CrystallographySymmetry  CrystallographyCrystal MorphologySlide 4TranslationsSlide 6Slide 7Unit Cell3-D translationsUnit cells have at least as much symmetry as the crystal (internal order > external order)Slide 113-D Translations and LatticesUnit cell typesUnit cells – counting motifs (atoms)Bravais LatticesSlide 16Slide 17Slide 183-D SpaceSymmetry operators, again… but we save the last ones for a reasonGlide PlanesScrew AxesSpace GroupsInternal – External OrderMineral ID informationWhy did we go through all this?How does that translate to what we see??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 ideasSymmetry  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 MorphologyGrowth 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 faceCrystallography•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 ideasTranslationsThis HAS symmetry, but was GENERATED by translation…Translations2-D translations = a netabUnit cellUnit cellUnit Cell: the basic repeat unit that, by translation only, generates the Unit Cell: the basic repeat unit that, by translation only, generates the entire pattern – can you pick more than 1 unit cell?entire pattern – can you pick more than 1 unit cell?How differ from motif ??How differ from motif ??TranslationsWhich unit cell is Which unit cell is correct ??correct ??Conventions:Conventions:1. Cell edges should, 1. Cell edges should, whenever possible, whenever possible, coincide with coincide with symmetry axes or symmetry axes or reflection planesreflection planes2. If possible, edges 2. If possible, edges should relate to each should relate to each other by lattice’s other by lattice’s symmetry.symmetry.3. The smallest possible 3. The smallest possible cell (the reduced cell) cell (the reduced cell) which fulfills 1 and 2 which fulfills 1 and 2 should be chosenshould be chosenUnit 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 symmetryUnit 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…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 categoriesName axes anglesTriclinica  b  c  90oMonoclinica  b  c = 90o 90oOrthorhombica  b  c = 90oTetragonala1 = a2  c = 90oHexagonal Hexagonal (4 axes)a1 = a2 = a3  c = 90o 120o Rhombohedrala1 = a2 = a3 90oIsometrica1 = a2 = a3 = 90o3-D Lattice Types++cc++aa++bbAxial convention:Axial convention:““right-hand rule”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 morphologyUnit cells – counting motifs (atoms)•Z represents the number of atoms the unit cell is comprised ofZ=1•Atom inside cell counts 1 each•Atom at face counts ½ each•Atom at edge counts ¼ each•Atom at corner counts 1/8 eachBravais 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 latticesabcPMonoclinicabcabcI = CabPTriclinicabcccaPOrthorhombicabcC F Iba1cPTetragonal a1 = a2cIa2a1a3PIsometrica1 = a2= a3a2F Ia1cP or Ca2RHexagonalRhombohedrala1a2c a1 = a2 = a3a.k.a. Trigonal3-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.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…Step 1: reflectStep 1: reflect(a temporary position)(a temporary