Lattices and Symmetry Scattering and Diffraction (Physics)
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1 Lattices and Symmetry Scattering and Diffraction (Physics) James A. Kaduk INEOS Technologies Analytical Science Research Services Naperville IL
2 Harry Potter and the Sorcerer s (Philosopher s) Stone Ron: Seeker? But first years never make the house team. You must be the youngest Quiddich player in Harry: a century. According to McGonagall. Fred/George: Well done, Harry. Wood s just told us. Ron: Fred and George are on the team, too. Beaters. Fred/George: Our job is to make sure you don t get bloodied up too bad. 2
3 Alastor Mad-Eye Moody Constant Vigilance Harry Potter and the Goblet of Fire (2005) 3
4 The crystallographer s world view Reality can be more complex! 4
5 Twinning at the atomic level International Tables for Crystallography, Volume D, p
6 PDB entry 1eqg = ovine COX-1 complexed with Ibuprofen 6
7 Atoms (molecules) pack together in a regular pattern to form a crystal. There are two aspects to this pattern: Periodicity Symmetry First, consider the periodicity 7
8 To describe the periodicity, we superimpose (mentally) on the crystal structure a lattice. A lattice is a regular array of geometrical points, each of which has the same environment (they are all equivalent). 8
9 A Primitive Cubic Lattice (CsCl) 9
10 A unit cell of a lattice (or crystal) is a volume which can describe the lattice using only translations. In 3 dimensions (for crystallographers), this volume is a parallelepiped. Such a volume can be defined by six numbers the lengths of the three sides, and the angles between them or three basis vectors. 10
11 A Unit Cell 11
12 Descriptions of the Unit Cell a, b, c, α, β, γ a, b, c x 1 a + x 2 b + x 3 c, 0 x n < 1 lattice points = ha + kb + lc, hkl integers domain of influence Dirichlet domain, Voronoi domain, Wigner-Seitz cell, Brillouin zone 12
13 A Brillouin Zone C. Kittel, Introduction to Solid State Physics, 6 th Edition, p. 41 (1986) 13
14 The unit cell is not unique (c:\myfiles\clinic\index2.wrl) 14
15 15
16 16
17 17
18 How do I pick the unit cell? Axis system (basis set) is right-handed Symmetry defines natural directions and boundaries Angles close to 90 Standard settings of space groups To make structural similarities clearer 18
19 The Reduced Cell 3 shortest non-coplanar translations Main Conditions (shortest vectors) Special Conditions (unique) May not exhibit the true symmetry 19
20 The Reduced Form a a A b b B c c C b c D a c E a b F 20
21 Positive Reduced Form, Type I Cell, all angles < 90, T = (a b)(b c)(c a) > 0 Main conditions: a a b b c c a c ½ a a b c ½ b b a b ½ a a Special conditions: if a a = b b then b c a c if b b = c c then a c a b if b c = ½ b b then if a c = ½ a a then if a b = ½ a a then a b 2 a c a b 2 b c a c 2 b c 21
22 Negative reduced Form, Type II Cell all angles 90, T = (a b)(b c)(c a) 0 Main Conditions: a a b b c c a c ½ a a b c ½ b b a b a a ( b c + a c + a b ) ½ ( a a + b b ) Special Conditions: if a a = b b then b c a c if b b = c c then a c a b if b c = ½ b b then a b = 0 if a c = ½ a a then a b = 0 if a b = ½ a a then a c = 0 if ( b c + a c + a b ) = ½ ( a a + b b ) then a a 2 a c + a b 22
23 There are 44 reduced forms. The relationships among the six terms determine the Bravais lattice of the crystal. J. K. Stalick and A. D. Mighell, NBS Technical Note 1229, A. D. Mighell and J. R. Rodgers, Acta Cryst., A36, (1980). 23
24 The Normalized Reduced Form and Cell: Mathematical Tools for Lattice Analysis Symmetry and Similarity, Alan D. Mighell, J. Res. Nat. Inst. Stand. Tech., 108(6), (2003). 24
25 25 International Tables for Crystallography, Volume F, Figure , p.52 (2001)
26 26
27 The mystery of the fifteenth Bravais lattice, A. Nussbaum, Amer. J. Phys., 68(10), (2000). 27
28 Symmetry Groups and Their Applications, W. Miller, Jr., Academic Press, New York (1972), Chapter 2. 1! 2 / m! mmm! 4 / mmm! m3 " " 31m! 6 / mmm 28
29 A = B = C Number Type D E F Bravais 1 I A/2 A/2 A/2 cf 2 I D D D hr 3 II cp 4 II -A/3 -A/3 -A/3 ci 5 II D D D hr 6 II D* D F ti 7 II D* E E ti 8 II D* E F oi * 2 D + E + F = A + B 29
30 A = B, no conditions on C Number Type D E F Bravais 9 I A/2 A/2 A/2 hr 10 I D D F mc 11 II tp 12 II 0 0 -A/2 hp 13 II 0 0 F oc 14 II -A/2 -A/2 0 ti 15 II D* D F of 16 II D D F mc 17 II D* E F mc * 2 D + E + F = A + B 30
31 B = C, no conditions on A Number Type D E F Bravais 18 I A/4 A/2 A/2 ti 19 I D A/2 A/2 oi 20 I D E E mc 21 II tp 22 II -B/2 0 0 hp 23 II D 0 0 oc 24 II D* -A/3 -A/3 hr 25 II D E E mc * 2 D + E + F = A + B 31
32 No conditions on A, B, C Number Type D E F Bravais 26 I A/4 A/2 A/2 of 27 I D A/2 A/2 mc 28 I D A/2 2D mc 29 I D 2D A/2 mc 30 I B/2 E 2E mc 31 I D E F ap 32 II op 40 II -B/2 0 0 oc 35 II D 0 0 mp 36 II 0 -A/2 0 oc 33 II 0 E 0 mp 38 II 0 0 -A/2 oc 34 II 0 0 F mp 42 II -B/2 -A/2 0 oi 41 II -B/2 E 0 mc 37 II D -A/2 0 mc 39 II D 0 -A/2 mc 43 II D E F mi 44 II D E F ap 32 2 D + E + F = A + B, plus 2D + F = B
33 Indexing programs can get caught in a reduced cell, and miss the (higher) true symmetry. It s always worth a manual check of your cell. 33
34 The metric symmetry can be higher than the crystallographic symmetry! (A monoclinic cell can have β = 90 ) 34
35 35
36 Definitions [hkl] <hkl> (hkl) {hkl} hkl indices of a lattice direction indices of a set of symmetryequivalent lattice directions indices of a single crystal face indices of a set of all symmetryequivalent crystal faces indices of a Bragg reflection 36
37 Now consider the symmetry 37
38 Point Symmetry Elements A point symmetry operation does not alter at least one point upon which it operates Rotation axes Mirror planes Rotation-inversion axes (rotation-reflection) Center Screw axes and glide planes are not point symmetry elements! 38
39 Symmetry Operations A proper symmetry operation does not invert the handedness of a chiral object Rotation Screw axis Translation An improper symmetry operation inverts the handedness of a chiral object Reflection Inversion Glide plane Rotation-inversion 39
40 Not all combinations of symmetry elements are possible. In addition, some point symmetry elements are not possible if there is to be translational symmetry as well. There are only 32 crystallographic point groups consistent with periodicity in three dimensions. 40
41 The 32 Point Groups (1) International Tables for Crystallography, Volume A, Table , p.819 (2002) 41
42 The 32 Point Groups (2) International Tables for Crystallography, Volume A, Table , p.819 (2002) 42
43 Symbols for Symmetry Elements (1) International Tables for Crystallography, Volume A, Table 1.4.5, p. 9 (2002) 43
44 Symbols for Symmetry Elements (2) International Tables for Crystallography, Volume A, Table 1.4.5, p. 9 (2002) 44
45 Symbols for Symmetry Elements (3) International Tables for Crystallography, Volume A, Table 1.4.2, p. 7 (2002) 45
46 2 Rotation Axis (ZINJAH) 46
47 3 Rotation Axis (ZIRNAP) 47
48 4 Rotation Axis (FOYTAO) 48
49 6 Rotation Axis (GIKDOT) 49
50 -1 Inversion Center (ABMQZD) 50
51 -2 Rotary Inversion Axis? 51
52 m Mirror Plane (CACVUY) 52
53 -3 Rotary Inversion Axis (DOXBOH) 53
54 -4 Rotary Inversion Axis (MEDBUS) 54
55 -6 Rotary Inversion Axis (NOKDEW) 55
56 2 1 Screw Axis (ABEBIS) 56
57 3 1 Screw Axis (AMBZPH) 57
58 3 2 Screw Axis (CEBYUD) 58
59 4 1 Screw Axis (ATYRMA10) 59
60 4 2 Screw Axis (HYDTML) 60
61 4 3 Screw Axis (PIHCAK) 61
62 6 1 Screw Axis (DOTREJ) 62
63 6 2 Screw Axis (BHPETS10) 63
64 6 3 Screw Axis (NAIACE) 64
65 6 4 Screw Axis (TOXQUS) 65
66 6 5 Screw Axis (BEHPEJ) 66
67 c Glide (ABOPOW) 67
68 n Glide (BOLZIL) 68
69 d (diamond) Glide (FURHUV) 69
70 What does all this mean? 70
71 Symmetry information is tabulated in International Tables for Crystallography, Volume A edited by Theo Hahn Fifth Edition
72 Guaifenesin, P (#19) 72
73 73
74 Copyright Birkbeck College, University of London. 74
75 Hermann-Mauguin Space Group Symbols the centering, and then a set of characters indicating the symmetry elements along the symmetry directions Lattice Primary Secondary Tertiary Triclinic None Monoclinic unique (b or c) Orthorhombic [100] [010] [001] Tetragonal [001] {100} {110} Hexagonal [001] {100} {110} Rhom. (hex) [001] {100} Rhom. (rho) [111] {1-10} Cubic {100} {111} {110} 75
76 Alternate Settings of Space Groups Triclinic none Monoclinic (a) b or c unique, 3 cell choices Orthorhombic 6 possibilities Tetragonal C or F cells Trigonal/hexagonal triple H cell Cubic Different Origins 76
77 An Asymmetric Unit A simply-connected smallest closed volume which, by application of all symmetry operations, fills all space. It contains all the information necessary for a complete description of the crystal structure. 77
78 78
79 Sub- and Super-Groups Phase transitions (second-order) Overlooked symmetry Relations between crystal structures Subgroups Translationengleiche (keep translations, lose class) Klassengleiche (lose translations, keep class) General (lose translations and class) 79
80 A Bärninghausen Tree for translationengleiche subgroups International Tables for Crystallography, Volume 1A, p. 396 (2004) 80
81 Mercury/ETGUAN (P #92) 81
82 82
83 83
84 Not all space groups are possible for protein crystals. 84
85 Space Group Frequencies in the Protein Data Bank, 17 June # Entries Space Group Number 85
86 Space Group Frequencies 100 PDB % CSD % ICSD % Frequency of O Space Group Number 86
87 Some Classifications of Space Groups Enantiomorphic, chiral, or dissymmetric absence of improper rotations (including 1, 2 = m, and 4 ) Polar two directional senses are geometrically or physically different 87
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