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 scale Symmetry Symmetry 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 systems 2 D Symmetry Operators Mirror Planes m reflection along a plane A line denotes mirror planes 2 D Symmetry Operators Rotation Axes 1 2 3 4 or 6 rotation of 360 180 120 90 or 60 around a rotation axis yields no change in orientation arrangement 2 fold 3 fold 4 fold 6 fold 2 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 combinations 4mm 3 D Symmetry Operators Mirror Planes m reflection along any plane in 3 D space 3 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 60 around a rotation axis through any angle yields no change in orientation arrangement 3 D Symmetry Operators Inversion i symmetry with respect to a point called an inversion center 1 1 3 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 functions 3 D Symmetry New Symmetry Elements 4 Rotoinversion d 4 fold rotoinversion 4 3 D Symmetry New Symmetry Elements 4 Rotoinversion d 4 fold rotoinversion 4 1 Rotate 360 4 3 D Symmetry New Symmetry Elements 4 Rotoinversion d 4 fold rotoinversion 4 1 Rotate 360 4 2 Invert 3 D Symmetry New Symmetry Elements 4 Rotoinversion d 4 fold rotoinversion 4 1 Rotate 360 4 2 Invert 3 D Symmetry New Symmetry Elements 4 Rotoinversion d 4 fold rotoinversion 4 3 Rotate 360 4 3 D Symmetry New Symmetry Elements 4 Rotoinversion d 4 fold rotoinversion 4 3 Rotate 360 4 4 Invert 3 D Symmetry New Symmetry Elements 4 Rotoinversion d 4 fold rotoinversion 4 3 Rotate 360 4 4 Invert 3 D Symmetry New Symmetry Elements 4 Rotoinversion d 4 fold rotoinversion 4 5 Rotate 360 4 3 D Symmetry New Symmetry Elements 4 Rotoinversion d 4 fold rotoinversion 4 5 Rotate 360 4 6 Invert 3 D Symmetry New Symmetry Elements 4 Rotoinversion d 4 fold rotoinversion 4 This is also a unique operation 3 D Symmetry New Symmetry Elements 4 Rotoinversion d 4 fold rotoinversion 4 A more fundamental representative of the pattern 3 D Symmetry New Symmetry Elements 4 Rotoinversion c 3 fold rotoinversion 3 5 3 1 This is unique 4 6 2 3 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 m 3 D Symmetry 3 D symmetry element combinations a Rotation axis parallel to a mirror Same as 2 D 2 m 2mm 3 m 3m also 4mm 6mm b Rotation axis mirror 2 m 2 m 3 m 3 m also 4 m 6 m c Most other rotations m are impossible Point 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 systems 3 D Symmetry The 32 3 D Point Groups Regrouped by Crystal System more later when we consider translations Crystal System No Center Center 1 1 Monoclinic 2 2 m 2 m Orthorhombic 222 2mm 2 m 2 m 2 m Tetragonal 4 4 422 4mm 42m 4 m 4 m 2 m 2 m Hexagonal 3 32 3m 3 3 2 m 6 6 622 6mm 62m 6 m 6 m 2 m 2 m 23 432 43m 2 m 3 4 m 3 2 m Triclinic Isometric Table 5 3 of Klein 2002 Manual of Mineral Science John Wiley and Sons Hexagonal class Rhombohedral form Hexagonal form Crystal Morphology habit Nicholas Steno 1669 Law of Constancy of Interfacial Angles 120o 120o 120o Quartz 120o 120o 120o 120o Crystal Morphology Diff planes have diff atomic environments Crystal Morphology Growth 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 face Note that the internal order of the atoms can be the same but the crystal habit can be different Crystal Morphology How 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 indexing Miller Index is the accepted indexing method It uses the relative intercepts of the face in question with the crystal axes Miller Indices
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