1.2 Symmetry and point groups

1.2 Symmetry and point groups - Symmetry Symmetry Symmetry...

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Unformatted text preview: Symmetry Symmetry Symmetry of minerals derive from their “highly ordered atomic arrangement” Patterns of atoms repeat “infinitely” in all directions Ordering of atoms may form external crystal faces – reflects internal symmetry Minerals usually lack crystal faces Still contain ordered internal arrangement Material without orderly arranged framework of atoms are not minerals Terms Terms Used to describe crystal faces: Euhedral – well developed crystal faces Subhedral – crystal faces present, but are not perfect Anhedral – without faces Crystals Crystals A homogeneous solid with long­range three dimensional order Minerals must be crystals – e.g., they are crystalline Some crystals are not minerals Why not? How would these solids not fit our definition of a mineral? What would be an example of this? Terms Terms Non crystalline material – no internal atomic symmetry present Amorphous – these are solids that lack an ordered arrangement of atoms, i.e. not crystalline Examples: glass – can be natural (e.g. frozen lavas) mineraloids ­ natural material that are not minerals because they have lost their atomic ordering. Processes that can create Processes that can create mineraloids Metamict – destruction of structure through radiation Fulgarite – melting of sand through lightning strike Clinker – melted material from fires in coal bed seams Observing internal ordering Observing internal ordering Internal ordering of atoms can be seen through symmetry present in minerals How is symmetry observed? Macroscopically: observable ONLY if euhedral Microscopically: usually observable with petrographic microscopes Atomic scale: Always observable with X­ray diffraction Granite Photomicrograph Granite Photomicrograph Quartz Feldspars Microscopic image Plagioclase K­spar Biotite Horneblende Accessory Mineral X­Ray diffractometer X­Ray diffractometer Observations at atomic scale Measures distance between atoms Determine three dimensional arrangement of atoms in solids All specimens of every mineral has identical symmetry If symmetry different between two specimens, then they HAVE to be different minerals E.g. calcite and aragonite Remember – easily observed so… An extremely powerful diagnostic tool BUT: Different minerals may have different symmetry What is symmetry? What is symmetry? Comparison: Correspondence in size, shape and position of parts on opposite sides of a dividing line or median plane or about a center or axis Action: A rigid motion of a geometric figure that determines a one­to­one mapping onto itself, i.e. a symmetry operation Two types of symmetry operations Two types of symmetry operations Translational – i.e. through a volume 1. Continues to infinity Repetition (also called “point symmetry”) 2. Around a point, line or plane Returns to original starting place Preview of Coming Attractions: Preview of Coming Attractions: 1. Describe translation operations and resulting shapes 1st in 2­D – easier Then in 3­D – needed to understand mineral classification scheme 2. We’ll find for 3­D symmetry there are only 14 unique translational shapes The 14 shapes define 6 CRYSTAL SYSTEMS Basis of organization (and classification) of minerals 3. Describe different repetition operations Translation in a plane (2­D) Translation in a plane (2­D) Repetition of a single point or pattern The resulting pattern is called a plane lattice Repeated in one and two directions Each point referred to as a lattice node The lattice extends to infinity in the plane Each plane lattice has a unique shape/pattern called a unit mesh A single lattice node: e.g. an atom or collection of atoms Translation in 1­D Figure 2.1 Plane lattice: translation in two directions The parallelogram is is the “unit mesh” Nomenclature Nomenclature Plane lattice axes parallel edges of unit mesh For unit meshes there are only 2 axes Axes used to define space Labeled “a” and “b”, or for negative directions –a and –b. Angle between axes is γ Plane symmetry operations Plane symmetry operations Only 5 unique plane lattices produced by infinite translation in 2 dimensions Only 4 unique shapes to the unit mesh One shape has two distinct lattices Primitive (p) – nothing at center of mesh Centered (c) – node at center of mesh Figure 2.2 ules for Plane lattice trans­ extends to lation infinity Unit mesh defines the symmetry (and chemistry if mineral) Square Rules for Plane lattice trans­ extends to lation infinity Unit mesh defines the symmetry (and chemistry if mineral) Hexagon (contains a rhombus) Parallelogram Primitive Rectangle Center Rectangle Note – these are the same shaped unit mesh Translation in 3­D Translation in 3­D Needed to build three dimensional objects: e.g., minerals Analogous to 2­D lattices, but slightly different nomenclature: Arrangement of nodes – Space Lattice (2­D = plane lattice) Volume outlined – Unit Cell (2D = unit mesh) Edges of unit cell – Crystallographic Axes (2­ D same) For unit cells there are three axes Unit Cell Unit Cell Smallest volume of crystalline structure that contains all: 1. 2. 3. Chemical information Structural (symmetry and atomic arrangement) information Edges define the lengths of and angles between crystallographic axes Crystallographic Axes Crystallographic Axes Axes are named a, b, and c (positive) or 1 2 3 a , a , a if the same length 2 They may be ­a, ­b, and –c, or –a , ­a , ­ 3 1 a (negative) Intersect at a point called origin Note that the axes are the same lengths as the edge of the unit cell Example of a Unit cell Translation in third dimension Unit mesh (parallelogram) a angle b = γ b angle c = α a angle c = β Figure 2.4 The crystallographic axes are DEFINED by the lengths of the edges of the unit cell P. 306 – olivine information P. 306 – olivine information Crystallographic information Bravais Lattices and Crystal Bravais Lattices and Crystal Systems If you take the 5 plane lattices and repeat systematically in 3­D, they make up 14 unique arrangement of nodes: Bravais lattices 14 Bravais lattices can be combined to 6 unique shapes that outline unit cells Crystal systems 6 Crystal systems 6 Crystal systems Triclinic Monoclinic Orthorhombic Hexagonal Tetragonal Isometric Each crystal system has a uniquely shaped unit cell Edge of the unit cell corresponds to “unit” lengths along the 3 edges of the unit cell Defines the lengths and arrangements of the crystallographic axes in the system Each mineral has specific and mostly unvarying “unit” length Unit lengths have absolute values, typically a few Å (1 Å = 0.1 nanometer = .0001 micron) Angles between the axes are also specific and mostly unvarying for each mineral Why 14 Bravais lattices, but only 6 Why 14 Bravais lattices, but only 6 systems? Systems have “extra” lattice nodes, but retain the unit shape of the system, e.g., Primative (p) – lattice points only at corners of unit cell Body centered (i) – additional lattice point at center of cell (innenzentrierte) Face centered (c) – additional lattice points on two opposite sides Face centered (f) – additional lattice points on every face Triclinic Monoclinic P P C Orthorhombic C P F I Tetragonal P Hexagonal P I 14 Bravais lattices Broken into 6 crystal systems Unit cells of each crystal system Isometric P I F Fig. 2­10 Arrangement of crystallographic Arrangement of crystallographic axes ­ Triclinic Triclinic – three inclined a ≠ b ≠ c and α ≠ β ≠ γ ≠ 90º Formed by translating an oblique plane lattice a distance “c” at an angle not equal to α or β There is no formal convention to how minerals correspond to the unit cell axes, generally: c is parallel to prominent elongation b is down and to right a is down and to front Triclinic bravais lattice a ≠ b ≠ c, α ≠ β ≠ γ ≠ 90º Fig. 2­5 Fig. 2­5 Monoclinic Monoclinic One inclined Translate rectangular plane lattice not perpendicular to other axes, and a distance not equal to a or b a ≠ b ≠ c, α = γ = 90º ≠ β 90º Monoclinic bravais lattice a ≠ b ≠ c, α = γ = 9 0 º ≠ β Fig. 2­6a and 2­7a Fig. 2­6a and 2­7a Orthorhombic Orthorhombic Translate rectangular plane lattice at right angles to other axes, but a distance not equal to a or b a ≠ b ≠ c, α = β = γ = 90º 90º No convention about actual lengths of unit cell No sides and axes. sides Commonly c < a < b, but not always Primitive Face Centered P C I F Body Centered Orthorhombic bravais lattice a ≠ b ≠ c, α = β = γ = 90º = 90º Face Centered Fig. 2­6 & 2­7 Fig. 2­6 & 2­7 Hexagonal Hexagonal Translate hexagonal plane lattice perpendicular to a1 and a2, but a distance that is not equal to a1 and a2. Angles between a axes are 120º Angle between a and c = 90º Hexagonal bravais lattice a1 = a2 = a3 ≠ c angles 120º & 90º Translation is 1/3C up and 2/3a*cos30 back Forms a rhombohedron, but also can use the hexagon as the unit cell (most common) Fig. 2­8 Fig. 2­8 Tetragonal Tetragonal Translate a square plane lattice a distance “c” not equal to a1 and a2 and perpendicular to the lattice 1 2 a = a ≠ c α = β = γ = 90º 90º Tetragonal bravais lattice a = b ≠ c, α = β = γ = 90º = 90º Fig. 2­9 Fig. 2­9 Isometric Isometric Translate a square plane lattice a distance equal to “a” and perpendicular to the lattice 1 2 3 a = a = a α= β= γ Isometric bravais lattice a = b = c, α = β = γ = 90º = 90º Fig. 2­9 Fig. 2­9 Summary – 6 Crystal Summary – 6 Crystal Systems Triclinic a ≠ b ≠ c α ≠ β ≠ γ Monoclinic a ≠ b ≠ c α = γ = 9 0 º ≠ β α= β= γ= 90º Angles are 120º and 90º α= β= γ= 90º α= β= γ= 90º Orthorhombic a ≠ b ≠ c Hexagonal a1 = a2 = a3 ≠ c Tetragonal a = b ≠ c Isometric a = b = c Repetition Symmetry Repetition Symmetry Crystallographic axes (i.e. unit cell dimensions) give minerals their symmetry Two types of symmetry: Translational – e.g. generates lattices (we’ve seen this) Point symmetry (repetition symmetry) – Defines a finite shape called a “form” Ultimately will get to defining 32 point groups, aka 32 crystal classes Point symmetry defined by specific operations Operations are procedures that generate the symmetry Limited number (4) of possible operations If you know the point symmetry you can identify which of the 6 crystal systems the mineral crystallizes in Eliminates roughly 5/6 of all minerals Point Symmetry Point Symmetry How point symmetry works: Repetition of motif around point, line or plane Motifs are geometric shape – represents crystal face, group of atoms, single atom etc. The motif is repeated until returned to original location An “operation” or “mapping” of motif onto itself Point Symmetry operations Point Symmetry operations Four types of symmetry operations Reflection Rotation Inversion Rotoinversion Reflection Reflection Created by a “mirror plane” Generates mirror image on opposite sides of plane Characteristics: Creates only one new motif E.g. reflection of new motif across plane ends up at the original location Handedness of motif changes (e.g. left to right hand) Notation = m Mirror image – reflection across a mirror plane {} Note – handedness changes with this face Fig. 2­11 Fig. 2­11 Rotation Rotation Motif is repeated by rotation around an axis Five possible rotation axes: One fold = 360° rotation Two fold = 180° rotation Three fold = 120° rotation Four fold = 90° rotation Six fold = 60° rotation Notation = An where n =1, 2, 3, 4, 6 depending on number of rotations. Fig. 2­12 Fig. 2­12 4­fold rotation, 90º 2­fold rotation, 180º 3­fold rotation, 120º Various rotation symmetry Note symbols for rotations 6­fold rotation, 60º Inversion Inversion Symmetry through a point, also “center of symmetry” A line drawn from a point through the center will hit an identical point equal distance on the opposite side of the center If a mineral has a center of symmetry, every point on mineral has a point exactly opposite point Notation = i Center of symmetry No center of symmetry, e.g. quartz, tourmaline, topaz All are piezoelectric Fig. 2­13 Fig. 2­13 Rotoinversion Rotoinversion Combination of rotation and inversion Rotation may be 1, 2, 3, 4, or 6 Notation Ān, where n = number of rotations Only Ā4 is unique, others can be duplicated with other operations Ā1= i Ā2 = m Ā3 = A3 + i Ā4 is unique, this symmetry cannot be produced any other way Ā6 = A3 + m = i = m = A3 + i Unique =A3 + m Fig. 2­14 Fig. 2­14 Multiple Point Symmetry Multiple Point Symmetry Shapes may have more than one type of symmetry If more than one, the different symmetries must relate to each other Example of multiple symmetry Example of multiple symmetry operations Consider a form with the following symmetry Center of symmetry Three 2­fold rotation axes Three mirror planes In shorthand = i, 3A2, 3m What does the form look like? Where are the symmetry elements? How do you write the symmetry elements? Orthorhombic system: a ≠ b ≠ c, α = β = γ = 90º 3 two fold rotations 3 mirrors 1 center of symmetry Why not some other system, e.g. isometric or triclinic? Fig. 2­15 Fig. 2­15 The total amount of symmetry (i, 3A2, 3m) also is described as: 2/m 2/m 2/m Describes 3 2­fold rotation axes perpendicular to mirrors This symmetry has to have a center in it. This type of notation called “Hermann­ Mauguin” 32 point groups 32 point groups In general, all the symmetry operations can be combined in only a 32 unique ways Called “Point Groups” First consider number of possible combinations in plane symmetry In two dimension, there are no inversions, only rotations and mirrors Combinations of rotations and mirrors produce only 10 unique combinations Point groups in 2­D: ­ only 10 possible combinations ­ symmetry operations only mirror and rotation Fig. 2­16 Fig. 2­16 In three dimensions, the rotations and mirrors are expanded because they also have inversions In this case there are 32 unique symmetry combinations Called 32 point groups Each mineral crystallizes with the symmetry of one of the point groups Called Crystal Classes Each crystal class falls into one of the 6 crystal systems The classes occur in the system depending on their common symmetry elements E.g. Table 2.2 Triclinic Monoclinic Orthorhombic Hexagonal Tetragonal Isometric Table 2­2 Table 2­2 • Symmetry of the 32 point groups • Common symmetry elements within crystal systems • Orientation of crystallographic axes Two additional symmetry Two additional symmetry operations Glide plane Screw Axis Translation plus a mirror Translation plus a rotation (2, 3, 4, or 6 fold) Note: 2­fold screw axis differs from glide plane Glide symmetry Silica tetrahedron, e.g. pyroxenes Fig. 2.19 & 2.20 Fig. 2.19 & 2.20 3­fold screw axis • May be 2, 3, 4, or 6 – fold • May be right of left spiral • Enantiomorphic forms How to use this to your advantage How to use this to your advantage Identify crystal class Determine if crystal has center of symmetry Identify all mirror planes Identify rotation axes and number of rotations Compile all symmetry P. 306 – olivine information P. 306 – olivine information Chemistry Crystallography Physical Properties Optical Properties ...
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This note was uploaded on 09/19/2011 for the course GLY 3200 taught by Professor Staff during the Fall '10 term at University of Florida.

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