Crystal symmetry and Space Groups

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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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