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1.2 Symmetry and point groups

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

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Unformatted text preview: Symmetry Symmetry of minerals derive from the "highly ordered atomic arrangement" Atomic framework is ordered patterns of atoms repeat "infinitely" in all directions Material without orderly arranged framework of atoms are not minerals Order may form crystal faces provides external order (symmetry) Minerals usually do not display external order 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 Used to describe macroscopic faces on crystals/minerals: Euhedral well developed crystal faces Subhedral crystal faces present, but are not perfect Anhedral without faces Terms Non crystalline material no 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 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 Internal ordering of atoms can be seen through symmetry present in minerals How is symmetry observered? Macroscopically: observable ONLY if euhedral Microscopically: usually observable with petrographic microscopes Atomic scale: Always observable with Xray diffraction Symmetry of particular mineral same for all specimens of that mineral If symmetry different between two specimens, then they HAVE to be different minerals Remember easily observed so... An extremely powerful diagnostic tool E.g. calcite and aragonite What is symmetry? Correspondence in size, shape and position of parts on opposite sides of a dividing line or median plane or about a center or axis A rigid motion of a geometric figure that determines a onetoone mapping onto itself, i.e. a symmetry operation Two types of symmetry operations 1. 2. Translational i.e. through a volume Repetition (also called "point symmetry") around a point, line or plane Preview of Coming Attractions: 1. Describe translation operations and resulting shapes 2. We'll find for 3D symmetry there are only 14 unique translational shapes 1st in 2D easier Then in 3D needed to understand minerals 3. Describe different repetition operations The 14 shapes define 6 CRYSTAL SYSTEMS Basis of organization (and classification) of minerals Translation in a plane (2D) Repetition of a single point The resulting pattern is called a plane lattice Repeated in one and two directions Each point referred to as a lattice node Each shape called a unit mesh Extends to infinity in the plane A single lattice node: e.g. an atom or collection of atoms Translation in 1D Plane lattice: translation in two directions A "unit mesh" Figure 2.1 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" Angle between axes is Plane symmetry operations Only 5 unique plane lattices produced by 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 Plane lattice extends to infinity Unit mesh defines the symmetry (and chemistry if mineral) Plane lattice extends to infinity Unit mesh defines the symmetry (and chemistry if mineral) Hexagon Square Parallelogram Primitive Rectangle Center Rectangle Note these are the same shaped unit mesh 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 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 Axes are named a, b, and c (positive) or 1 2 3 a , a , a if the same length They may be a, b, and c (negative) Intersect at a point called origin Angles between axes named , , The organization and arrangement of axes are very specific 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 = b angle c = a angle c = Figure 2.4 The crystallographic axes are DEFINED by the edges of the unit cell P. 306 olivine information Crystallographic information 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: 14 Bravais lattices can be combined to 6 unique shapes to unit cells Bravais lattices 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 crystallographic axis of the system Angles between the axes are also specific and mostly unvarying for each mineral Each mineral has specific and mostly unvarying "unit" length These are the "unit" axial lengths 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 P Monoclinic P C Orthorhombic P C Unit cells of each crystal system F I Tetragonal P I Hexagonal P Isometric P I F Fig. 210 Arrangement of crystallographic axes Triclinic Triclinic three inclined There is no formal convention to how minerals correspond to the unit cell axes, generally: a b c and 90 Formed by translating an oblique plane lattice a distance "c" at an angle not equal to or c is parallel to prominent elongation b is down and to right a is down and to front Triclinic bravais lattice a b c, 90 Fig. 25 Monoclinic Translate rectangular plane lattice not perpendicular to other axes, and a distance not equal to a or b a b c, = = 90 Monoclinic bravais lattice a b c, = = 9 0 Fig. 26a and 27a Orthorhombic Translate rectangular plane lattice at right angles to other axes, but a distance not equal to a or b a b c, = = = 90 No convention about actual lengths of unit cell sides and axes. Commonly c < a < b, but not always P C I F Body Centered Orthorhombic bravais lattice a b c, = = = 90 = 90 Face Centered Fig. 26 & 27 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 120 Angle between a and c = 90 Hexagonal bravais lattice a1 = a2 = a3 c angles 120 & 90 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 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 = = = 90 Tetragonal bravais lattice a = b c, = = = 90 = 90 Fig. 29 Isometric Translate a square plane lattice a distance equal to "a" and perpendicular to the lattice 1 2 3 a = a = a = = Isometric bravais lattice a = b = c, = = = 90 = 90 Fig. 29 Summary 6 Crystal Systems Triclinic Monoclinic a b c a b c = = 9 0 = = = 90 Angles are 120 and 90 = = = 90 = = = 90 Orthorhombic a b c Hexagonal Tetragonal Isometric a1 = a2 = a3 c a = b c a = b = c 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 If you know the point symmetry Operations are procedures used to generate the symmetry Number of operations possible is limited you can identify which of the 6 crystal systems the mineral crystallizes in Eliminates roughly 5/6 of all minerals 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 Four types of symmetry operations Reflection Rotation Inversion Rotoinversion Reflection Created by a "mirror plane" Generates mirror image on opposite sides of plane Characteristics: Creates only one new motif Notation = m Handedness of motif changes (e.g. left to right hand) E.g. reflection of new motif across plane ends up at the original location Mirror image reflection across a mirror plane {} Note handedness changes with this face Fig. 211 Rotation Motif is repeated by rotation around an axis Five possible rotation axes: Notation = An where n =1, 2, 3, 4, 6 depending on number of rotations. One fold = 360 rotation Two fold = 180 rotation Three fold = 120 rotation Four fold = 90 rotation Six fold = 60 rotation Fig. 212 Various rotation symmetry 4fold rotation, 90 Note symbols for rotations 2fold rotation, 180 3fold rotation, 120 6fold rotation, 60 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 Rotoinversion Combination of rotation and inversion Rotation may be 1, 2, 3, 4, or 6 Notation n, where n = number of rotations Only 4 is unique, others can be duplicated with other operations 1= i 2 = m 3 = A3 + i 4 is unique, this symmetry cannot be produced any other way 6 = A3 + m = i = m = A3 + i Unique =A3 + m Fig. 214 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 operations Consider a form with the following symmetry What does the form look like? Where are the symmetry elements? How do you write the symmetry elements? Center of symmetry Three 2fold rotation axes Three mirror planes In shorthand = i, 3A2, 3m Orthorhombic system: a b c, = = = 90 3 two fold rotations 3 mirrors 1 center of symmetry Why not some other system, e.g. isometric or triclinic? Fig. 215 The total amount of symmetry in the form could be 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 In general, all the symmetry operations can be combined in only a 32 unique ways First consider number of possible combinations in plane symmetry Called "Point Groups" 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 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 Table 22 Symmetry of the 32 point groups Common symmetry elements within crystal systems Orientation of crystallographic axes 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 Chemistry Crystallography Physical Properties Optical Properties ...
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