“International Tables of Crystallography” Physics 590 “International Tables of Crystallography” “Everything you wanted to know about beautiful flies, but were afraid to ask.” Schönflies Gordie Miller (321 Spedding) Proposed Plan What basic information is found on the space group pages… Stoichiometry of the unit cell (Wyckoff sites) Site (point) symmetry of atoms in solids Solid-solid phase transitions (group-subgroup relationships) Diffraction conditions – what to expect in a XRD powder pattern. References Space Groups for Solid State Scientists, G. Burns and A. M. Glazer (Little mathematical formalism; prose style) International Tables for Crystallography (http://it.iucr.org/) Bilbao Crystallographic Server (www.cryst.ehu.es) (Comprehensive resources for all space groups)

BaFe2As2 What can we learn from the International Tables? c b a Space Group: I4/mmm Lattice Constants: a = 3.9630 Å c = 13.0462 Å Asymmetric Unit: Ba (2a): 0 0 0 Fe (4d): ½ 0 ¼ As (4e): 0 0 0.3544 a b c Intensity (Arb. Units) (hkl) Indices h + k + l = even integer (2n) (013) (112) (200) (116) (213) (015) (004) (215) (028) (002) (011) 2θ (Cu Kα)

Typical Space Group Pages… Symbolism Diffraction Extinction Conditions Point Symmetry Features Stoichiometry Structure of Unit Cell Subgroup/Supergroup Relationships

NOTE: Symbolism In Schönflies notation, what does the symbol S2 mean? Point Group of the Space Group Crystal System Space Group Molecules Solids Symmetry Operation Schönflies Notation International Notation Proper Rotation (by 2π/n) Cn (C2, C3, C4, …) “n” (2, 3, 4, …) Identity E = C1 1 Improper Rotation Sn = h  Cn (S3, S4, S5, …) Inversion (x,y,z)  (–x,–y,–z) i = S2 Mirror plane  Principal Axis h = S1 /m (n/m is the designator: 4/m) Mirror plane  Principal Axis v , d (= S1) m = NOTE: S2 = h  C2 x y , (z) In Schönflies notation, what does the symbol S2 mean? (x,y,z) S2 = inversion C2 rotation followed by sh  C2 axis In International notation, what does the symbol mean? x y (z) 2-fold (C2) rotation followed by inversion ( ) (x,y,z) , Why are the symbols S2 and not used?

Symbolism: Crystal Systems What rotational symmetries are consistent with a lattice (translational symmetry)? C1 C2 (2π/2) C3 (2π/3) C4 (2π/4) C6 (2π/6) Crystal System Minimum Symmetry Primitive Unit Cell Lattice Types Triclinic None a  b  c;      Monoclinic One 2-fold axis (b-axis) a  b  c;  =  = 90,   90 Orthorhombic Three orthogonal 2-fold axes a  b  c;  =  =  = 90 Tetragonal One 4-fold axis (c-axis) a = b  c;  =  =  = 90 Cubic Four 3-fold axes a = b = c;  =  =  = 90 Trigonal One 3-fold axis a = b = c;  =  =  a = b  c;  =  = 90,  = 120 Hexagonal One 6-fold axis (c-axis)  = angle between b and c  = angle between a and c  = angle between a and b c a b

“Centered Lattices” Symbolism: Bravais Lattices 7 Crystal Systems = 7 Primitive Lattices (Unit Cells): P Crystal System Minimum Symmetry Primitive Unit Cell Lattice Types Triclinic None a  b  c;      P Monoclinic One 2-fold axis (b-axis) a  b  c;  =  = 90,   90 P C Orthorhombic Three orthogonal 2-fold axes a  b  c;  =  =  = 90 P C (A) I F Tetragonal One 4-fold axis (c-axis) a = b  c;  =  =  = 90 P I Cubic Four 3-fold axes a = b = c;  =  =  = 90 P I F Trigonal One 3-fold axis a = b = c;  =  =  a = b  c;  =  = 90,  = 120 R Hexagonal One 6-fold axis (c-axis) (rhombohedral) “Centered Lattices”  = angle between b and c  = angle between a and c  = angle between a and b ? c a b I F C B A Body- (All) Face- Base-

Symbolism: Point Groups Schönflies Notation Type Symbol Features Uniaxial n Single rotation axis Cn nh + mirror plane  Cn axis nv + n mirror planes || Cn axis Low Symmetry 1 Asymmetric (NO symmetry) s Mirror plane only i Inversion center only Dihedral n Rotation axis Cn + n C2 axes  Cn axis nd nh Polyhedral T, Th , Td Tetrahedral; 4 C3 axes (cube body-diagonals) O, Oh Octahedral; 4 C3 axes + 3 C4 axes (cube faces) I, Ih Icosahedral; 6 C5 axes

Symbolism: Crystallographic Point Groups Allowed Rotations = C1 C2 C3 C4 C6 32 Point Groups Crystal System Schönflies Symbol International Symbol Order / Inversion? Full Abbrev. Directions Triclinic 1 1 1 / No (Holohedral) i 2 / Yes Monoclinic s 1m1 or 11m m [010] or [001] 2 / No 2 121 or 112 2 2h 12/m1 or 112/m 2/m 4 / Yes Orthorhombic 2v 2mm [100][010][001] 4 / No 2 222 2h 2/m 2/m 2/m mmm 8 / Yes Tetragonal 4 4 [001]{100}{110} 4 4h 4/m 2d 8 / No 4v 4mm 4 422 4h 4/m 2/m 2/m 4/mmm 16 / Yes Yes: Laue Groups b c m2m or mm2 a b c c a b a+b a–b

Cmm2 For the following space group symbol What is the crystal class? Questions for Friday… For the following space group symbol What is the crystal class? What is the lattice type? Which face(s) are centered? (c) What is the point group of the space group using Schönflies notation? (d) Does the point group contain the inversion operation? Cmm2 Orthorhombic Base (C)-centered ab-faces C2v No

Symbolism: Crystallographic Point Groups (cont.) System Schönflies Symbol International Symbol Order / Inversion? Full Abbrev. Directions Trigonal 3 3 [001]{100}{210} 3 / No 6 6 / Yes 3v 3m or 3m1 6 / No 3 32 or 321 (Holohedral) 3d 12 / Yes Hexagonal 6 6 3h 6h 6/m 3h 12 / No 6v 6mm 6 622 6h 6/m 2/m 2/m 6/mmm 24 / Yes Cubic T 23 {100}{111}{110} Th Td 24 / No O 432 Oh 48 / Yes c a b 31m 312 c a b a b c

4 / m m m Symbolism: Symmetry Operations (4h = 4/mmm = 4/m 2/m 2/m) Schönflies International Coordinates (1) E 1 x, y, z (𝑥,𝑦,𝑧) (9) i –x, –y, –z ( 𝑥 , 𝑦 , 𝑧 ) (2) C2 = C42 2 0,0,z –x, –y, z ( 𝑥 , 𝑦 ,𝑧) (10) σh m x,y,0 x, y, –z (𝑥,𝑦, 𝑧 ) (3) C4 4+ 0,0,z –y, x, z ( 𝑦 ,𝑥,𝑧) (11) S43 4 + 0,0,z; 0,0,0 y, –x, –z (𝑦, 𝑥 , 𝑧 ) (4) C43 4– 0,0,z y, –x, z (𝑦, 𝑥 ,𝑧) (12) S4 4 − 0,0,z; 0,0,0 –y, x, –z ( 𝑦 ,𝑥, 𝑧 ) (5) C2 2 0,y,0 –x, y, –z ( 𝑥 ,𝑦, 𝑧 ) (13) σv m x,0,z x, –y, z (𝑥, 𝑦 ,𝑧) (6) 2 x,0,0 x, –y, –z (𝑥, 𝑦 , 𝑧 ) (14) m 0,y,z –x, y, z ( 𝑥 ,𝑦,𝑧) (7) C2 2 x,x,0 y, x, –z (𝑦,𝑥, 𝑧 ) (15) σd m x, 𝑥 ,z –y, –x, z ( 𝑦 , 𝑥 ,𝑧) (8) 2 x, 𝑥 ,0 –y, –x, –z ( 𝑦 , 𝑥 , 𝑧 ) (16) m x,x,z y, x, z (𝑦,𝑥,𝑧) Proper Rotations Improper Rotations – + 4 / m m m , , x y + – , , + – + – 33 Matrices: , , – + – + – + , , + – (Determinant = +1) (Determinant = -1)

Symmorphic Space Groups General Position Special Positions Point Group = {Symmetry operations intersecting in one point} (32) Space Group = {Essential Symmetry Operations}  {Bravais Lattice} (230)

“Ba2Fe4As4” BaFe2As2 Symmorphic Space Groups Z = 2 General Position Ba (2a): 4/mmm (D4h) Fe (4d): 𝟒 m2 (D2d) As (4e): 4mm (C4v) General Position “Ba2Fe4As4” Special Positions Z = 2 Space Group: I4/mmm Lattice Constants: a = 3.9630 Å c = 13.0462 Å Asymmetric Unit: Ba (2a): 0 0 0 Fe (4d): ½ 0 ¼ As (4e): 0 0 0.3544 BaFe2As2

Point Group of the Space Group Space Groups (230) Symmorphic Space Groups (73): {Essential Symmetry Operations} is a group. Point Group of the Space Group Nonsymmorphic Space Groups (157): {Essential Symmetry Operations} is a not a group.

Point Group of the Space Group Space Group Operations: Screw Rotations and Glide Reflections Screw Rotations: Rotation by 2/n (Cn) then Displacement j/n lattice vector || Cn axis (allowed integers j = 1,…, n–1) Symbol = nj 21, 31, 32, 41, 42, 43, 61, 62, 63, 64, 65 I41/amd 4/mmm Point Group of the Space Group P42/ncm Glide Reflections: Reflection then Displacement 1/2 lattice vector || reflection plane Axial: a, b, c (lattice vectors = a, b, c) Diagonal: n (vectors = a+b, a+c, b+c) Diamond: d (vectors = (a+b+c)/2, (a+b)/2, (b+c)/2, (b+c)/2) Nonsymmorphic Space Groups (157)

The Origin! Si: 0, 0, 0