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III Crystal Symmetry 3-3 Point group and space group A.Point group Symbols of the 32 three dimensional point groups Rotation axis X x

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Ratation axis with mirror plane normal to it X/m x x x x Rotation axis with mirror plane (planes) parallel to it Xm mirror

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Rotation axis with diad axis (axes) normal to it X2 x x

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Rotation axis with mirror plane (planes) normal to it and mirror plane (planes) parallel to it X/mm mirror x x

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System and point group Position in point group symbolStereographic representation PrimarySecondaryTertiary Only one symbol which denotes all directions in the crystal. Monoclinic 2, m, 2/m The symbol gives the nature of the unique diad axis (rotation and/or inversion). 1 st setting: z-axis unique 2 nd setting: y-axis unique 1 st setting 2 nd setting

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System and point group Position in point group symbolStereographic representation PrimarySecondaryTertiary Orthorhombic 222, mm2, mmm Diad (rotation and/or inversion) along x-axis Diad (rotation and/or inversion) along y-axis Diad (rotation and/or inversion) along z-axis Tetrad (rotation and/or inversion) along z-axis Diad (rotation and/or inversion) along x- and y-axes

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System and point group Position in point group symbolStereographic representation PrimarySecondaryTertiary Triad or hexad (rotation and/or inversion) along z-axis Diad (rotation and/or inversion) along x-, y- and u-axes Diad (rotation and/or inversion) normal to x-, y-, u-axes in the plane (0001) Diads or tetrad (rotation and/or inversion) along axes Triads (rotation and/or inversion) along axes Diads (rotation and/or inversion) along axes

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Crystal System Symmetry Direction PrimarySecondaryTertiary TriclinicNone Monoclinic[010] Orthorhombic[100][010][001] Tetragonal[001][100]/[010][110] Hexagonal/ Trigonal [001][100]/[010] Cubic [100]/[010]/ [001] [111][110]

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Triclinic Rotation axis X 1

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Monoclinic 1 st setting X 2 2 mirror 2

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Monoclinic 2 st setting X2 Xm 2/m 2 1

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Orthorhombic Rotation axis X X2 Xm 222 2mm (2D) = mm2 2/m Already discussed

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Tetragonal Rotation axis X 4 X2 Xm 422 4mm

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Trigonal Rotation axis X 3 X2 Xm 32 3m3m 2/m

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Hexagonal Rotation axis X 6 X2 Xm 622 6mm

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Cubic Rotation axis X 23 X2 Xm 432 2/m

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Examples of point group operation #1 Point group 222 (1) At a general position [x y z], the symmetry is 1, Multiplicity = 4 x y The multiplicity tells us how many atoms are generated by symmetry if we place a single atom at that position.

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(2)At a special position [100], the symmetry is 2. Multiplicity = 2 At a special position [010], the symmetry is 2. Multiplicity = 2 At a special position [001], the symmetry is 2. Multiplicity = 2

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#2 Point group 4 (1)At a general position [x y z], the symmetry is 1. Multiplicity = 4

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(2) At a special position [001], the symmetry is 4. Multiplicity = 1

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(1)At a general position [x y z], the symmetry is 1. Multiplicity = 4

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4h1xyz, -x-yz, y-x-z, -yx-z 2gm0 ½ z, ½ 0 -z 2fm½ ½ z, ½ ½ -z 2em0 0 z, 0 0 -z 1d½ ½ ½ 1c½ ½ 0 1b0 0 ½ 1a000

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Transformation of vector components i.e.

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The angular relations between the axes may be specified by drawing up a table of direction cosines. Old axes New axes

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For example: #1 Point group 4 The direction cosines for the first operation is Old axes New axes

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After symmetry operation, the new position is [x y z] in new axes. We can express it in old axes by i.e. or

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14 plane lattices + 32 point groups 230 Space groups Crystal Class Bravais Lattices Point Groups TriclinicP MonoclinicP, C2, m, 2/m OrthorhombicP, C, F, I222, mm2, 2/m 2/m 2/m TrigonalP, R HexagonalP TetragonalP, I IsometricP, F, I

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B. Space group Table for all space groups Look at the notes! SpaceGrps/3dspgrp.htm Good web site to read about space group

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The first character: P: primitive A, B, C: A, B, C-base centered F: Face centered I: Body centered R: Romohedral Symmetry elements in space group (1)Point group (2)Translation symmetry + point group Translational symmetry operations

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Glide plane also exists for 3D space group with more possibility Symmetry planes normal to the plane of projection Symmetry planeGraphical symbolTranslationSymbol Reflection planeNone m Glide plane1/2 along line a, b, or c Glide plane 1/2 normal to plane a, b, or c Double glide plane 1/2 along line & 1/2 normal to plane e Diagonal glide plane 1/2 along line & 1/2 normal to plane n Diamond glide plane 1/4 along line & 1/4 normal to plane d

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Projection plane Symmetry planes normal to the plane of projection

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1/8 Symmetry planeGraphical symbolTranslationSymbol Reflection planeNone m Glide plane1/2 along arrow a, b, or c Double glide plane 1/2 along either arrow e Diagonal glide plane 1/2 along the arrow n Diamond glide plane 1/8 or 3/8 along the arrows d 3/8 Symmetry planes parallel to plane of projection The presence of a d-glide plane automatically implies a centered lattice!

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Glide planes ---- translation plus reflection across the glide plane * axial glide plane (glide plane along axis) ---- translation by half lattice repeat plus reflection ---- three types of axial glide plane i. a glide, b glide, c glide (a, b, c)

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e.g. b glide, b --- graphic symbol for the axial glide plane along y axis c.f. mirror (m) graphic symbol for mirror

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c glide glide plane symbol

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b, underneath the glide plane or

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ii. Diagonal glide (n) If glide plane is perpendicular to the drawing plane (xy plane), the graphic symbol is If glide plane is parallel to the drawing plane, the graphic symbol is

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iii. Diamond glide (d)

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Symmetry Element Graphical SymbolTranslationSymbol IdentityNone 1 2-fold ⊥ page None 2 2-fold in pageNone 2 2 sub 1 ⊥ page 1/ sub 1 in page1/ foldNone 3 3 sub 11/ sub 22/ foldNone 4 4 sub 11/ sub 21/ sub 33/ foldNone 6 6 sub 11/ sub 21/ sub 31/2 6 3 Symbols of symmetry axes

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Symmetry Element Graphical SymbolTranslationSymbol 6 sub 42/ sub 55/6 6 5 InversionNone 1 3 barNone 3 4 barNone 4 6 barNone 6 = 3/m 2-fold and inversion None 2/m 2 sub 1 and inversion None 2 1 /m 4-fold and inversion None 4/m 4 sub 2 and inversion None 4 2 /m 6-fold and inversion None 6/m 6 sub 3 and inversion None 6 3 /m

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i. All possible screw operations screw axis --- translation τ plus rotation

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6262

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Symmorphic space group is defined as a space group that may be specified entirely by symmetry operation acting at a common point (the operations need not involve τ) as well as the unit cell translation. (73 space groups) Nonsymmorphic space group is defined as a space group involving at least a translation τ.

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Cubic – The secondary symmetry symbol will always be either 3 or – 3 (i.e. Ia3, Pm3m, Fd3m) Tetragonal – The primary symmetry symbol will always be either 4, (-4), 4 1, 4 2 or 4 3 (i.e. P , I4/m, P4/mcc) Hexagonal – The primary symmetry symbol will always be a 6, (-6), 6 1, 6 2, 6 3, 6 4 or 6 5 (i.e. P6mm, P6 3 /mcm) Trigonal – The primary symmetry symbol will always be a 3, (-3) 3 1 or 3 2 (i.e P31m, R3, R3c, P312) Orthorhombic – All three symbols following the lattice descriptor will be either mirror planes, glide planes, 2-fold rotation or screw axes (i.e. Pnma, Cmc2 1, Pnc2) Monoclinic – The lattice descriptor will be followed by either a single mirror plane, glide plane, 2-fold rotation or screw axis or an axis/plane symbol (i.e. Cc, P2, P2 1 /n) Triclinic – The lattice descriptor will be followed by either a 1 or a (- 1).

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Examples Space group P1

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Space group P112

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Space group P121

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Space group P112 1

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Condition limiting possible reflections

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Space group P12 1 1

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Space group B112

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(x,y,z)(x,y,z) (1/2+x,y,1/2+z) x y

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1. Generating a Crystal Structure from its Crystallographic Description What can we do with the space group information contained in the International Tables? 2. Determining a Crystal Structure from Symmetry & Composition

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Example: Generating a Crystal Structure htm Description of crystal structure of Sr 2 AlTaO 6 Atomxyz Sr0.25 Al0.0 Ta0.5 O

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From the space group tables bin/cryst/programs/nph-wp-list?gnum=225 32f3mxxx, -x-xx, -xx-x, x-x-x, xx-x, -x-x-x, x-xx, -xxx 24e4mmx00, -x00, 0x0, 0-x0,00x, 00-x 24dmmm0 ¼ ¼, 0 ¾ ¼, ¼ 0 ¼, ¼ 0 ¾, ¼ ¼ 0, ¾ ¼ 0 8c¼ ¼ ¼, ¼ ¼ ¾ 4b½ ½ ½ 4a000

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Sr 8c; Al 4a; Ta 4b; O 24e 40 atoms in the unit cell stoichiometry Sr 8 Al 4 Ta 4 O 24 Sr 2 AlTaO 6 F: face centered (000) (½ ½ 0) (½ 0 ½) (0 ½ ½) 8c: ¼ ¼ ¼ (¼¼¼) (¾¾¼) (¾¼¾) (¼¾¾) ¼ ¼ ¾ (¼¼¾) (¾¾¾) (¾¼¼) (¼¾¼) ¾ + ½ = 5/4 =¼ Sr Al 4a: (000) (½ ½ 0) (½ 0 ½) (0 ½ ½) (000) (½½0) (½0½) (0½½)

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Ta 4b: ½ ½ ½ (½½½) (00½) (0½0) (½00) O 24e: ¼ 0 0 (¼00) (¾½0) (¾0½) (¼½½) (000) (½½0) (½0½) (0½½) ¾ 0 0 (¾00) (¼½0) (¼0½) (¾½½) x00 -x00 0 ¼ 0 (0¼0) (½¾0) (½¼½) (½¾½) 0x00x0 0-x0 0 ¾ 0 (0¾0) (½¼0) (½¾½) (0¼½) 0 0 ¼ (00¼) (½½¼) (½0¾) (0½¾) 00x 00-x 0 0 ¾ (00¾) (½½¾) (½0¼) (0½0¼)

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Sr ion is surrounded by 12 O Sr-O distance: d = 2.76 Å

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Determining a Crystal Structure from Symmetry & Composition

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First step: calculate the number of formula units per unit cell : Formula Weight SrTiO 3 = (16.00) = g/mol (M) Unit Cell Volume = (3.90 cm) 3 = 5.93 cm 3 (V) (5.1 g/cm 3 ) (5.93 cm 3 ) : weight in a unit cell ( g/mole) / (6.022 /mol) : weight of one molecule of SrTiO 3

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number of molecules per unit cell : 1 SrTiO 3. (5.1 g/cm 3 ) (5.93 cm 3 )/ ( g/mole/6.022 /mol) = e4mmx00, -x00, 0x0, 0-x0,00x, 00-x 3d4/mmm½ 0 0, 0 ½ 0, 0 0 ½ 3c4/mmm0 ½ ½, ½ 0 ½, ½ ½ 0 1b½ ½ ½ 1a000 From the space group tables (only part of it) list?gnum=221

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Sr: 1a or 1b; Ti: 1a or 1b Sr 1a Ti 1b or vice verse O: 3c or 3d Evaluation of 3c or 3d: Calculate the Ti-O bond distances: d 3c) = 2.76 Å (0 ½ ½) D 3d) = 1.95 Å (½ 0 0, Better) Atomxyz Sr0.5 Ti000 O0.500

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Reference to note chapter 3-2 page 26 Another example from the note

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6e4mmx00, -x00, 0x0, 0-x0,00x, 00-x 3d4/mmm½ 0 0, 0 ½ 0, 0 0 ½ 3c4/mmm0 ½ ½, ½ 0 ½, ½ ½ 0 1b½ ½ ½ 1a000 From the space group tables (only part of it) list?gnum=221

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#1 Simple cubic

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CsCl Vital Statistics FormulaCsCl Crystal SystemCubic Lattice TypePrimitive Space GroupPm3m, No. 221 Cell Parametersa = Å, Z=1 Atomic Positions Cl: 0, 0, 0 Cs: 0.5, 0.5, 0.5 (can interchange if desired) Density3.99 #2 CsCl structure

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#3 BaTiO 3 structure Temperature 183 K rhombohedral (R3m) 278 K Orthorhombic (Amm2) 393 K Tetragonal (P4mm).

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