Crystallographic Symmetry Jeremy Karl Cockcroft
Why bother? To describe crystal structures Simplifies the description, e.g. NaCl structure Requires coordinates for just 2 atoms + space group symmetry! To solve crystal structures Relate diffraction (reciprocal-space) symmetry to crystal (real-space) symmetry To relate crystal structures Phase transitions To index a powder pattern Exploit symmetry of the unit cell
Types of Symmetry Rotational symmetry about a line Rotary-inversion symmetry about a line Translational symmetry Screw symmetry Glide symmetry
Rotational Symmetry Rotation anticlockwise 360 / n Symbols: 1, 2, 3, 4, 6
Symmetry Operators Need to distinguish between symmetry elements, symmetry operators, & coordinates
Symmetry Operators Advantage of non-orthogonal axes
Symmetry Operators Off-centre (non-origin) axes
Rotary-inversion Symmetry Rotation anticlockwise 360 / n + Inversion Symbols: 1, 2, 3, 4, 6 written with bar above digit n = 1 enantiomorphs
Mirror Symmetry Rotary-inversion axis with n = 2
Higher-order Rotary-inversion Staggered v. eclipsed C 2 H 6 3 and 6
Translation Symmetry
Unit Cell & Lattices Define unit cell (6 parameters) Lattice obtained by adding unit translations in x, y, and z
Choices
Coordinate Systems Cartesian: r = Xi + Yj + Zk X, Y, Z Fractional real space: r = xa + yb + zc x, y, z Integer reciprocal space: r* = ha* + kb* + lc* h, k, l
7 Crystal Systems Combination of rotational (or rotaryinversion) symmetry with a lattice Triclinic Monoclinic Orthorhombic Tetragonal Trigonal Hexagonal Cubic 1 1-fold 1 2-fold 3 2-fold 1 4-fold 1 3-fold 1 6-fold 4 3-fold
14 Bravais Lattices Combination of 7 crystal systems with lattice centring operation P, A, B, C, I, F, R
Screw Symmetry Combination of rotational symmetry with translational: n m (360 /n R + m/n T) Enables efficient packing of atoms/molecules
Helical Symmetry e.g. 3 1 v. 3 2 Others 4 1 and 4 3, 6 1 and 6 5, 6 2 and 6 4 Note 2 1, 4 2, 6 3 are not helical
Glide Symmetry Combination of 2-fold rotary-inversion (m) with translation: a, b, c, n, d Also enables efficient packing of atoms/molecules
Symmetry Symbols Planes perpendicular to the screen plane Planes parallel to the screen plane
Symmetry Symbols Axes perpendicular to the plane
Symmetry Symbols Axes parallel to the plane
Point Groups Local symmetry at a point in space Combination of rotation and rotaryinversion axes to form a mathematical group Only use 1, 2, 3, 4, 6 and 1, 2, 3, 4, 6 32 crystallographic point groups 11 centrosymmetric Diffraction symmetry Laue classes
Point Groups 11 centrosymmetric point groups 1 2/m mmm 4/m 4/mmm 3 3m 6/m 6/mmm m 3 m 3m 11 enantiomorphic point groups Rotation axes only 1 2 222 4 422 3 32 6 622 23 432 10 polar point groups Leave more than one common point unchanged 1 2 m mm2 4 4mm 3 3m 6 6mm
Example Combination of m x, m y, and 2 z gives mm2
Molecular Symmetry May be higher than crystallographic point group symmetry e.g. C 60 Point group 5 3 2/m
e.g. 2/m Diffraction Symmetry
Reflection Multiplicity Single-crystal diffraction Individual reflections measured Powder diffraction Reflections related by point group symmetry are superimposed Multiplicity is the number of symmetry equivalent reflections Depends on diffraction symmetry Depends on class of reflection
Space Groups Combination of symmetry elements to form a mathematical group All must contain at least the identity (1) operation plus the unit translations: t(1,0,0), t(0,1,0), t(0,0,1) 230 combinations Classified by crystal system & crystal class
Crystal Class Point group derived from a space group by setting all the translation components of the symmetry operators to zero e.g. 2 1 2, a m,... Crystal class + Inversion symmetry Diffraction symmetry
Space Group Diagrams
Special Positions Points in space where the symmetry is higher than 1 e.g. SF 6
Asymmetric Units
Asymmetric Units
Asymmetric Units
Asymmetric Units Space occupied by molecule can be used!
Triclinic Space Groups
Monoclinic Space Groups
Space Group Determination Geometrical implications, e.g. a b c, α β γ 90 triclinic a b c, α = γ = 90, β 90 monoclinic a b c, α = β = γ = 90 orthorhombic a = b c, α = β = γ = 90 tetragonal a = b = c, α = β = γ = 90 cubic Beware of experimental error Symmetry may be lower than expected
Systematic Absences Random v. systematic zero intensity
Systematic Absences Random v. systematic zero intensity I(h00) = 0 when h = 2n + 1 (i.e. h odd) Reflection Conditions I(h00) 0 when h = 2n (i.e. h even)
Centred Lattices Reciprocal Space v. Real Space
Centred Lattices Reciprocal Space v. Real Space
Centred Lattices Translation in 3 dimensions gives rise to reflection condition in 3 dimensions P none none A x, ½+y, ½+z hkl: k + l = 2n B ½+x, y, ½+z hkl: h + l = 2n C ½+x, ½+y, z hkl: h + k = 2n I ½+x, ½+y, ½+z hkl: h + k + l = 2n F = A + B + C hkl: h + k = 2n, h + l = 2n, & h + k = 2n
Translation Along an Axis No translation, e.g. 2 axis No reflection condition
Translation Along an Axis With translation, e.g. 2 1 axis Reflection condition 00l: l = 2n
Glide Planes Translational component gives rise to reflection conditions in a diffraction plane Reflection condition h0l: l = 2n
Single-Crystal v Powder Diffraction Single crystal Lattice reflection conditions obvious Glide plane reflection conditions obvious Screw axis reflection conditions often obvious Powder Lattice reflection conditions obvious Glide plane reflection conditions often obvious Screw axis reflection conditions tricky
Web Site 3-D Symmetry Elements http://pd.chem.ucl.ac.uk/pdnn/symm1/symindex.htm Point Groups http://pd.chem.ucl.ac.uk/pdnn/symm2/indexpnt.htm Space Groups http://pd.chem.ucl.ac.uk/pdnn/symm3/spgindex.htm Space-Group Determination http://pd.chem.ucl.ac.uk/pdnn/symm4/condex.htm