Crystal Structure NaCl Well defined surfaces reflects some internal order

Crystal Structure In 1913, William L. Bragg NaCl discovered ordered diffraction pattern of x-ray going through crystals The symmetry of the diffraction pattern reflects the symmetry of the crystal NaCl

Crystal Structure Transition from single molecule to a crystal NaCl Transition from single molecule to a crystal From single ionic NaCl molecule Into a lattice

Crystal Structure 1D lattice of alternating ions The binding energy (N – number of molecules) is the interaction energy of the ion i with all other ions nearest neighbors otherwise 1D lattice of alternating ions

Crystal Structure The binding energy (N – number of molecules) is the interaction energy of the ion i with all other ions nearest neighbors otherwise z – number of nearest neighbors Madelung constant

Determining crystal parameters Madelung constant should be positive 1D lattice of alternating ions

Nomenclature Ideal crystal – infinite repetition of identical structural units in space Lattice – regular periodic array of points in space (called a net, in 2D) Translation vectors, a1 a2 a3 r’ = r + n1a1 + n2a2 + n3a3 Lattice looks the same from every r’

Basis – set of atoms associated with each lattice point general crystal = lattice + basis http://frank.mtsu.edu/~njsmith/modernii/lec10I.html

Primitive translation vectors – every point in lattice reached using integer ni There are many different possible choices Primitive cell parallelpiped of primitive vectors minimum volume cell that fills space one lattice point Volume of Cell:

construction: intersection of bisectors of lines to Wigner-Seitz cell – region of space about a lattice point closer to that point than to any other construction: intersection of bisectors of lines to neighboring lattice points http://www.cours.phy.ulaval.ca/cours/17323.gbedard/05_Cours_9_DC_RR.html BCC Of particular significance when discussing phonon and electron wavevectors http://dcssi.istm.cnr.it/CORSO%20IPERTESTUALE/StatoSolido/Strutture_Cristalline/wigner_seitz.htm FCC

Bravais Latice

Symmetry Operations Symmetry operation – brings crystal back into itself Besides translation, point symmetry operations (point operation – leave one point fixed): Rotation (possible: 2p, 2p/2, 2p/3, 2p/4, 2p/6; there is no 2p/5, 2p/7) Reflection (mirror planes) Inversion Compound operations (rotation-reflection, rotation-inversion)

Symmetry Operations

Point Group Nomenclature What distinguishes the crystal systems is point group symmetries To discuss symmetries we need to know how they are classified Two different nomenclature systems Schoenflies International (Hermann-Mauguin) Considering rotation, reflection, rotation-reflection Space group adds symmetry operations of screw axis and glide plane. Screw axis – rotation plus translation along axis Glide plane – reflection in a plane plus translation parallel to plane

Schoenflies / international Notation

Any symmetry operation on lattice can be made from translation + point operation (see Ashcroft, 113) Point groups – set of point operations Space groups – set of point operations + translations (i.e. full symmetry group)

Bravais Lattices Based on symmetry, can classify lattices into archetypes known as Bravais lattices In 2D, 5 types In 3D, 14 types 14 space groups – Bravais lattices (M.L. Frankenheim, 1842; corrected by A. Bravais, 1845)

Lattice: 7 distinct point groups 14 distinct space groups Lattice + basis: 32 point groups 230 space groups For visualization and examples: http://cst-www.nrl.navy.mil/lattice/spcgrp/

Bravais Latice

http://frank.mtsu.edu/~njsmith/modernii/lec10I.html

Naming vectors and planes Lattice direction: The direction defined by the lattice point r = n1a1 + n2a2 + n3a3 is [n1 n2 n3] Lattice planes: Miller indices (h k l) a) Intercepts of plane with axes: x1, x2, x3 b) Miller indices: smallest three integers proportional to 1/ x1, 1/ x2, 1/ x3 h:k:l = 1/ x1:1/ x2:1/ x3 For negative numbers use:

Lattice planes: Miller indices (h k l)

http://en.wikipedia.org/wiki/Miller_index http://www.geo.umn.edu/courses/2301/fall2003/miller_indices.jpg

(hkl) define family of lattice planes [parallel, equally spaced, contain all the points of the lattice] {hkl} means (hkl) and all (hkl) equivalent in terms of the crystal symmetry Example: for cubic crystal http://leung.uwaterloo.ca/CHEM/750/Lectures%202007/SSNT-3-Surface%20Structure%20I.htm This is a note