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Symmetry Operations

V(X). V(X). V(X 1 ). V(X 2 ). x. x. X 2. X 2. X 1. X 1. Symmetry Operations. brief remark about the general role of symmetry in modern physics. conservation of momentum. change of momentum. V(X 1 ) = V(X 2 ). translational symmetry. Emmy Noether 1918: Symmetry in nature .

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Symmetry Operations

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  1. V(X) V(X) V(X1) V(X2) x x X2 X2 X1 X1 Symmetry Operations brief remark about the general role of symmetry in modern physics conservation of momentum change of momentum V(X1) = V(X2) translational symmetry Emmy Noether 1918: Symmetry in nature conservation law 1882 in Erlangen, Bavaria, Germany 1935 in Bryn Mawr, Pennsylvania, USA

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

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

  4. b a T=n1a+n2b+n3c where n1, n2, n3 arbitrary integers Example: crystal from hexagonal unit cell square unit cell there is often more than one reasonable choice of a repeat unit (orunit cell) most obvious symmetry of crystalline solid Translational symmetry n2=1 3D crystalline solid 3 translational basis vectors a, b, c n1=2 -by parallel extensions the basis vectors form a parallelepiped, the unit cell, of volume V=a(bxc) translational operation -connects positions with identical atomic environments

  5. r lattice crystal structure basis concept of translational invariance is more general physical property at r (e.g.,electron density)is also found at r’=r+T Set of operations T=n1a+n2b+n3c defines r’ space lattice or Bravais lattice purely geometrical concept + =

  6. identical atomic arrangement y y r r’=r+ a2 no primitive unit cell x x r r’=r+0.5 a4 no primitive translation vector 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 Primitive basis: minimum number of atoms in the primitive (smallest) unit cell which is sufficient to characterize crystal structure No integer!

  7. a3=a(½,0,½) a1=a(½, ½,0) a2=a(0, ½,½) a3=(½,- ½,½) a1=(½, ½,-½) a2=(-½, ½,½) 2 important examples for primitive and non primitive unit cells face centered cubic 1atom/Vprimitive 4 atoms/Vconventinal Primitive cell: rhombohedron = body centered cubic 1atom/Vprimitive 2 atoms/Vconventinal

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

  9. = n -fold rotation axis • Rotation about an axis H2o = 2 -fold rotation axis NH3 SF5 Cl Cr(C6H6)2 Click for more animations and details about point group theory

  10. point inversion • Glide = reflection + translation = rotation + translation • Screw

  11. 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

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

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

  14. 17 space groups in 2D Plane lattices and their symmetries 4mm Point-group symmetry of lattice:2 2mm 2mm 6 mm 5 two-dimensional lattice types Crystal=lattice+basismay 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

  15. oblique lattice in 2D triclinic lattice in 3D Three-dimensional crystal systems Special relations between axes and angles 14 Bravais (or space) lattices

  16. 7 crystal systems

  17. Many important solids share a few relatively simple structures 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

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