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 60566James.Kaduk@innovene.com

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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).

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°)

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