1 Lattices and Symmetry Scattering and Diffraction (Physics) James A. Kaduk Innovene USA LLC Naperville IL 60566 [email protected]

$Page 1: 1 Lattices and Symmetry Scattering and Diffraction (Physics) James A. Kaduk Innovene USA LLC Naperville IL 60566 James.Kaduk@innovene.com$
1

Lattices and SymmetryScattering and Diffraction (Physics)

James A. KadukInnovene USA LLCNaperville IL 60566

[email protected]

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.

3

The crystallographer’s world view

Reality can be more complex!

4

Twinning at the atomic level

International Tables for Crystallography, Volume D, p. 438

5

PDB entry 1eqg = ovine COX-1 complexed with Ibuprofen

6

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

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

A Primitive Cubic Lattice (CsCl)

9

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

A Unit Cell

11

a, b, c, , , a, 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

12

A Brillouin Zone

Kittel, Solid State Physics

13

The unit cell is not unique(c:\MyFiles\Clinic\index2.wrl)

14

15

16

17

How do I pick the unit cell?

• Axis system (basis set) is right-handed

• Symmetry defines natural directions

• Angles close to 90°

• Standard settings of space groups

• To make structural similarities clearer

18

The Reduced Cell

• 3 shortest non-coplanar translations

• Main Conditions (shortest vectors)

• Special Conditions (unique)

• May not exhibit the true symmetry

19

The Reduced Form

a·a

A

b·b

B

c·c

C

b·c

D

a·c

E

a·b

F

20

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·b

a·c ½ a·a 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 a·b 2 a·c

if a·c = ½ a·a then a·b 2 b·c

if a·b = ½ a·a then a·c 2 b·c

21

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 = 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

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, 1986.A. D. Mighell and J. R. Rodgers, Acta Cryst., A36, 321-326 (1980).

23

International Tables for Crystallography, Volume F, Figure 2.1.3.3, p.52 (2001)

24

25

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 0 0 0 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

26

A = B, no conditions on C


9 I A/2 A/2 A/2 hR

10 I D D F mC

11 II 0 0 0 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

27

B = C, no conditions on A


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 0 0 0 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

28

No conditions on A, B, CNumber 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 0 0 0 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

† 2|D + E + F| = A + B, plus |2D + F| = B

29

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.

30

The metric symmetry can be higher than the crystallographic symmetry!

(A monoclinic cell can have β = 90°)

31http://www.haverford.edu/physics-astro/songs/bravais.htm

32

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

33

Now consider the symmetry…

34

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!

35

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

36

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.

37

The 32 Point Groups (1)

International Tables for Crystallography, Volume A, Table 12.1.4.2, p.819 (2002)

38

The 32 Point Groups (2)

International Tables for Crystallography, Volume A, Table 12.1.4.2, p.819 (2002)

39

Symbols for Symmetry Elements (1)

International Tables for Crystallography, Volume A, Table 1.4.5, p. 9 (2002)

40



41



42

2 Rotation Axis (ZINJAH)

43

3 Rotation Axis (ZIRNAP)

44

4 Rotation Axis (FOYTAO)

45

6 Rotation Axis (GIKDOT)

46

-1 Inversion Center (ABMQZD)

47

-2 Rotary Inversion Axis?

48

m Mirror Plane (CACVUY)

49

-3 Rotary Inversion Axis (DOXBOH)

50

-4 Rotary Inversion Axis (MEDBUS)

51

-6 Rotary Inversion Axis (NOKDEW)

52

21 Screw Axis (ABEBIS)

53

31 Screw Axis (AMBZPH)

54

32 Screw Axis (CEBYUD)

55

41 Screw Axis (ATYRMA10)

56

42 Screw Axis (HYDTML)

57

43 Screw Axis (PIHCAK)

58

61 Screw Axis (DOTREJ)

59

62 Screw Axis (BHPETS10)

60

63 Screw Axis (NAIACE)

61

64 Screw Axis (TOXQUS)

62

65 Screw Axis (BEHPEJ)

63

c Glide (ABOPOW)

64

n Glide (BOLZIL)

65

d (diamond) Glide (FURHUV)

66

What does all this mean?

67

Symmetry information is tabulated in International Tables for

Crystallography, Volume A edited by Theo Hahn Fifth

Edition 2002

68

Guaifenesin, P212121 (#19)

69

70

© Copyright 1997-1999. Birkbeck College, University of London.

71

Hermann-Mauguin Space Group Symbolsthe 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}

72

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

73

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.

74

75

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)

76

A Bärninghausen Treefor translationengleiche subgroups

International Tables for Crystallography, Volume 1A, p. 396 (2004)

77

Mercury/ETGUAN (P41212 #92)

78

79

80

Not all space groups are possible for protein crystals.

81

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

# Entries

1

10

100

1000

10000

82

Space Group Frequencies

Space Group Number

0 20 40 60 80 100 120 140 160 180 200 220

Frequency of Occurrence, %

0.01

0.1

1

10

100

PDB % CSD % ICSD %

83

Some Classifications of Space Groups

• Enantiomorphic, chiral, or dissymmetric – absence of improper rotations (including , = m, and )

• Polar – two directional senses are geometrically or physically different

84

Basic Diffraction Physics

85

Bragg’s Law2sinndλθ=

1sin2dθλ=

86

Bragg’s Law

V. K. Pecharsky and P. Y. Zavalij, Fundamentals of Powder Diffraction and Structural Characterization of Materials, p. 148 (2003)

87

Optical Diffraction

PSSC Physics, Figure 18-A, p. 202-203 (1965)

88

Optical Diffraction

PSSC Physics, Figure 18-B, p. 202-203 (1965)

89

Optical Diffraction

D. Halliday and R. Resnick, Physics, p. 1124 (1962)

90

Oscillation direction of the electron

X-ray beam

Electric vector ofThe incident beam

Electron

Scattering by One Electron

91

Scattering by One Electron

• Inelastic ( = Compton scattering = a component of the background) and elastic

• Phase difference between incident and scattered beams is π

• The scattered energy (intensity) is:2220221sineleIIrmcϕ⎛⎞=⎜⎟⎝⎠

92

Electromagnetic Waves

(0;)cos2zEtzAπλ==

E λ

A

A

z

93

During a time t the wave travels over a distance tc = tλν

Therefore at time t, the field strength at position z is what it was at t = 0 and

position z – tλν:1(,)cos2()cos2cos2EtzAztzzAtAtcπλνλπνπνλ=−⎛⎞⎛⎞=−=−⎜⎟⎜⎟⎝⎠⎝⎠

94

(,0)cos2cosEtAtAtπνω==

95

Consider a new wave displaced by a distance Z from the original wave:

Z corresponds to a phase shift2π(Z/λ) = α

E Z’

z

new wave

original wave

96

(,0)cos(,0)cos()orignewEtAtEtAtωω==+

97

cos()coscossinsincoscossincos(90)AtAtAtAtAtωωωωω+=−=++°

cos()cossiniAtAiAAeω+=+=

imaginary axis

real axisA

Acos

Asin

98

Scattering by Two Electrons

1

2

pq

r 2

1 s

s0

99

Scattering by Two Electrons

• Let the magnitudes of s0 and s = 1/λ

• Diffracted beams 1 and 2 have the same magnitude, but differ in phase because of the path difference p + q

100

00sincos(90)1cos(90)()prrrqpqθθλθλλλλ==−=−=⋅=−⋅+=⋅−rsrsrss

101

The phase difference of wave 2 with respect to wave 1 is:002()2πλπλ⋅−−=⋅=−rssrSSss

s0 s0

s S

θ

“reϕλectiν πλνe”

102

2sin2sinθθλ==Ss

103

Scattering by an Atom

r(r)

ρ(-r)

-r

+r

nucleus

104

The Atomic Scattering Factor[2][2][2]()()2()cos[2]iiifedeeddπππrrrπ⋅⋅−⋅=⎡⎤=+⎣⎦=⋅∫∫∫rSrrSrSrrrrrrrrSr

105

Atomic Scattering Factors

V. K. Pecharsky and P. Y. Zavalij, Fundamentals of Powder Diffraction and Structural Characterization of Materials, p. 213 (2003)

106

Scattering from a Row of Atoms

M. J. Buerger, X-ray Crystallography, Fig. 14B, p. 32 (1942)

107

Scattering from a Row of Atomscoscoscoscos(coscos)coscosOQPRmOQaPRaaamammaλνμνμλνμλλνμ−===−=−==+

108

Scattering from a Row of Atoms

M. J. Buerger, X-ray Crystallography, Fig. 15, p. 33 (1942)

109

Scattering by a Plane of Atoms

M. J. Buerger, X-ray Crystallography, Fig. 16, p. 34 (1942)

110

Scattering by a Unit Cell

J. Drenth, Principles of Protein X-ray Crystallography, Fig. 4.12 p. 80 (1999)

111

Scattering by a Unit Cell221()jjijjnijjfefeππ⋅⋅===∑rSrSfFS

112

Scattering by a Crystal

The scattering of this unit cellwith O as the origin is F(S)

The scattering of this unit cellwith O as the origin is:

F(S)exp[2πitaS]exπ[2πiubS]exπ[2πivcS]

O a

c

a

c

ta+ub+vc

113

For a unit cell with its own origin at ta + ub + vc

the scattering is

and the total scattering by the crystal is

222()itiuiveeeπππ⋅⋅⋅×××aSbScSFS

312222000()()nnnitiuivtuveeeπππ⋅⋅⋅====×××∑∑∑aSbScSKSFS

114

Scattering by a Crystal

2π .a S =0t

=1t

=2t=3t=4t

=5t

=6t

=7t

=8t

115

A crystal does not scatter X-rays unlesshkl⋅=⋅=⋅=aSbScS

These are the Laue conditions

116

Now remember111hkl⋅=⋅=⋅=aSbScS

s0 s0

s S

θ

“ ”reflecting plane

117

The “reflecting planes” are lattice planes

reflecting planesdirection of Salong this line

a

b

a/h

b/kd

118

Consider just one direction

1proj on 12sin2sinor, since (or ) can be any integer2sindhhdddhndλθλθλθ==⋅=====aaSSS

119

The Reciprocal Lattice

The idea of a reciprocal lattice predates crystallography. It was invented by J. W. Gibbs in the late 1880s, and its utility for

describing diffraction data was realized by P. P. Ewald in 1921.

120

The Reciprocal Lattice

For any lattice with basis vectors a, b, and c, construct another lattice with basis vectors a*, b*, and c* such

thata·a* = b·b* = c·c* = 1 and

a·b* = a·c* = b·a* = b·c* = c·a* = c·b* = 0Therefore,

a* = K(b×c) and K = 1/[a·(b×c)]K is 1/V if a, b, and c form a right-handed system.

121

Why do we care?

122

Remember the Laue conditions:hkl⋅=⋅=⋅=aSbScS

S a reflecting plane = a lattice plane. The

equation of such a plane through the

origin ishx + ky + lz = 0

123

The reflecting plane contains general vectors and lattice vectors:

r = xa + yb + zcrL = ua + vb + wc

u, v, and w integers

124

S is perpendicular to any vector in the plane, or

S·(r – rL) = 0S·r = S· rL

S· rL = n(the planes don’t have to pass through the origin)

125

r = ua + vb + wc

S·a = hS·b = kS·c = l

126

Consider the possibility of a different basis set for S:

S = UA + VB + WCr = ua + vb + wc

127

2coscos1cos1cos1coscoscoscos1cos1cos1coscos11coscoscos1coscoscos1hklhhhaaahklkkkbbbabclllcccd++=

Triclinic

128

(UA + VB + WC)·(ua + vb + wc) = n

()()()uUVWvnwuUVWvnw⎛⎞⎛⎞⎜⎟⎜⎟=⋅ ⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⋅⋅⋅⎛⎞⎛⎞⎜⎟⎜⎟ =⋅⋅⋅ ⎜⎟⎜⎟⎜⎟⎜⎟⋅⋅⋅⎝⎠⎝⎠ABabcCAaAbAcBaBbBcCaCbCc

129

Since U, V, and W are general integers, the ends of S are points on a lattice

reciprocal to the direct lattice.

real unit cell

reciprocal unit cell

b

a

O

b*

a*

01

1011

130

The Ewald Construction

P

OMs0

S

2θ

s

reciprocal lattice

1/λ

131

Mosaicity (Mosaic Spread)

B. D. Cullity and S. R. Stock, Elements of X-ray Diffraction, p. 175 (2001).

132

Intensity of a Diffraction Spot

2322int022()()creIhklVILpTFhklVmcλω⎛⎞=⎜⎟⎝⎠

133

The Lorentz Factor

• With an angular speed of rotation ω a r.l.p. at a distance 1/d from the origin moves with a linear speed v = (1/d)ω

• For passage through the Ewald sphere, we need the component v = (1/d)ωcosθ = (ωsin2θ)/λ

• The time to pass through the surface is proportional to

1sin2λωθ⎛⎞⎜⎟⎝⎠

134

The Polarization Factor

• P = sin2φ where φ is the angle between the polarization direction of the incident beam and the scattering direction

• φ = 90 - 2θ

• For an unpolarized incident beam, P = (1 + cos22θ)/2

• Synchrotron radiation is polarized, so check with your beamline staff!

135

The Polarization Factor

beam

θ

90° - 2θp

psin2θ

136

Transmission (Absorption)()01diiiTAIeIXμμμρ ρ−=−==∑

137

(Extinction)

138

Calculation of Electron Density312222000()()nnnitiuivtuvKFeeeπππ⋅⋅⋅====×××∑∑∑aSbScSSS

A more accurate expression is2()()icrrealcrystalKedvπr⋅=∫rSSr

This operation is a Fourier transformation

139

Calculation of Electron Density222()()()1()()()1()()icrreciprocaliihxkylzhklWedveVxyzxyzhxkylzxyzhkleVπππrrr−⋅−⋅−++=⎛⎞=⎜⎟⎝⎠⋅=++⋅=⋅+⋅+⋅=++⎛⎞=⎜⎟⎝⎠∫∑∑∑∑rSSrhhrSrFhrSabcSaSbScSF

140

Calculation of Electron Density1112()000()[2()()]1/2int()()()()1()()()()ihxkylzxyzihklihxkylzihklhklhklVxyzedxdydzhklFhklexyzFhkleVIhklFhklLpTππrr++===−+++===⎡⎤=⎢⎥⎣⎦∫∫∫∑∑∑FF

141

Anomalous (Resonant) Scattering

()()()normally ()()and ()IhklIhklFhklFhklhklhklαα ===−

142


http://physics.nist.gov/PhysRefData/XrayMassCoef/cover.html

143


Near an absorption edge

fanom = f + Δf + f”

f

fanomalous f”

fΔf

144


FPH(+)

FP(+)

FH(+) without anomalous scattering

FH(+) with anomalous scattering

FP(-)

FPH(-)

FH(-) without anomalous scattering

FH(-) with anomalous scattering

145

How do I determine f’ and f”?

• Calculate them by programs such as FPRIME

• Measure the NEXAFS/XAFS, and calculate them using the (classical or quantum) Kramers-Kroning transform220222"22'"(',0)'''aedgmcddgfdffPdκκκωπμeωπωωωωωπωω∞⎡⎤=⎢⎥⎣⎦⎡⎤=⎢⎥⎣⎦=−∑∫

146

Symmetry in the Diffraction Pattern()2(***)()or in matrix notationin this notation, the structure factor is()()TTirealcellhxkylzhklxyzxhklyzFedvπρ ⋅++=++++=⋅⎛⎞⎜⎟=⋅⎜⎟⎜⎟⎝⎠=∫hrabcabchrhrhr

147

A symmetry operation can be represented by a combination of a rotation/inversion/reflection and a translation. The rotation… can be represented by a matrix R and the

translation by a vector t. By symmetry, ρ(R·r+t) = ρ(r).

148

F(h) can also be written:2()2222()()()()()()()TTTTTTirealcelliirealcellTTTiirealcellFedveedvFeedvπππππrrr⋅⋅+⋅⋅⋅⋅⋅⋅=⋅+=⋅=⋅=∫∫∫hRrththRrhtRhrhRrtrhRRhhr

149

The integral is just F(RT·h), so

2()()()()2()()TiTTTTFeFIIππ⋅=⋅=⋅+⋅=⋅hthRhhRhhthRh

150

For a 21 axis along b22()1000010 and 1/20001therefore()()[]TTiirealhalfthecellFeedvππr⋅⋅+⎛⎞⎛⎞⎜⎟⎜⎟==⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠=+∫hrhRrtRthr

151

For the (0k0) reflections, h = kb*, sohT·r = hT·R·r = 0 + ky + 0

and hT·t = k/2This simplifies:

2(00)[1]()ikikyrealhalfthecellFkeedvππr=+∫r

If k is odd, F(0k0) = 0

152

For C-centering()21/21/2, so /2/20and()[1]()TTihkirealhalfthecellhkFeedvππr+⋅⎛⎞⎜⎟=⋅=+⎜⎟⎜⎟⎝⎠=+∫hrththr

153

The Patterson Function

222Let ()()1()()cos(2)or equivalently (no anomalous dispersion)1()()iPuvwPPFVPFeVππ⋅=⎛⎞=⋅⎜⎟⎝⎠⎛⎞=⎜⎟⎝⎠∑∑hhuhuuhhuuh

154

What does the Patterson function mean?

Consider an alternate expression:()()()realPdvrr=+∫rurru

155

More about Structure Factors2111()For noncentrosymmetric structures, it is useful:()cos(2)sin(2)()()jnijjnnjjjjjjfefifAiBπππ⋅=====⋅+⋅=+∑∑∑rSFSFSrSrSSS

156

|F(S)| decreases with increasing |S|

• The falloff in f (greater interference)

• Static/dynamic disorder (thermal motion)

220(sin)/22where 8BiiffeBuθλπ−==

157

The falloff makes statistical analysis awkward. Intensities are normally measured on an arbitrary

scale. For large numbers of well-distributed S22222sin/202()(,)but()iiiBiiFIabsfffeθλ−===∑SS

158

The Wilson Plot222sin/020222()(,)()ln[()/()]ln2sin/BiiiiIKIabsKefIfKBθλθλ−===−∑∑SSS

http://www.ysbl.york.ac.uk/~mgwt/CCP4/EJD/bms/bms10.html

159

Normalized Structure Factors

221/221/2sin/02()()/()()jjBjjEFfFefθλ⎛⎞=⎜⎟⎝⎠⎡⎤=⎢⎥⎣⎦∑∑SSS

Documents

1 Lattices and Symmetry Scattering and Diffraction (Physics) James A. Kaduk Innovene USA LLC Naperville IL 60566 [email protected]