UT Arlington GEOL 2313 - Introduction to Crystallography

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Introduction to Crystallography!• Translational Symmetry!– Plane lattices!– Space lattices and the Unit Cell!– Bravais Lattices and Crystal Systems!• Point Symmetry!– Reflection!– Rotation!– Inversion!• Point Groups and Space Groups!• Crystal Forms!• Crystal Habit, Faces, and Crystallinity!Translational Symmetry!• Plane Lattices: linear translations may be used to define a plane lattice!– Spots represent “atoms”!– Start with simple linear translation in 1D - a!– Then impose unequal translation in a second direction - b!– Arbitrary angle a^b – is not equal to 90° and is denoted as !Lattice lines!Lattice nodes!All figures on these slides are modified from Nesse (2000)!• Translation in 2D yields five unique Plane Lattices and associated Unit Meshes:!– Square Lattice -> Square mesh!– Rectangle Lattice -> Rectangle primitive (p) mesh!– Diamond Lattice -> Rectangle centered (c) mesh!– Hexagonal Lattice -> Hexagon (rhombus) mesh!– Oblique Lattice -> Parallelogram mesh!Examples of patterns of Plane Lattices!a) Square mesh in square lattice!b) Rectangular mesh in rectangular lattice!c) Rectangular mesh in diamond lattice!d) Hexagonal mesh in hexagonal lattice!Extension of Plane Lattices to 3D!• The 5 unique 2D plane lattices may be extended to 3D by translation “above” and below the original plane!• Gives rise to a 3D system of crystallographic axes: a, b, and c whose angles are referred to as , , and $– a^b defined by angle $– b^c defined by angle $– a^c defined by angle $• Resulting 3D array is referred to as a Space Lattice and associated Unit Cell!Note origin of crystallographic axes is at their intersection and that a is positive (out of page) and –a is negative (into page)!Bravais Lattices – 14 unique space lattices!Bravais Lattices may be grouped by the similarity of their unit cell shape. !In order of increasing symmetry, this gives rise to the six basic crystal systems: triclinic, monoclinic, orthorhombic, hexagonal (rhombohedral), tetragonal, and isometric or cubic.!Triclinic primitive (P)!Note that two alternate definitions of a unit cell are shown: a) rectangle in dark gray and b) parallelogram in light gray!Bravais Lattices – primitive cells modified!The primitive (P) Bravais Lattices in each crystal system may be modified to include a node in the center of the volume or in the center of the face of the primitive unit cell – new lattices are called body centered (I) or face centered (F) for a specific system.!Crystal Systems – summary of 6 unit cell volumes!Geometric Definitions of the!6 Crystal Systems:!1) Triclinic (P)!a ≠ b ≠ c;  ≠ ≠ $2) Monoclinic (P, C)!a ≠ b ≠ c;  ≠ and  = 90°$3) Orthorhombic (P, C, F, I)!a ≠ b ≠ c;  =  =  = 90°$4) Tetragonal (P, I)!a = b ≠ c;  =  =  = 90°$5) Hexagonal (P, R)!a = b ≠ c;  =  = 90° and  = 120°$6) Isometric (P, I, F)!a = b = c;  =  =  = 90°!Point Symmetry!• Operations based on how a single motif may be repeated. Three basic operations:!– Reflection -> mirror planes!– Rotation -> angular motion about a fixed axis!– Inversion -> reflection through a point rather than a plane like a mirror!– Compound Symmetry operations, which combine all three!Compound Symmetry Operations!Simplest combination is 1-fold rotation with inversion –!This operation is trivial because it is the same as a simple inversion:!1A = iOther combinations are non-trivial, e.g. a 2-fold axis combined with inversion is the same as a mirror plane: !2A = mSymmetry Notation – Hermann-Mauguin symbols!Orthorhombic mineral contains the following symmetry elements: !i, 3A2, 3m – that is one central inversion point, three 2-fold rotation axes, and three perpendicular mirrors.!The Hermann-Mauguin designation is 2/m 2/m 2/m, which means that there are three mirrors perpendicular to three 2-fold axes. Inversion point is thus implicit.!Two Dimensional Point Groups – 10 total!Only mirrors (m) and rotations (An) may be combined in 2D. Unique combinations of both point symmetry operators gives rise to 10 Point Groups. Inversion requires 3D. When this is combined with the 10 point groups we get 32 unique 3D point groups.!Space Groups – Combination of point and translational symmetry operators!• Translational symmetry operations yield!– 14 Bravais Lattices (3D)!– Divided into 6 crystal systems based on shape of the unit cell!• Point symmetry operations yield$– 32 Point Groups (3D)$• Combining simple point and translation symmetry elements together yields!– 73 Space Groups!• Two additional symmetry operations are possible!– Glides – translation plus reflection by a mirror!– Screws – translation plus rotation!– Combining these two operations yields 157 additional space groups!• Final total is 230 possible ways to repeat a motif in 3D or 230 Space Groups!Complex Symmetry Operations: Glides and Screws!Glide: Translation + mirror Single chain silicates - pyroxenes!Screw: Translation + rotation !3D network silicates - quartz!Before the advent of X-ray crystallography, mineralogists attempted to elucidate the symmetry elements in specific minerals by examining well-formed examples of crystals:!Nicolas Steno (1669) observed that the angles between crystal faces for a mineral were always the same. Hence Steno’s Law:!The angle between equivalent faces of crystals of the same mineral are the same.!Steno’s Law – Constancy of Interfacial Angles!Motifs from Escher and Schaschl!Escher Butterfly Pattern!Schaschl Iselberg Pattern!What symmetry elements can you recognize in these drawings?!From http://serc.carleton.edu/files/NAGTWorkshops!Laws of Haüy and Bravais!Crystal faces grow in rational orientations relative to the crystal lattice. Common faces are often parallel to the surfaces that define the unit cell.!Thus minerals with isometric (cubic) unit cells often form crystals with external morphology that is cubic – hexagonal unit cells yield crystals with hexagonal external morphology, etc.!These observations have led to two related laws of crystallography:!Law of Haüy: Crystal faces make simple rational intercepts on crystal axes!Law of Bravais: Common crystal faces are parallel to lattice planes that have high lattice


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UT Arlington GEOL 2313 - Introduction to Crystallography

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