Symmetry Operations brief remark about the general role of symmetry in modern physics V(X) x V(X 1 ) X1X1 V(X 2 ) X2X2 change of momentum V(X) x conservation of momentum X1X1 X2X2 V(X 1 ) = V(X 2 ) translational symmetry Emmy Noether Emmy Noether 1918: Symmetry in nature conservation law 1882 in Erlangen, Bavaria, Germany 1935 in Bryn Mawr, Pennsylvania, USA

Breaking the symmetry with magnetic field Hamiltonian invariant with respect to rotation Example for symmetry in QM angular momentum conserved J good quantum number E Proton and Neutron 2 states of one particle breaking the Isospin symmetry Magnetic phase transition T>T C T<T C Zeeman splitting

called basis Symmetry in perfect single crystals ideally perfect single crystal infinite three-dimensional repetition of identical building blocks basis single atom simple molecule very complex molecular structure Quantity of matter contained in the unit cell Volume of space ( parallelepiped ) fills all of space by translation of discrete distances

Example: crystal from square unit cell hexagonal unit cell there is often more than one reasonable choice of a repeat unit (or unit cell) most obvious symmetry of crystalline solid Translational symmetry 3D crystalline solid3 translational basis vectors a, b, c translational operation T=n 1 a+n 2 b+n 3 cwhere n 1, n 2, n 3 arbitrary integers -connects positions with identical atomic environments a b n 1 =2 n 2 =1 -by parallel extensions the basis vectors form a parallelepiped, the unit cell, of volume V=a(bxc)

concept of translational invariance is more general physical property at r (e.g.,electron density) is also found atr’=r+T Set of operationsT=n 1 a+n 2 b+n 3 c r’ defines space lattice or Bravais lattice purely geometrical concept + = lattice basis crystal structure r

lattice and translational vectors a, b,c are primitive if every point r’ equivalent to r identical atomic arrangement is created by T according to r’=r+T r x y x y r r’=r+0.5 a 4 No integer! no primitive translation vector no primitive unit cell Primitive basis: minimum number of atoms in the primitive (smallest) unit cell which is sufficient to characterize crystal structure r’=r+ a 2

2 important examples for primitive and non primitive unit cells f ace c entered c ubic b ody c entered c ubic a 1 =(½, ½,-½)a 2 =(-½, ½,½) a 3 =(½,- ½,½) a 1 =a(½, ½,0)a 2 =a(0, ½,½) a 3 =a(½,0,½) Primitive cell: rhombohedron = 1atom/V primitive 4 atoms/V conventinal 1atom/V primitive 2 atoms/V conventinal

Lattice Symmetry Symmetry of the basis point group symmetry has to be consistent with symmetry of Bravais lattice Limitation of possible structures Operations (in addition to translation) which leave the crystal lattice invariant No change of the crystal after symmetry operation Reflection at a plane ( point group of the basis must be a point group of the lattice )

Rotation about an axis H2oH2o NH 3 SF 5 Cl Cr(C 6 H 6 ) 2 = 2 -fold rotation axis = n -fold rotation axis Click for more animations and details about point group theory

point inversion Glide = reflection + translation Screw = rotation + translation

Notation for the symmetry operations Origin of the Symbols after Schönflies: E:identity from the German Einheit =unity C n :Rotation (clockwise) through an angle 2π/n, with n integer  : mirror plane from the German Spiegel=mirror  h :horizontal mirror plane, perpendicular to the axis of highest symmetry  v :vertical mirror plane, passing through the axis with the highest symmetry * rotation by 2  /n degrees + reflection through plane perpendicular to rotation axis *

n-fold rotations with n=1, 2, 3,4 and 6 are the only rotation symmetries consistent with translational symmetry ! ? ? ? ? ? ? ? Intuitive example: pentagon

Two-dimensional crystal with lattice constant a in horizontal direction a 1 2 Row A (m-1) a m α α Row B X 1’ m’ If rotation by α is a symmetry operation 1’and m’ positions of atoms in row B X=p a p integer! = (m-1)a – 2a+ 2a cos α = (m-3)a + 2a cos α p-m integer p-m 1 0/2π =1-fold -21/2π/3=6-fold -3 0π/2=4-fold -4-1/22π/3=3-fold -5π=2-fold order of rotation

Plane lattices and their symmetries 5 two-dimensional lattice types Point-group symmetry of lattice: 2 2mm 4mm 6 mm 10 types of point groups ( 1, 1m, 2, 2mm,3, 3mm, 4, 4mm, 6, 6mm )possible basis: Combination of point groups and translational symmetry 17 space groups in 2D Crystal=lattice+basis may have lower symmetry

Three-dimensional crystal systems oblique lattice in 2D triclinic lattice in 3D Special relations between axes and angles14 Bravais (or space) lattices

7 crystal systems

There are 32 point groups in 3D, each compatible with one of the 7 classes 32 point groups and compound operations applied to 14 Bravais lattices 230 space groups or structures exist Many important solids share a few relatively simple structures