Transcript Slide 1
Symmetry Operations brief remark about the general role of symmetry in modern physics V(X) V(X) dV Fx dx V(X1) V(X2) Fx dV 0 dx conservation of momentum change of momentum dp x Fx dt dp x 0 dt px const. V(X1) = V(X2) X1X2 x X1 translational symmetry X2 Emmy Noether 1918: Symmetry in nature conservation law 1882 in Erlangen, Bavaria, Germany 1935 in Bryn Mawr, Pennsylvania, USA x Example for symmetry in QM Hamiltonian invariant with respect to rotation Breaking the symmetry with magnetic field E mJ=-1 mJ=0 angular momentum conserved J good quantum number Zeeman splitting Proton and Neutron 2 states of one particle mJ=+1 B=0 B0 breaking the Isospin symmetry Magnetic phase transition T>TC T<TC Symmetry in perfect single crystals ideally perfect single crystal infinite three-dimensional repetition of identical building blocks called basis 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 hexagonal unit cell square unit cell there is often more than one reasonable choice of a repeat unit (or unit cell) most obvious symmetry of crystalline solid n2=1 3D crystalline solid n1=2 b a Translational symmetry 3 translational basis vectors a, b, c -by parallel extensions the basis vectors form a parallelepiped, the unit cell, of volume V=a(bxc) translational operation T=n1a+n2b+n3c where n1, n2, n3 arbitrary integers -connects positions with identical atomic environments concept of translational invariance is more general is also found at r’=r+T physical property at r (e.g.,electron density) Set of operations T=n1a+n2b+n3c defines r’ r space lattice or Bravais lattice purely geometrical concept lattice crystal structure basis + = lattice and translational vectors a, b,c are primitive if every point r’ equivalent to r is created by T according to r’=r+T y r identical atomic arrangement r’=r+ a2 no primitive unit cell x Primitive basis: minimum number of atoms in the primitive (smallest) unit cell which is sufficient to characterize crystal structure y r r’=r+0.5 a4 x No integer! no primitive translation vector 2 important examples for primitive and non primitive unit cells face centered cubic 1atom/Vprimitive 4 atoms/Vconventinal a1=a(½, ½,0) a2=a(0, ½,½) a3=a(½,0,½) Primitive cell: rhombohedron Vprimitive a1 a2 a3 = body centered cubic 1atom/Vprimitive Vprimitive 2 atoms/Vconventinal 1 3 1 a Vconventional 2 2 a1=(½, ½,-½) a2=(-½, ½,½) a3=(½,- ½,½) 1 3 1 a Vconventional 4 4 Lattice Symmetry Symmetry of the basis point group symmetry Limitation of possible structures has to be consistent with symmetry of Bravais lattice (point group of the basis must be a point group of the lattice) No change of the crystal after symmetry operation Operations (in addition to translation) which leave the crystal lattice invariant • Reflection at a plane • Rotation about an axis 2 n H2o 2 2 = 2 -fold rotation axis = n -fold rotation axis NH3 SF5 Cl Cr(C6H6)2 Click for more animations and details about point group theory • point inversion ( x, y, z) (x,y,z) • Glide = reflection + translation • Screw = rotation + translation Notation for the symmetry operations * * rotation by 2/n degrees + reflection through plane perpendicular to rotation axis Origin of the Symbols after Schönflies: E:identity from the German Einheit =unity Cn :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 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 Row A 1 a 2 (m-1) α Row B m a α m’ 1’ X If rotation by α is a symmetry operation X=p a p integer! cos 1 p-m integer 1 1’ and m’ positions of atoms in row B = (m-1)a – 2a + 2a cos α 3pm cos 2 = (m-3)a + 2a cos α order of p-m cos rotation 2 -1 1 0/2π =1-fold 1 -2 1/2 π/3 2 / 6 =6-fold -3 -4 -5 0 -1/2 -1 π/2 2 / 4 =4-fold 2π/3 2 / 3 =3-fold π 2 / 2 =2-fold Plane lattices and their symmetries 4mm Point-group symmetry of lattice: 2 2mm 2mm 6 mm 5 two-dimensional lattice types Crystal=lattice+basis may have lower symmetry possible basis: 10 types of point groups (1, 1m, 2, 2mm,3, 3mm, 4, 4mm, 6, 6mm) Combination of point groups and translational symmetry 17 space groups in 2D Three-dimensional crystal systems a b c, oblique lattice in 2D Special relations between axes and angles triclinic lattice in 3D 14 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