# Crystal symmetry and Space Groups

Crystal symmetry and Space Groups

DC Levendis 26 March 20103 Dimensional Space Group NotationSpace Group symbolsP222 (16)Space group numberSymmetry operatorsLattice centering (P, F, I, C, R) RDefine using diagrams & fractional coordinatesExample: Triclinic (only two symmetry operators compatible,identity and centre of inversion) Why?Example: Monoclinic (only one two fold axis and or one mirrorplane compatible with or without identity and centre of inversion)n)Why?3 Dimensional Space Group NotationMore examples:P2/mP222(10) monoclinic (assignment)(16) orthorhombicPmmm (47) orthorhombicI4(79) tetragonal2

DC Levendis 26 March 2010Examples of triclinic crystalse.g. tetrazole: P1 (1 general equivalent position;1 molecule per unit cell; i.e. Z=1). (draw the spacegroup diagram) See .CIF fileCentre of Inversion/symmetry• In point groups defined by the symbol i• In crystallography text defined by a – (or bar)and in crystallography graphics defined by acircleoo o o4

DC Levendis 26 March 2010Examples of triclinic crystalsP-1 (space group number 2, 2 general equivalentpositions, 2 molecules per unit cell (i.e. Z=2)•Example: Z=2 for ethyne-nitrobenzene, the moleculeis on a general position). In this case, all of the atomsof the molecule make up one asymmetric unitExamples of triclinic crystalsP-1 (space group number 2)P-1 has 2 general equivalent positions in the unit cell.Example: phenyl-diboronic acid has one ONEmeolecule per unit cell; Z=1:Therefore the molecule must occupy a specialposition. In this case, each general equivalentposition is made up of only half a molecule5

DC Levendis 26 March 2010Examples of triclinic crystalsExercise: Explain the difference betweena special position and a generalequivalent position in a crystalSpace Group DiagramsDraw the space group diagram for thetriclinic space group, P-1Draw the space group diagram for themonoclinic space group, P2/mMirror planeperpendicular to the2-fold axisDemonstrate which new symmetry operator (and generalequivalent position) is generated by the combination of the2-fold rotation axis and the mirror plane.Write down the matrix representation of this relationship.6

DC Levendis 26 March 2010Space Group P2 (monoclinic)In this casethe 2-foldrotation positions isaround bGeneral equivalentSpecial positions at (0,y,0)for a 2-fold rotation axisalong b.Space GroupPm8

DC Levendis 26 March 2010Crystal symmetryWhen considering crystals we also need the translational symmetrySpace Group DiagramsDraw the space group diagram for thetriclinic space group, P-1Draw the space group diagram for themonoclinic space group, P2/mMirror planeperpendicular to the2-fold axis9

DC Levendis 26 March 2010Space Group DiagramsMatrix representation of symmetryelementsUse matrix representation for thesymmetry equivalent positions in thetriclinic space group, P-1Use matrix representation for thesymmetry equivalent positions in themonoclinic space group, P2/mMirror planeperpendicular to the2-fold axisSome examplessee CIF files EX1-EX510

DC Levendis 26 March 2010Screw axes• Sometimes also referred to asrototranslation axes• Involve a rotation of n around an unit cellaxis followed by translation along that axisby r/nSymmetry elements of the first kindDo not change the handedness of the object towhich the symmetry is being applied• Rotation axes• Screw axes11

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