRecognition of Mineral SymmetryExternal Crystal Form (Morphology)External Crystal Form (Morphology)- macroscopic observations/measurements of - macroscopic observations/measurements of crystal facescrystal faces- 32 crystal classes (point groups) possible symmetry - 32 crystal classes (point groups) possible symmetry elements or combinations of elementselements or combinations of elements- 1- 1stst Half of Chapter 5 (p. 170-213) Half of Chapter 5 (p. 170-213)Internal Atomic ArrangementInternal Atomic Arrangement - determined from X-ray diffraction- determined from X-ray diffraction- takes into account 3D translation- takes into account 3D translation- 14 lattice types (Bravais lattices)- 14 lattice types (Bravais lattices)- Combination of 32 point groups and 14 lattice - Combination of 32 point groups and 14 lattice types yields 230 space groups types yields 230 space groups - 2- 2ndnd Half of Chapter 5 (p. 213-239) Half of Chapter 5 (p. 213-239)External Symmetry ElementsExternal Symmetry ElementsCrystal MorphologyCrystal MorphologyRotation Rotation Reflection Reflection Center of Symmetry Rotation with Center of Symmetry Rotation with InversionInversionExternal Symmetry ElementsExternal Symmetry ElementsMotif PatternsMotif PatternsRotationRotationReflectionReflectionCenter of Center of SymmetrySymmetryRotation with Rotation with InversionInversionRotational AxesRotational Axes As an external symmetry element, As an external symmetry element, rotation increments (n) range from 1 to ∞rotation increments (n) range from 1 to ∞When considering the When considering the limits of translation, limits of translation, rotation increments arerotation increments arelimited to 1, 2, 3, 4, limited to 1, 2, 3, 4, and 6-fold and 6-foldRotation with Inversion Rotation with Inversion (Rotoinversion)(Rotoinversion)Equivalent to other symmetry Equivalent to other symmetry operationsoperationsEquivalent to Center Equivalent to Center of Symmetryof SymmetryCombination of Symmetry Elements – Combination of Symmetry Elements – Multiple Rotational AxesMultiple Rotational Axes Axes at 90º Axes at 90º (except (except 3-fold axes in cubic 3-fold axes in cubic symmetry at 54º44’)symmetry at 54º44’) Axes intersect at Axes intersect at pointpointPossible symmetry Possible symmetry combinations: combinations: 422, 622, 222, 32, 422, 622, 222, 32, 23, 43223, 432(View 422 Symmetry.ai)(View 422 Symmetry.ai)Combination of Symmetry Elements –Combination of Symmetry Elements – Multiple Rotational Axes with 32 SymmetryMultiple Rotational Axes with 32 Symmetry- motif projections do not require a second set of 2-fold axesCombination of Symmetry Elements – Combination of Symmetry Elements – Multiple Rotational Axes in the Cubic SystemMultiple Rotational Axes in the Cubic System 432 Point Group432 Point GroupNext LectureNext LectureExternal Symmetry (Continued)External Symmetry (Continued)Combination of Rotation Axes and Combination of Rotation Axes and Mirror PlanesMirror Planes32 Crystal Classes/Point Groups32 Crystal Classes/Point Groups6 Crystal Systems6 Crystal SystemsCrystal MorphologyCrystal MorphologyRead: Chapter 5, p. 181-192Read: Chapter 5, p.