UVM GEOL 110 - Lecture 10 - Crystal Symmetry

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Crystal Chem  Crystallography• Chemistry behind minerals and how they are assembled– Bonding properties and ideas governing how atoms go together– Mineral assembly – precipitation/ crystallization and defects from that• Now we will start to look at how to look at, and work with, the repeatable structures which define minerals.– This describes how the mineral is assembled on a larger scaleSymmetrySymmetry Introduction• Symmetry defines the order resulting from how atoms are arranged and oriented in a crystal• Study the 2-D and 3-D order of minerals• Do this by defining symmetry operators (there are 13 total)  actions which result in no change to the order of atoms in the crystal structure• Combining different operators gives point groups –which are geometrically unique units.• Every crystal falls into some point group, which are segregated into 6 major crystal systems2-D Symmetry Operators• Mirror Planes (m) – reflection along a planeA line denotes mirror planes2-D Symmetry Operators• Rotation Axes (1, 2, 3, 4, or 6) – rotation of 360, 180, 120, 90, or 60º around a rotation axis yields no change in orientation/arrangement2-fold3-fold4-fold6-fold2-D Point groups• All possible combinations of the 5 symmetry operators: m, 2, 3, 4, 6, then combinations of the rotational operators and a mirror yield 2mm, 3m, 4mm, 6mm• Mathematical maximum of 10 combinations4mm3-D Symmetry Operators• Mirror Planes (m) – reflection along any plane in 3-D space3-D Symmetry Operators• Rotation Axes (1, 2, 3, 4, or 6 a.k.a. A1, A2, A3, A4, A6) – rotation of 360, 180, 120, 90, or 60º around a rotation axis through any angle yields no change in orientation/arrangement3-D Symmetry Operators• Inversion (i) – symmetry with respect to a point, called an inversion center113-D Symmetry Operators• Rotoinversion (1, 2, 3, 4, 6 a.k.a. A1, A2, A3, A4, A6) – combination of rotation and inversion. Called bar-1, bar-2, etc.• 1,2,6 equivalent to other functions3-D SymmetryNew Symmetry Elements4. Rotoinversiond. 4-fold rotoinversion ( 4 )3-D SymmetryNew Symmetry Elements4. Rotoinversiond. 4-fold rotoinversion ( 4 )1: Rotate 360/43-D SymmetryNew Symmetry Elements4. Rotoinversiond. 4-fold rotoinversion ( 4 )1: Rotate 360/42: Invert3-D SymmetryNew Symmetry Elements4. Rotoinversiond. 4-fold rotoinversion ( 4 )1: Rotate 360/42: Invert3-D SymmetryNew Symmetry Elements4. Rotoinversiond. 4-fold rotoinversion ( 4 )3: Rotate 360/43-D SymmetryNew Symmetry Elements4. Rotoinversiond. 4-fold rotoinversion ( 4 )3: Rotate 360/44: Invert3-D SymmetryNew Symmetry Elements4. Rotoinversiond. 4-fold rotoinversion ( 4 )3: Rotate 360/44: Invert3-D SymmetryNew Symmetry Elements4. Rotoinversiond. 4-fold rotoinversion ( 4 )5: Rotate 360/43-D SymmetryNew Symmetry Elements4. Rotoinversiond. 4-fold rotoinversion ( 4 )5: Rotate 360/46: Invert3-D SymmetryNew Symmetry Elements4. Rotoinversiond. 4-fold rotoinversion ( 4 )This is also a unique operation3-D SymmetryNew Symmetry Elements4. Rotoinversiond. 4-fold rotoinversion ( 4 )A more fundamental representative of the pattern3-D SymmetryNew Symmetry Elements4. Rotoinversionc. 3-fold rotoinversion ( 3 )This is unique1652343-D Symmetry Operators• Mirror planes ┴rotation axes (x/m) – The combination of mirror planes and rotation axes that result in unique transformations is represented as 2/m, 4/m, and 6/m3-D Symmetry3-D symmetry element combinationsa. Rotation axis parallel to a mirrorSame as 2-D2 || m = 2mm3 || m = 3m, also 4mm, 6mmb. Rotation axis mirror2 m = 2/m3 m = 3/m, also 4/m, 6/mc. Most other rotations + m are impossiblePoint Groups• Combinations of operators are often identical to other operators or combinations – there are 13 standard, unique operators• I, m, 1, 2, 3, 4, 6, 3, 4, 6, 2/m, 4/m, 6/m• These combine to form 32 unique combinations, called point groups• Point groups are subdivided into 6 crystal systems3-D SymmetryThe 32 3-D Point GroupsRegrouped by Crystal System(more later when we consider translations)Crystal System No Center CenterTriclinic 1 1Monoclinic 2, 2 (= m) 2/mOrthorhombic 222, 2mm 2/m 2/m 2/mTetragonal 4, 4, 422, 4mm, 42m 4/m, 4/m 2/m 2/mHexagonal 3, 32, 3m 3, 3 2/m6, 6, 622, 6mm, 62m 6/m, 6/m 2/m 2/mIsometric 23, 432, 43m 2/m 3, 4/m 3 2/mTable 5.3 of Klein (2002) Manual of Mineral Science, John Wiley and SonsHexagonal classRhombohedralformHexagonalformCrystal Morphology (habit)Nicholas Steno (1669): Law of Constancy of Interfacial AnglesQuartz120o120o120o120o120o120o120oCrystal MorphologyDiff planes have diff atomic environmentsCrystal MorphologyGrowth of crystal is affected by the conditions and matrix from which they grow. That one face grows quicker than another is generally determined by differences in atomic density along a crystal faceNote that the internal order of the atoms can be the same but the crystal habit can be different!Crystal MorphologyHow do we keep track of the faces of a crystal?Face sizes may vary, but angles can't Thus it's the orientation & angles that are the best source of our indexingMiller Index is the accepted indexing methodIt uses the relative intercepts of the face in question with the crystal axesMiller


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UVM GEOL 110 - Lecture 10 - Crystal Symmetry

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