Lattices and Symmetry Scattering and Diffraction...
Transcript of Lattices and Symmetry Scattering and Diffraction...
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Lattices and SymmetryScattering and Diffraction (Physics)
James A. KadukINEOS Technologies
Analytical ScienceResearch Services
Naperville IL [email protected]
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Harry Potter and the Sorcerer’s(Philosopher’s) Stone
Ron: Seeker? But first years never make the houseteam. 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 getbloodied up too bad.
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Alastor “Mad-Eye” Moody – “Constant Vigilance”
Harry Potter and the Goblet of Fire (2005)
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The crystallographer’s world view
Reality can be more complex!
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Twinning at the atomic level
International Tables for Crystallography, Volume D, p. 438
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PDB entry 1eqg = ovine COX-1complexed with Ibuprofen
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Atoms (molecules) pack together in aregular pattern to form a crystal.
There are two aspects to this pattern:
PeriodicitySymmetry
First, consider the periodicity…
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To describe the periodicity, wesuperimpose (mentally) on the
crystal structure a lattice.A lattice is a regular array of
geometrical points, each of whichhas the same environment (they
are all equivalent).
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A Primitive Cubic Lattice (CsCl)
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A unit cell of a lattice (or crystal) isa volume which can describe the
lattice using only translations. In 3dimensions (for crystallographers),
this volume is a parallelepiped.Such a volume can be defined by six
numbers – the lengths of the threesides, and the angles between them –
or three basis vectors.
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A Unit Cell
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a, b, c, α, β, γa, b, cx1a + x2b + x3c, 0 ≤ xn < 1lattice points = ha + kb + lc,
hkl integersdomain of influence – Dirichlet domain, Voronoi domain, Wigner-Seitz cell, Brillouin zone
Descriptions of the Unit Cell
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A Brillouin Zone
C. Kittel, Introduction to Solid State Physics, 6th Edition, p. 41 (1986)
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The unit cell is not unique(c:\MyFiles\Clinic\index2.wrl)
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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
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The Reduced Cell
• 3 shortest non-coplanar translations• Main Conditions (shortest vectors)• Special Conditions (unique)
• May not exhibit the true symmetry
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The Reduced Form
a·bF
a·cE
b·cD
c·cC
b·bB
a·aA
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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 b·c ≤ ½ b·ba·c ≤ ½ a·a a·b ≤ ½ a·a
Special conditions:if a·a = b·b then b·c ≤ a·cif b·b = c·c then a·c ≤ a·bif b·c = ½ b·b then a·b ≤ 2 a·cif a·c = ½ a·a then a·b ≤ 2 b·cif a·b = ½ a·a then a·c ≤ 2 b·c
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Negative reduced Form, Type II Cellall angles ≥ 90°, T = (a·b)(b·c)(c·a) ≤ 0
Main Conditions:a·a ≤ b·b ≤ c·c |b·c| ≤ ½ b·b|a·c| ≤ ½ a·a |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 = 0if |a·c| = ½ a·a then a·b = 0if |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|
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There are 44 reduced forms. Therelationships among the six termsdetermine the Bravais lattice of
the crystal.
J. K. Stalick and A. D. Mighell,NBS Technical Note 1229, 1986.A. D. Mighell and J. R. Rodgers,
Acta Cryst., A36, 321-326 (1980).
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“The Normalized Reduced Form andCell: Mathematical Tools for Lattice
Analysis – Symmetry andSimilarity”, Alan D. Mighell, J. Res.
Nat. Inst. Stand. Tech., 108(6),447-452 (2003).
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25International Tables for Crystallography, Volume F, Figure 2.1.3.3, p.52 (2001)
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“The mystery of the fifteenthBravais lattice”, A. Nussbaum,
Amer. J. Phys., 68(10),950-954 (2000).
http://ojps.aip.org/ajp/
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Symmetry Groups and TheirApplications, W. Miller, Jr., Academic
Press, New York (1972), Chapter 2.
1 2 / 4 / 3
31 6 /
m mmm mmm m
m mmm
! ! ! !
" "
!
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A = B = C
oIFED*II8
tIEED*II7
tIFDD*II6
hRDDDII5
cI-A/3-A/3-A/3II4
cP000II3
hRDDDI2
cFA/2A/2A/2I1
BravaisFEDTypeNumber
* 2|D + E + F| = A + B
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A = B, no conditions on C
mCFED*II17
mCFDDII16
oFFDD*II15
tI0-A/2-A/2II14
oCF00II13
hP-A/200II12
tP000II11
mCFDDI10
hRA/2A/2A/2I9
BravaisFEDTypeNumber
* 2|D + E + F| = A + B
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B = C, no conditions on A
mCEEDII25
hR-A/3-A/3D*II24
oC00DII23
hP00-B/2II22
tP000II21
mCEEDI20
oIA/2A/2DI19
tIA/2A/2A/4I18
BravaisFEDTypeNumber
* 2|D + E + F| = A + B
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No conditions on A, B, C
aPFEDII44
mIFED†II43
mC-A/20DII39
mC0-A/2DII37
mC0E-B/2II41
oI0-A/2-B/2II42
mPF00II34
oC-A/200II38
mP0E0II33
oC0-A/20II36
mP00DII35
oC00-B/2II40
oP000II32
aPFEDI31
mC2EEB/2I30
mCA/22DDI29
mC2DA/2DI28
mCA/2A/2DI27
oFA/2A/2A/4I26
BravaisFEDTypeNumber
† 2|D + E + F| = A + B, plus |2D + F| = B
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Indexing programs can get “caught” ina reduced cell, and miss the (higher)true symmetry. It’s always worth a
manual check of your cell.
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The metric symmetry can be higherthan the crystallographic symmetry!
(A monoclinic cell can have β = 90°)
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35http://www.haverford.edu/physics-astro/songs/bravais.htm
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Definitions[hkl] indices of a lattice direction<hkl> indices of a set of symmetry-
equivalent lattice directions(hkl) indices of a single crystal face{hkl} indices of a set of all symmetry-
equivalent crystal faceshkl indices of a Bragg reflection
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Now consider the symmetry…
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Point Symmetry Elements
• A point symmetry operation does not alter atleast one point upon which it operates– Rotation axes– Mirror planes– Rotation-inversion axes (rotation-reflection)– Center
Screw axes and glide planes are notpoint symmetry elements!
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Symmetry Operations• A proper symmetry operation does not invert the
handedness of a chiral object– Rotation– Screw axis– Translation
• An improper symmetry operation inverts thehandedness of a chiral object– Reflection– Inversion– Glide plane– Rotation-inversion
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Not all combinations of symmetryelements are possible. In addition,
some point symmetry elements are notpossible if there is to be translationalsymmetry as well. There are only 32
crystallographic point groupsconsistent with periodicity in three
dimensions.
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The 32 Point Groups (1)
International Tables for Crystallography, Volume A, Table 12.1.4.2, p.819 (2002)
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The 32 Point Groups (2)
International Tables for Crystallography, Volume A, Table 12.1.4.2, p.819 (2002)
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Symbols for Symmetry Elements (1)
International Tables for Crystallography, Volume A, Table 1.4.5, p. 9 (2002)
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Symbols for Symmetry Elements (2)
International Tables for Crystallography, Volume A, Table 1.4.5, p. 9 (2002)
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Symbols for Symmetry Elements (3)
International Tables for Crystallography, Volume A, Table 1.4.2, p. 7 (2002)
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2 Rotation Axis (ZINJAH)
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3 Rotation Axis (ZIRNAP)
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4 Rotation Axis (FOYTAO)
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6 Rotation Axis (GIKDOT)
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-1 Inversion Center(ABMQZD)
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-2 Rotary Inversion Axis?
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m Mirror Plane (CACVUY)
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-3 Rotary Inversion Axis (DOXBOH)
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-4 Rotary Inversion Axis (MEDBUS)
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-6 Rotary Inversion Axis (NOKDEW)
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21 Screw Axis (ABEBIS)
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31 Screw Axis (AMBZPH)
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32 Screw Axis (CEBYUD)
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41 Screw Axis (ATYRMA10)
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42 Screw Axis (HYDTML)
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43 Screw Axis (PIHCAK)
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61 Screw Axis (DOTREJ)
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62 Screw Axis (BHPETS10)
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63 Screw Axis (NAIACE)
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64 Screw Axis (TOXQUS)
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65 Screw Axis (BEHPEJ)
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c Glide (ABOPOW)
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n Glide (BOLZIL)
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d (diamond) Glide (FURHUV)
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What does all this mean?
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Symmetry information is tabulated inInternational Tables for
Crystallography, Volume Aedited by Theo Hahn
Fifth Edition 2002
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Guaifenesin, P212121 (#19)
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© Copyright 1997-1999. Birkbeck College, University of London.
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Hermann-Mauguin Space Group Symbolsthe centering, and then a set of characters indicating the
symmetry elements along the symmetry directions
{110}{111}{100}Cubic{1-10}[111]Rhom. (rho){100}[001]Rhom. (hex)
{110}{100}[001]Hexagonal{110}{100}[001]Tetragonal[001][010][100]Orthorhombic
unique (b or c)MonoclinicNoneTriclinic
TertiarySecondaryPrimaryLattice
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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
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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 acomplete description of the crystal structure.
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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)
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A Bärninghausen Treefor translationengleiche subgroups
International Tables for Crystallography, Volume 1A, p. 396 (2004)
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Mercury/ETGUAN (P41212 #92)
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Not all space groups are possible forprotein crystals.
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Space Group Frequencies in theProtein Data Bank, 17 June 2003
Space Group Number
0 20 40 60 80 100 120 140 160 180 200 220
# E
ntrie
s
1
10
100
1000
10000
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Space Group Frequencies
Space Group Number
0 20 40 60 80 100 120 140 160 180 200 220
Fre
qu
en
cy o
f Occu
rren
ce, %
0.01
0.1
1
10
100
PDB %
CSD %
ICSD %
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Some Classifications of Space Groups
• Enantiomorphic, chiral, or dissymmetric –absence of improper rotations(including , = m, and )
• Polar – two directional senses aregeometrically or physically different
1̄ 2̄ 4̄