III Crystal Symmetry 3-3 Point group and space group A.Point group Symbols of the 32 three dimensional point groups Rotation axis X Rotation-Inversion axis X x X X + centre (inversion): Include X for odd order X2 or Xm even: only for even rotation symmetry X2 + centre; Xm +centre: the same result (Include Xm for odd order) Ratation axis with mirror plane normal to it X/m x x mirror Rotation axis with mirror plane (planes) parallel to it Xm x mirror x Rotation axis with diad axis (axes) normal to it x X2 x Rotation-inversion axis with diad axis (axes) normal to it X2 X X Rotation-inversion axis with mirror plane (planes) parallel to it Xm X mirror X Rotation axis with mirror plane (planes) normal to it and mirror plane (planes) parallel to it X/mm x mirror x mirror System and point group Position in point group symbol Primary Secondary Tertiary Triclinic 1, 1 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). 1st setting: z-axis unique 2nd setting: y-axis unique Stereographic representation 1st setting 2nd setting System and point group Orthorhombic 222, mm2, mmm Position in point group symbol Primary Secondary Tertiary Diad (rotation Diad Diad and/or (rotation (rotation inversion) and/or and/or along x-axis inversion) inversion) along y-axis along z-axis Tetragonal Tetrad Diad Diad 4, 4, 4/m, (rotation (rotation (rotation 422, 4mm, and/or and/or and/or 42m, 4/mmm inversion) inversion) inversion) along z-axis along x- and along [110] y-axes and [110] axis Stereographic representation System and point group Trigonal and Hexagonal 3, 3, 32, 3m, 3m, 6, 6, 6/m, 622, 6mm, 6m2, 6/mmm Cubic 23, m3, 432,43m, m3m Position in point group symbol Stereographic Primary Secondary Tertiary representation Triad or Diad Diad hexad (rotation (rotation (rotation and/or and/or and/or inversion) inversion) inversion) along x-, y- normal to x-, along z-axis and u-axes y-, u-axes in the plane (0001) Diads or Triads Diads tetrad (rotation (rotation (rotation and/or and/or and/or inversion) inversion) inversion) along <111> along <110> along <100> axes axes axes Symmetry Direction Crystal System Primary Triclinic None Monoclinic [010] Orthorhombic Secondary Tertiary [100] [010] [001] Tetragonal [001] [100]/[010] [110] Hexagonal/ Trigonal [001] [100]/[010] [120]/[110] Cubic [100]/[010]/ [001] [111] [110] Rotation axis X Triclinic 1 Rotation-Inversion axis X 1 X + centre Include X (odd order) 1 For odd order includes X already! Monoclinic 1st setting X 2 X 2=m X + centre Include X (odd order) 2 đ 2 mirror 2 mirror 2=m 2 đ Monoclinic 2st setting X2 12 ≡ 2 Xm 1m ≡ m X2 or Xm even X2 + centre, Xm +centre Include Xm (odd order) 2/m 2/m 1 2 Orthorhombic 2/m Rotation axis X Rotation-Inversion axis X 2/m X + centre Include X (odd order) 2/m X2 Xm X2 or Xm even 222 2mm (2D) = mm2 2/m X2 + centre, Xm +centre mmm ≡ Include Xm (odd order) 2/m2/m2/m Already discussed Tetragonal Rotation axis X 4 Rotation-Inversion axis X 4 X + centre Include X (odd order) 4 đ X2 422 Xm 4mm X2 or Xm even 42đ X2 + centre, Xm +centre 4/mmm ≡ Include Xm (odd order) 4/m 2/m 2/m Trigonal Rotation axis X 3 Rotation-Inversion axis X 2/m X + centre Include X (odd order) 3 X2 32 Xm 3m X2 or Xm even X2 + centre, Xm +centre Include Xm (odd order) 2/m 3đ ≡ 32 đ Hexagonal Rotation axis X 6 Rotation-Inversion axis X 6 X + centre Include X (odd order) 6 đ X2 622 Xm 6mm X2 or Xm even 6đ2 X2 + centre, Xm +centre 6/mmm ≡ Include Xm (odd order) 6/m 2/m 2/m Cubic Rotation axis X Rotation-Inversion axis X X + centre Include X (odd order) 23 2/m m3≡ 2/m 3 X2 432 Xm 2/m X2 or Xm even 43đ X2 + centre, Xm +centre m3m ≡ Include Xm (odd order) 4/m 3 2/m Examples of point group operation #1 Point group 222 (1) At a general position [x y z], the symmetry is 1, Multiplicity = 4 The multiplicity tells us how many atoms are generated by symmetry if we place a single atom at that position. y x (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 #2 Point group 4 (1) At a general position [x y z], the symmetry is 1. Multiplicity = 4 (2) At a special position [001], the symmetry is 4. Multiplicity = 1 #3 Point group 4 (1)At a general position [x y z], the symmetry is 1. Multiplicity = 4 (2) At a special position [001], the symmetry is đ. Multiplicity = 2 (3) At a special position [000], the symmetry is 4. Multiplicity = 1 P4 4 2 2 2 1 1 1 1 h g f e d c b a 1 m m m 4 4 4 4 xyz, -x-yz, y-x-z, -yx-z 0 ½ z, ½ 0 -z ½ ½ z, ½ ½ -z 0 0 z, 0 0 -z ½½½ ½½0 00½ 000 Transformation of vector components Original vector is P =[đ1 , đ2 , đ3 ]= đĨ, đĻ, đ§ i.e. P=đĨx+đĻy+đ§z When symmetry operation transform the original axes x, y, z to the new axes x′, y′, z′ New vector after transformation of axes becomes P ′ = [đ′1, đ′2, đ′3 ] = đĸ, đŖ, đ¤ i.e. P′ = đĸ x ′ + đŖy′ + đ¤ z′ The angular relations between the axes may be specified by drawing up a table of direction cosines. x New axes x′ y′ z′ a11 = cosx′x a21 = cosy′x a31 = cosz′x Old axes y a12 = cosx′y a22 = cosy′y a32 = cosz′y z a13 = cosx′z a23 = cosy′z a33 = cosz′z Then đĸ = đĨ ∗ cosx′x + đĻ ∗ cosx′y + đ§ ∗ cosx′z i.e. đ′1 = a11 ∗ đ1 + a12 ∗ đ2 + a13 ∗ đ3 In a dummy notation đ′1 = a1j ∗ đj Similarly đ′2 = a2j ∗ đj đ′3 = a3j ∗ đj i.e. đ′i = aij ∗ đj Moreover, by repeating the argument for the reverse transformation and we have đĨ = đĸ ∗ cosx ′ x + đŖ ∗ cosy ′ x + đ¤ ∗ cosz ′ x đ1 = aj1 ∗ đ′j Similarly, đ2 = aj2 ∗ đ′j đ3 = aj3 ∗ đ′j i.e. “old” in terms of “new” đi = aji ∗ đ′j For example: #1 Point group 4 The direction cosines for the first operation is x′= - y a11 = 0 Old axes y a12 =−1 y′= x z′= z a21 = 1 a31 = 0 a22 = 0 a32 = 0 x New axes z a13 = 0 a23 = 0 a33 =1 After symmetry operation, the new position is [x y z] in new axes. We can express it in old axes by đi = aji ∗ đ′j = đ′j ∗ aji 0 −1 0 đ3 ] = [đĨ đĻ đ§] 1 0 0 0 0 1 = [đĻ đĨ đ§ ] i.e. [đ1 đ2 or đ1 y 0 1 0 x đ2 = −1 0 0 y = x đ3 z 0 0 1 z 14 plane lattices + 32 point groups īŽ 230 Space groups Triclinic Bravais Lattices P Monoclinic P, C 2, m, 2/m Orthorhombic P, C, F, I 222, mm2, 2/m 2/m 2/m Trigonal P, R Hexagonal P Tetragonal P, I 3, 3, 32, 3m, 32/m 6, 6, 6/m, 622, 6mm, 6m2, 6/m 2/m 2/m 4, 4, 4/m, 422, 4mm, 42m, 4/m 2/m 2/m Isometric P, F, I Crystal Class Point Groups 1, 1 23, 2/m3, 432, 43m, 4/m32/m B. Space group Table for all space groups Look at the notes! Good web site to read about space group http://www.uwgb.edu/dutchs/SYMMETRY/3d SpaceGrps/3dspgrp.htm http://img.chem.ucl.ac.uk/sgp/mainmenu.htm Symmetry elements in space group (1)Point group (2)Translation symmetry + point group Translational symmetry operations The first character: P: primitive A, B, C: A, B, C-base centered F: Face centered I: Body centered R: Romohedral Glide plane also exists for 3D space group with more possibility Symmetry planes normal to the plane of projection Symmetry plane Graphical symbol Translation Symbol Reflection plane None m Glide plane 1/2 along line 1/2 normal to plane 1/2 along line & 1/2 normal to plane 1/2 along line & 1/2 normal to plane 1/4 along line & 1/4 normal to plane a, b, or c Glide plane Double glide plane Diagonal glide plane Diamond glide plane a, b, or c e n d Symmetry planes normal to the plane of projection Projection plane Symmetry planes parallel to plane of projection Symmetry plane Graphical symbol Translation Symbol Reflection plane None m Glide plane 1/2 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 1/8 The presence of a d-glide plane automatically implies a centered lattice! 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) 1 2 1 2 along line in plane ≡ along line parallel to projection plane e.g. b glide , b --- graphic symbol for the axial glide plane along y axis c.f. mirror (m) graphic symbol for mirror 1 2 If the axial glide plane is normal to projection plane, the graphic symbol change to zˆ c yˆ b xˆ c glide a glide plane⊥đ§ axis If b glide plane is ⊥đ§ axis zˆ glide plane symbol yˆ xˆ b , underneath the glide plane đ 2 c glide: along z axis or đ+đ+đ 2 along [111] on rhombohedral axis ii. Diagonal glide (n) đ+đ đ+đ đ+đ , , 2 2 2 or đ+đ+đ 2 (tetragonal, cubic system) 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 iii. Diamond glide (d) đ+đ đ+đ+đ , 4 4 (tetragonal, cubic system) Symbols of symmetry axes Symmetry Element Identity 2-fold ⊥ page 2-fold in page 2 sub 1 ⊥ page 2 sub 1 in page 3-fold 3 sub 1 3 sub 2 4-fold 4 sub 1 4 sub 2 4 sub 3 6-fold 6 sub 1 6 sub 2 6 sub 3 Graphical Symbol Translation Symbol None None None None 1/2 1/2 None 1/3 2/3 None 1/4 1/2 3/4 None 1/6 1/3 1/2 1 2 2 21 21 3 31 32 4 41 42 43 6 61 62 63 Symmetry Element 6 sub 4 6 sub 5 Inversion 3 bar 4 bar 6 bar 2-fold and inversion 2 sub 1 and inversion 4-fold and inversion 4 sub 2 and inversion 6-fold and inversion 6 sub 3 and inversion Graphical Symbol Translation Symbol 2/3 5/6 None None None None 64 65 1 3 4 6 = 3/m None 2/m None 21/m None 4/m None 42/m None 6/m None 63/m i. All possible screw operations screw axis --- translation τ plus rotation screw Rn along c axis = counterclockwise rotation 360/R o + translation n/R c 2 21 4 41 3 31 42 43 32 6 61 62 63 64 65 62 ī˛ T 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. •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), 41, 42 or 43 (i.e. P41212, I4/m, P4/mcc) •Hexagonal – The primary symmetry symbol will always be a 6, (-6), 61, 62, 63, 64 or 65 (i.e. P6mm, P63/mcm) •Trigonal – The primary symmetry symbol will always be a 3, (-3) 31 or 32 (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, Cmc21, 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, P21/n) •Triclinic – The lattice descriptor will be followed by either a 1 or a (1). Examples Space group P1 P1 No. 1 P1 1 Triclinic C11 Origin on 1 Number Wyckoff Point Coordinates of equivalent Condition of notation symmetry positions limiting positions possible reflections 1 a 1 x, y, z No conditions http://img.chem.ucl.ac.uk/sgp/large/001az1.htm Space group P1 P1 No. 2 Ci1 P1 1 Triclinic Origin on 1 Number Wyckoff Point Coordinates of equivalent Condition of notation symmetry positions limiting positions possible reflections 2 i 1 x, y, zīŧx, y, z General: No conditions 1 h 1 1 1 g 1 0, , 1 f 1 1 e 1 1 d 1 1 c 1 b 1 0, 0, a 1 0, 0, 0 1 1 2 1 2 1 2 1 2 1 , , 1 2 2 1 1 2 2 1 , 0, 1 2 , ,0 2 , 0, 0 1 0, , 0 2 1 2 Special: No conditions http://img.chem.ucl.ac.uk/sgp/large/002az1.htm Space group P112 P112 C21 No. 3 Ist setting P112 2 Monoclinic Origin on 2; unique axis c Number Wyckoff Point Coordinates of equivalent Condition of notation symmetry positions limiting positions possible reflections 2 e 1 x, y, z; x, y, z General: hkl hk0 00l No conditions 1 d 2 1 , ,z 1 c 2 2 1 Special: , 0, z No conditions 1 b 2 0, , z 1 a 2 2 1 2 1 2 0, 0, z http://img.chem.ucl.ac.uk/sgp/large/003az1.htm Space group P121 P121 C21 No. 3 P121 2 Monoclinic nd Origin on 2; unique axis b 2 setting Number Wyckoff Point Coordinates of equivalent Condition of notation symmetry positions limiting positions possible reflections 2 e 1 x, y, z; x, y, z General: hkl h0l 0k0 No conditions 1 d 2 1 c 2 2 1 1 b 2 0, y, 1 a 2 1 2 , y, 1 2 , y, 0 1 2 0, y, 0 Special: No conditions http://img.chem.ucl.ac.uk/sgp/large/003ay1.htm Space group P1121 P21 C22 No. 4 Ist setting P1121 2 Monoclinic Origin on 21; unique axis c Number Wyckoff Point Coordinates of equivalent Condition of notation symmetry positions limiting positions possible reflections 2 a 1 x, y, z; x, y, 1 2 +z General: hkl: No conditions hk0: No conditions 00l: l=2n http://img.chem.ucl.ac.uk/sgp/large/004az1.htm Explanation: Condition limiting possible reflections #1 Consider the diffraction condition from plane (h k 0) Two atoms at x, y, z; x, y, 1 2 +z The diffraction amplitude F can be expressed as fi ∗ e−2π i [h k l]∗[x y z ] F= i fi ∗ e−2π i [h k 0]∗[x y z ] = i = fi ∗ e−2π i [h k 0]∗[x y z ] + fi ∗ e−2π i [h k 0]∗[ x = fi ∗ e−2π i(hx +ky ) + fi ∗ e−2π i −hx −ky = fi ∗ e−2π i(hx +ky ) + e2π i(hx +ky ) y 1/2 +z ] = fi ∗ 2 cos 2πi hx + ky = 2fi Therefore, no conditions can limit the (h, k, 0) diffraction #2 For the planes (00l) Two atoms at x, y, z; x, y, 12+z The diffraction amplitude F can be expressed as fi ∗ e−2π i [h k l]∗[x i y i z i ] F= i fi ∗ e−2π i [0 0 l]∗[x i y i z i ] = i = fi ∗ e−2π i [0 0 l ]∗[x y z ] + fi ∗ e−2π i [0 0 l ]∗[ x −2π ilz −2π i = fi ∗ e + fi ∗ e = fi ∗ e−2π ilz ∗ 1 + e−πil = fi ∗ 1 + e−πil y 1/2 +z ] l +lz 2 If l=2n, then F=2fi If l=2n+1, then F=0 Therefore, the condition l=2n limit the (0, 0 ,l) diffraction. Space group P1211 P21 C22 No. 4 P1211 2 Monoclinic nd Origin on 21; unique axis b 2 setting Number Wyckoff Point Coordinates of equivalent Condition of notation symmetry positions limiting positions possible reflections 2 a 1 1 x, y, z; x, +y, z 2 General: hkl: No conditions h0l: No conditions 0k0: k=2n Space group B112 B2 No. 5 B112 2 Monoclinic C23 Ist setting Origin on 2; unique axis c Number Wyckoff Point Coordinates of equivalent Condition of notation symmetry positions limiting positions possible reflections 4 c 1 x, y, z; x, y, z General: hkl: h+l=2n +( 1 1 1 0 2 2 hk0: h=2n 00l: l=2n 2 b 2 0, , z Special: 2 a 2 0, 0, z as above only 2 (1/2+đĨ,đĻ,1/2+z) (đĨ,đĻ,z) (x,y,z) y x (1/2+x,y,1/2+z) What can we do with the space group information contained in the International Tables? 1. Generating a Crystal Structure from its Crystallographic Description 2. Determining a Crystal Structure from Symmetry & Composition Example: Generating a Crystal Structure http://chemistry.osu.edu/~woodward/ch754/sym_itc. htm Description of crystal structure of Sr2AlTaO6 Space Group = Fm3m; a = 7.80 Å Atomic Positions Atom Sr Al Ta O x 0.25 0.0 0.5 0.25 y 0.25 0.0 0.5 0.0 z 0.25 0.0 0.5 0.0 From the space group tables http://www.cryst.ehu.es/cgibin/cryst/programs/nph-wp-list?gnum=225 32 f 3m 24 e 4mm 24 d mmm 8 4 4 c b a 43m m3m m3m xxx, -x-xx, -xx-x, x-x-x, xx-x, -x-x-x, x-xx, -xxx x00, -x00, 0x0, 0-x0,00x, 00-x 0 ¼ ¼, 0 ¾ ¼, ¼ 0 ¼, ¼ 0 ¾, ¼ ¼ 0, ¾ ¼ 0 ¼¼¼,¼¼¾ ½½½ 000 Sr 8c; Al 4a; Ta 4b; O 24e 40 atoms in the unit cell stoichiometry Sr8Al4Ta4O24 īŽ Sr2AlTaO6 F: face centered īŽ (000) (½ ½ 0) (½ 0 ½) (0 ½ ½) (000) (½½0) (½0½) (0½½) Sr 8c: ¼ ¼ ¼ īŽ (¼¼¼) (¾¾¼) (¾¼¾) (¼¾¾) ¼ ¼ ¾ īŽ (¼¼¾) (¾¾¾) (¾¼¼) (¼¾¼) Al ¾ + ½ = 5/4 =¼ 4a: 0 0 0 īŽ (000) (½ ½ 0) (½ 0 ½) (0 ½ ½) (000) (½½0) (½0½) (0½½) Ta 4b: ½ ½ ½ īŽ (½½½) (00½) (0½0) (½00) O (000) (½½0) (½0½) (0½½) x00 24e: ¼ 0 0 īŽ (¼00) (¾½0) (¾0½) (¼½½) -x00 ¾ 0 0 īŽ (¾00) (¼½0) (¼0½) (¾½½) 0x0 0 ¼ 0 īŽ (0¼0) (½¾0) (½¼½) (½¾½) 0-x0 0 ¾ 0 īŽ (0¾0) (½¼0) (½¾½) (0¼½) 00x 0 0 ¼ īŽ (00¼) (½½¼) (½0¾) (0½¾) 00-x 0 0 ¾ īŽ (00¾) (½½¾) (½0¼) (0½0¼) Bond distances: Al ion is octahedrally coordinated by six O Al-O distance d = 7.80 Å ī´ 0.25 − 0 2 + 0 − 0 2 + 0 − 0 2 = 1.95 Å Ta ion is octahedrally coordinated by six O Ta-O distance d = 7.80 Å ī´ 0.25 − 0.5 2 + 0.5 − 0.5 2 + 0.5 − 0.5 = 1.95 Å Sr ion is surrounded by 12 O Sr-O distance: d = 2.76 Å 2 Determining a Crystal Structure from Symmetry & Composition Example: Consider the following information: Stoichiometry = SrTiO3 Space Group = Pm3m a = 3.90 Å Density = 5.1 g/cm3 First step: calculate the number of formula units per unit cell : Formula Weight SrTiO3 = 87.62 + 47.87 + 3ī´ (16.00) = 183.49 g/mol (M) Unit Cell Volume = (3.90ī´10-8 cm)3 = 5.93 ī´ 10-23 cm3 (V) (5.1 g/cm3)ī´(5.93 ī´ 10-23 cm3) : weight in a unit cell (183.49 g/mole) / (6.022 ī´1023/mol) : weight of one molecule of SrTiO3 īŽ (5.1 g/cm3)ī´(5.93 ī´ 10-23 cm3)/ (183.49 g/mole/6.022 ī´1023/mol) = 0.99 īŽ number of molecules per unit cell : 1 SrTiO3. From the space group tables (only part of it) 6 e 4mm 3 3 1 1 d c b a 4/mmm 4/mmm m3m m3m x00, -x00, 0x0, 0-x0,00x, 00-x ½ 0 0, 0 ½ 0, 0 0 ½ 0½½,½0½,½½0 ½½½ 000 http://www.cryst.ehu.es/cgi-bin/cryst/programs/nph-wplist?gnum=221 Sr: 1a or 1b; Ti: 1a or 1b īŽ Sr 1a Ti 1b or vice verse O: 3c or 3d Evaluation of 3c or 3d: Calculate the Ti-O bond distances: d (O @ 3c) = 2.76 Å (0 ½ ½) D (O @ 3d) = 1.95 Å (½ 0 0, Better) Atom Sr Ti O x 0.5 0 0.5 y 0.5 0 0 z 0.5 0 0 Another example from the note The usage of space group for crystal structure identification Space group P 4/m 3 2/m Reference to note chapter 3-2 page 26 From the space group tables (only part of it) 6 e 4mm 3 3 1 1 d c b a 4/mmm 4/mmm m3m m3m x00, -x00, 0x0, 0-x0,00x, 00-x ½ 0 0, 0 ½ 0, 0 0 ½ 0½½,½0½,½½0 ½½½ 000 http://www.cryst.ehu.es/cgi-bin/cryst/programs/nph-wplist?gnum=221 #1 Simple cubic Number of Wyckoff Point positions notation symmetry 1 A m3m Coordinates of equivalent positions 0, 0, 0 #2 CsCl structure atoms Number of Wyckoff Point Coordinates of equivalent positions notation symmetry positions Cl 1 a m3m 0, 0, 0 Cs 1 b m3m 1 2 1 1 2 2 , , CsCl Vital Statistics Formula Crystal System Lattice Type Space Group Cell Parameters Atomic Positions Density CsCl Cubic Primitive Pm3m, No. 221 a = 4.123 Å, Z=1 Cl: 0, 0, 0 Cs: 0.5, 0.5, 0.5 (can interchange if desired) 3.99 #3 BaTiO3 structure atoms Number of Wyckoff Point Coordinates of equivalent positions notation symmetry positions Ba 1 a m3m 0, 0, 0 Ti 1 b m3m 1 O 3 c 4/mmm 0, , ; 2 1 2 1 2 183 K rhombohedral (R3m) Temperature 278 K Orthorhombic (Amm2) 1 , , 1 2 2 1 1 2 2 1 , 0, ; 1 2 , ,0 2 393 K Tetragonal (P4mm). Cubic (Pm3m)