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 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 Crystal Axes generally taken as parallel to the edges intersections of prominent crystal faces b a c Miller Indices Why would it be important to describe what face is exposed for a crystal Symmetry Crystallography Preceding discussion related to the shape of a crystal Now we will consider the internal order of a mineral How are these different 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 Crystallography Motif the fundamental part of a symmetric design that when repeated creates the whole pattern In 3 D translation defines operations which move the motif into infinitely repeating patterns M C Escher s works are based on these ideas Translations This HAS symmetry but was GENERATED by translation Translations 2 D translations a net a b Unit cell Unit Cell the basic repeat unit that by translation only generates the entire pattern can you pick more than 1 unit cell How differ from motif Translations Which unit cell is correct Conventions 1 Cell edges should whenever possible coincide with symmetry axes or reflection planes 2 If possible edges should relate to each other by lattice s symmetry 3 The smallest possible cell the reduced cell which fulfills 1 and 2 should be chosen Unit Cell How to choose a unit cell if more than one unit cell is a possibility Rule Must represent the symmetry elements of the whole 3 D translations Operations which move a motif create the lattice a representation of the moves which create the pattern in plane or 3 D space Unit cell is a representation of the crystal such that it can be repeated by moving it to make that pattern If a crystal has symmetry the unit cell must have at least that much symmetry Unit cells have at least as much symmetry as the crystal internal order external order Here is why there are no 5 fold rotation axes If the unit cell cannot be repeated that way to make a lattice then a crystal cannot have that symmetry 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 Internal order the same here however 3 D Translations and Lattices Different ways to combine 3 non parallel non coplanar axes Really deals with translations compatible with 32 3 D point groups or crystal classes 32 Point Groups fall into 6 categories c 3 D Lattice Types Name Triclinic axes angles a b c 90o o Monoclinic a b c 90 90 Orthorhombic a b c 90o a1 a2 c 90o a 1 a2 a3 c 90o 120o a1 a 2 a 3 90o a1 a 2 a 3 o Tetragonal Hexagonal Hexagonal 4 axes Rhombohedral Isometric 90 o a b Axial convention right hand rule Unit cell types Correspond to 6 distinct shapes named after the 6 crystal systems In each representations include ones that are Primitive P distance between layers is equal to the distance between points in a layer Body centered I extra point in the center End centered A B C extra points on opposite faces named depending on axial relation Face centered F extra points at each face Cannot tell between P I A B C F without X ray diffraction Can often tell point group system or class and unit cell shape from xstal morphology Unit cells counting motifs atoms Z represents the number of atoms the unit cell is comprised of Atom inside cell counts 1 each Z 1 Atom at face counts each Atom at edge counts each Atom at corner counts 1 8 each c c c b a b b P a Triclinic a b c P a I Monoclinic a b c c a b P C F Orthorhombic a b c I C c c a2 a1 a2 P a1 I P or C R Hexagonal Rhombohedral a1 a2 a3 a1 a2 c Tetragonal a1 a2 c a k a Trigonal a3 a2 a1 P F I Isometric a1 a2 a3 3 D Space Possible translations of a lattice point yield the 6 crystal class shapes by moving a point in space a b c or x y z coordinates Those movements have to preserve the symmetry elements and are thus limited in the number of possible shapes they will create Bravais Lattices Assembly of the lattice points in 3 D results in 14 possible combinations Those 14 combinations may have any of the 6 crystal system class symmetries These 14 possibilities are the Bravais lattices Symmetry operators again but we save the last ones for a reason Must now define 2 more types of symmetry operators Space group operators Glide Plane Screw Axes These are combinations of simple translation and mirror planes or rotational axes Glide Planes Combine translation with a mirror plane 3 different types Axial glide plane a b c Diagonal glide plane n Diamond glide plane d Diagonal and diamond glides are truly 3 D repeat Step 2 translate Step 1 reflect a temporary position Screw Axes Combine translation with a rotation Can have 2 3 4 or 6 fold rotation 360 n and translation of some magnitude rational fraction of t unit cell edge length Subscripts indicate the magnitude of translation 42 indicates a 4 fold axis translated every 2 4 t 43 indicates a 4 fold axis translated every 3 4 t 4 indicates a 4 fold axis translated every t The screw axes can be either right handed advances away from observer when rotated clockwise or left handed advances away from observer when rotated counterclockwise Space Groups Atomic structure Point groups 32 3 D symmetry combinations lattice type the 14 different Bravais lattices glide planes screw axes 230 space groups They are represented Start with lattice type P I F R A B or C …