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$: Scattering and Diffractionagni.phys.iit.edu/~howard/ACASchool/lectures07/... · International Tables for Crystallography, Volume D, p. 438. 6. PDB entry 1eqg = ovine COX-1 complexed$
1

Lattices and Symmetry Scattering and Diffraction (Physics)

James A. KadukINEOS Technologies

Analytical Science Research Services

Naperville IL [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

Alastor

“Mad-Eye” Moody –

“Constant Vigilance”

Harry Potter and the Goblet of Fire

(2005)

4

The crystallographer’s world view

Reality can be more complex!

5

Twinning at the atomic level

International Tables for Crystallography, Volume D, p. 438

6

PDB entry 1eqg = ovine COX-1 complexed

with Ibuprofen

7

Atoms pack together in a regular pattern to form a crystal. There are two aspects to this pattern:

PeriodicitySymmetry

First consider the periodicity…

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).

9

A Primitive Cubic Lattice (CsCl)

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.

11

A Unit Cell

12

a, b, c, α, β, γ a, b, c

x1

a + x2

b + x3

c, 0 ≤

xn

< 1 lattice points = ha + kb + lc,

hkl

integers domain of influence

–

Dirichlet

domain, Voronoi

domain, Wigner-Seitz cell, Brillouin

zone

Descriptions of the Unit Cell

13

A Brillouin

Zone

C. Kittel, Introduction to Solid State Physics, 6th

Edition, p. 41 (1986)

14

Bilbao

Crystallographic Server

http://www.cryst.ehu.es/

15

The unit cell is not unique (c:\MyFiles\ACA\

ACA_big_07\index2.wrl)

16

17

18

19

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

20

The Reduced Cell

•

3 shortest non-coplanar translations•

Main Conditions (shortest vectors)

•

Special Conditions (unique)

•

May not exhibit the true symmetry

21

The Reduced Form

a·aA

b·bB

c·cC

b·cD

a·cE

a·bF

22

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·aSpecial 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

23

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 |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 = 0if ( |b·c| + |a·c| + |a·b| ) = ½

( a·a + b·b ) then a·a ≤

2 |a·c| + |a·b|

24

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).

25

“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), 447-452 (2003).

26International Tables for Crystallography, Volume F, Figure 2.1.3.3, p.52 (2001)

27

28

“The mystery of the fifteenth Bravais

lattice”, A. Nussbaum,

Amer. J. Phys.,

68(10), 950-954 (2000).

http://ojps.aip.org/ajp/

29

Symmetry Groups and Their Applications, W. Miller, Jr., Academic

Press, New York (1972), Chapter 2.

1 2 / 4 / 3

31 6 /

m mmm mmm m

m mmm

⊂ ⊂ ⊂ ⊂∩ ∩

⊂

30

The 44 Reduced Forms

31

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

32

A

= B, no conditions on CNumber Type D E F Bravais

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

33

B

= C, no conditions on ANumber 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 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

34

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

35

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.

36

The metric symmetry

can be higher than the crystallographic symmetry!

(A monoclinic cell can have β

= 90°)

37

Centered Cells conventional crystallographic basis/cell

•

Point group symmetry of the lattice (holohedry)-1, 2/m, mmm, 4/mmm, -3m, 6/mmm, m-3m

•

Basis vectors (and sides) contain symmetry elements

•

2, 3, or 4 lattice points / unit cell•

P, A, B, C, R, I, F

38

39http://www.haverford.edu/physics-astro/songs/bravais.htm

40

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 faces

hkl

indices of a Bragg reflection

41

Now consider the symmetry…

42

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!

43

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

44

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.

45

The 32 Point Groups (1)

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

46

The 32 Point Groups (2)

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

47

Symbols for Symmetry Elements (1)

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

48



49



50

2 Rotation Axis (ZINJAH)

51

3 Rotation Axis (ZIRNAP)

52

4 Rotation Axis (FOYTAO)

53

6 Rotation Axis (GIKDOT)

54

-1 Inversion Center (ABMQZD)

55

-2 Rotary Inversion Axis?

56

m Mirror Plane (CACVUY)

57

-3 Rotary Inversion Axis (DOXBOH)

58

-4 Rotary Inversion Axis (MEDBUS)

59

-6 Rotary Inversion Axis (NOKDEW)

60

21 Screw Axis (ABEBIS)

61

31 Screw Axis (AMBZPH)

62

32 Screw Axis (CEBYUD)

63

41 Screw Axis (ATYRMA10)

64

42 Screw Axis (HYDTML)

65

43 Screw Axis (PIHCAK)

66

61 Screw Axis (DOTREJ)

67

62 Screw Axis (BHPETS10)

68

63 Screw Axis (NAIACE)

69

64 Screw Axis (TOXQUS)

70

65 Screw Axis (BEHPEJ)

71

c Glide (ABOPOW)

72

n Glide (BOLZIL)

73

d (diamond) Glide (FURHUV)

74

What does all this mean?

75

Symmetry information is tabulated in International Tables for

Crystallography, Volume A edited by Theo Hahn

Fifth Edition 2002

76

Guaifenesin, P21

21

21

(#19)

77

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

http://img.chem.ucl.ac.uk/sgp/mainmenu.htm

79

http://www.aps.anl.gov/Xray_Science_ Division/Powder_Diffraction_

Crystallography/SymmetryRietveld.html

80

Hermann-Mauguin

Space Group Symbols the centering, and then a set of characters indicating the

symmetry elements along the symmetry directions

Lattice Primary Secondary TertiaryTriclinic 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}

81

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

82

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.

83

84

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)

85

A Bärninghausen

Tree for

translationengleiche

subgroups

International Tables for Crystallography,

Volume 1A, p. 396 (2004)

86

Mercury/ETGUAN (P41

21

2

#92)

87

88

89

Not all space groups are possible for protein crystals.

90

Space Group Frequencies in theProtein Data Bank, 17 June 2003

Space Group Number0 20 40 60 80 100 120 140 160 180 200 220

# En

trie

s

1

10

100

1000

10000

91

Space Group Frequencies

Space Group Number0 20 40 60 80 100 120 140 160 180 200 220

Freq

uenc

y of

Occ

urre

nce,

%

0.01

0.1

1

10

100

PDB % CSD % ICSD %

92

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

1̄ 2̄ 4̄

93

Basic Diffraction Physics

94

Bragg’s Law

2 sinn dλ θ=

1 sin2d

θλ

=

95

Bragg’s Law

V. K. Pecharsky

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

96

Optical Diffraction

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

97

Optical Diffraction

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

98

Optical Diffraction

D. Halliday

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

99

Oscillation directionof the electron

X-ray beam

Electric vector ofThe incident beam

Electron

ϕ

Scattering by One Electron

100

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:22

20 2 2

1 sineleI I

r mcϕ

⎛ ⎞= ⎜ ⎟

⎝ ⎠

101

Electromagnetic Waves

( 0; ) cos 2 zE t z A πλ

= =

E λ

AA

z

102

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( , ) cos 2 ( )

cos 2 cos 2

E t z A z t

z zA t A tc

π λνλ

π ν πνλ

= −

⎛ ⎞ ⎛ ⎞= − = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

103

( ,0) cos 2cos

E t A tA t

πνω

==

104

Consider a new wave displaced by a distance Z

from the original wave:

Z

corresponds to a phase shift 2π(Z/λ) = α

E Z

z

new wave

original wave

105

( ,0) cos

( ,0) cos( )orig

new

E t A t

E t A t

ω

ω α

=

= +

106

cos( ) cos cos sin sincos cos sin cos( 90 )

A t A t A tA t A t

ω α α ω α ωα ω α ω

+ = −= + + °

cos( ) cos sini

A t A iAAe α

ω α α α+ = +

=

imaginary axis

real axisAα

Acosα

Asinα

107

Scattering by Two Electrons

1

2

pq

r2

•1 s

s0 θ

108

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

109

0

0

sincos(90 )

1 cos(90 )

( )

p rr

r

qp q

θθ

λ θλ

λλ

λ

== −

= −

= ⋅= − ⋅

+ = ⋅ −

r sr s

r s s

110

The phase difference of wave 2 with respect to wave 1 is:

0

0

2 ( ) 2πλ πλ⋅ −

− = ⋅

= −

r s s r S

S s ss0 s0

s S

θ

“reflecting plane”

111

2 sin2sin

θθ

λ

=

=

S s

112

Scattering by an Atomρ(r)

ρ(-r)

-r

+r

nucleus

113

The Atomic Scattering Factor

[2 ]

[2 ] [ 2 ]

( )

( )

2 ( ) cos[2 ]

i

i i

f e d

e e d

d

π

π π

ρ

ρ

ρ π

⋅

⋅ − ⋅

=

⎡ ⎤= +⎣ ⎦

= ⋅

∫

∫

∫

r S

r

r S r S

r

r

r r

r r

r r S r

114

Atomic Scattering Factors

V. K. Pecharsky

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

115

Scattering from a Row of Atoms

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

116


cos

cos

cos cos(cos cos )

cos cos

OQ PR mOQa

PRa

a a ma m

ma

λ

ν

μ

ν μ λν μ λ

λν μ

− =

=

=

− =− =

= +

117


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

118

Scattering by a Plane of Atoms

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

119

Scattering by a Unit Cell

J. Drenth, Principles of Protein X-ray Crystallography,

Fig. 4.12 p. 80 (1999)

120

Scattering by a Unit Cell2

2

1( )

j

j

ij j

ni

jj

f e

f e

π

π

⋅

⋅

=

=

=∑

r S

r S

f

F S

121

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[2πita·S]exp[2πiub·S]exp[2πivc·S]

O a

c

ac

ta+ub+vc

122

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

the scattering is

and the total scattering by the crystal is

2 2 2( ) it iu ive e eπ π π⋅ ⋅ ⋅× × ×a S b S c SF S

31 22 2 2

0 0 0( ) ( )

nn nit iu iv

t u ve e eπ π π⋅ ⋅ ⋅

= = =

= × × ×∑ ∑ ∑a S b S c SK S F S

123

Scattering by a Crystal

2πa.S t=0

t=1

t=2t=3t=4

t=5

t=6

t=7

t=8

124

Ewald

–

oscillators Laue

–

spacing?

125

The Fourier transform is an equation to calculate the frequency, amplitude and

phase of each sine wave needed to make up any given signal x(t):

( ) ( ) 2i ftF f x t e dtπ+∞ −

−∞= ∫

126

Brightness image / Fourier transform

http://cns-alumni.bu.edu/~slehar/fourier/fourier.html

127

Kevin Cowtan’s

Fourier Duck http://www.ysbl.york.ac.uk/~cowtan/fourier/magic.html

128

Kevin Cowtan’s

Fourier Cat http://www.ysbl.york.ac.uk/~cowtan/fourier/magic.html

129

Magnitudes from duck and phases from cat

130

Magnitudes from cat and phases from duck

131

A tail of two cats http://www.ysbl.york.ac.uk/~cowtan/fourier/coeff.html

Magnitudes only

132

A similar structure – tail-less Manx cat

133

Known magnitudes and Manx phases “an |Fo| map”

Despite the fact that the phases contain more structural information about the image than the magnitudes, the missing tail is restored at about half of its original weight. This occurs only when the phases are almost correct. The factor of one half arises because we are making the right correction parallel to the estimated phase, but no correction perpendicular to the phase (and <cos2>=1/2). There is also some noise in the image.

134

“A 2|Fo|-|Fc| map”

135

A crystal does not scatter X-rays unless

hkl

⋅ =⋅ =⋅ =

a Sb Sc S

These are the Laue

conditions

136

Now remember

1

1

1

h

k

l

⋅ =

⋅ =

⋅ =

a S

b S

c S

s0 s0

s S

θ

“reflecting plane”

137

The “reflecting planes” are lattice planes

reflecting planesdirection of Salong this line

a

b

a/h

b/kd

138

Consider just one direction

1proj on ; 1, ,

2sin2 sin

or, since (or ) can be any integer2 sin

d but so dh h

d

dd h

n d

λθ

λ θ

λ θ

= ⋅ = =

=

=

=

a aS SS

139

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.

140

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.

141

Why do we care?

142

Remember the Laue

conditions:

hkl

⋅ =⋅ =⋅ =

a Sb Sc S

S ⊥

a reflecting plane = a lattice plane. The

equation of such a plane through the

origin ishx

+ ky

+ lz

= 0

143

The reflecting plane contains general vectors and lattice vectors:

r = xa + yb + zc

rL = ua + vb + wc u, v, and w

integers

144

S is perpendicular to any vector in the plane, or

S·(r –

rL ) = 0 S·r = S·

rL

S·

rL = n (the planes don’t have to

pass through the origin)

145

r = ua + vb + wc

S·a = h S·b = k S·c = l

146

Consider the possibility of a different basis set for S:

S = UA + VB + WC

r = ua + vb + wc

147

2

cos cos 1 cos 1 cos

1 cos cos cos cos 1

cos 1 cos 1 cos cos11 cos cos

cos 1 coscos cos 1

hkl

h h ha a a

h k lk k kb b ba b c

l l lc c c

d

γ β β γ

α γ α γ

α β β α

γ βγ αβ α

+ +

=

Triclinic

148

(UA + VB + WC)·(ua + vb + wc) = n

( )

( )

( )u

U V W v nw

uU V W v n

w

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⋅ =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⋅ ⋅ ⋅⎛ ⎞⎛ ⎞

⎜ ⎟⎜ ⎟⋅ ⋅ ⋅ =⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⋅ ⋅ ⋅⎝ ⎠⎝ ⎠

AB a b cC

A a A b A cB a B b B cC a C b C c

149

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

150

The Ewald

Construction

P

OMs0

S

2θ

s

reciprocal lattice

1/λ

151

152

Mosaicity

(Mosaic Spread)

B. D. Cullity

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

153

Intensity of a Diffraction Spot

23 22

int 02 2( ) ( )creI hkl V I LpT F hkl

V mcλω

⎛ ⎞= ⎜ ⎟

⎝ ⎠

154

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 1

sin 2λω θ⎛ ⎞⎜ ⎟⎝ ⎠

155

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!

156

The Polarization Factorbeam

θ

90° - 2θp

psin2θ

157

Transmission (Absorption)

( )0

1

d

iii

T AI eI

X

μ

μμ ρ ρ

−

= −

=

= ∑

158

(Extinction)

159

Calculation of Electron Density31 2

2 2 2

0 0 0( ) ( )

nn nit iu iv

t u vK F e e eπ π π⋅ ⋅ ⋅

= = =

= × × ×∑ ∑ ∑a S b S c SS S

A more accurate expression is2( ) ( ) i

cr realcrystal

K e dvπρ ⋅= ∫ r SS r

This operation is a Fourier transformation

160

Calculation of Electron Density2

2

2 ( )

( ) ( )

1( ) ( )

( )

1( ) ( )

icr reciprocal

i

i hx ky lz

h k l

W e dv

eVx y z

x y zhx ky lz

xyz hkl eV

π

π

π

ρ

ρ

ρ

− ⋅

− ⋅

− + +

=

⎛ ⎞= ⎜ ⎟⎝ ⎠

⋅ = + + ⋅= ⋅ + ⋅ + ⋅= + +

⎛ ⎞= ⎜ ⎟⎝ ⎠

∫

∑

∑ ∑ ∑

r S

S

r h

h

r S

r F h

r S a b c Sa S b S c S

F

161

Calculation of Electron Density1 1 1

2 ( )

0 0 0

( )

[ 2 ( ) ( )]

1/ 2

int

( ) ( )

( ) ( )1( ) ( )

( )( )

i hx ky lz

x y z

i hkl

i hx ky lz i hkl

h k l

hkl V xyz e dxdydz

hkl F hkl e

xyz F hkl eV

I hklF hklLpT

π

α

π α

ρ

ρ

+ +

= = =

− + + +

=

=

=

⎡ ⎤= ⎢ ⎥⎣ ⎦

∫ ∫ ∫

∑∑∑

F

F

162

Anomalous (Resonant) Scattering

( )( )( )

normally ( )

( )

and ( )

I hkl I hkl

F hkl F hkl

hkl hklα α

=

=

= −

163


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

164

Anomalous (Resonant) ScatteringNear an absorption edge

fanom

= f

+ Δf

+ f”

f

fanomalous f”

fΔf

165


FPH(+)

FP(+)

FH(+) without anomalous scattering

FH(+) with anomalous scattering

FP(-)

FPH(-)

FH(-) without anomalous scattering

FH(-) with anomalous scattering

166

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

transform

2 2

0

2 2

2

"2

2 ' "( ',0)' ''

ae dg

mc ddgfd

ff P dκ

κ

κ ω

πμε ωπ ω

ω

ω ω ωπ ω ω

∞

⎡ ⎤= ⎢ ⎥⎣ ⎦

⎡ ⎤= ⎢ ⎥⎣ ⎦

=−∑ ∫

167

Symmetry in the Diffraction Pattern

( )

2

( * * *)( )or in matrix notation

in this notation, the structure factor is

( ) ( )T

T

ireal

cell

hx ky lz h k l x y z

xh k l y

z

F e dvπρ ⋅

+ + = + + + + = ⋅

⎛ ⎞⎜ ⎟ = ⋅⎜ ⎟⎜ ⎟⎝ ⎠

= ∫ h r

a b c a b c h r

h r

h r

168

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).

169

F(h) can also be written:2 ( )

2 2

2 2 ( )

( ) ( )

( )

( )

( ) ( )

T

T T

T T T

ireal

cell

i ireal

cell

T T T

i ireal

cell

F e dv

e e dv

F e e dv

π

π π

π π

ρ

ρ

ρ

⋅ ⋅ +

⋅ ⋅ ⋅

⋅ ⋅ ⋅

= ⋅ +

=

⋅ = ⋅

=

∫

∫

∫

h R r t

h t h R r

h t R h r

h R r t

r

h R R h

h r

170

The integral is just F(RT·h), so

2( ) ( )( ) ( ) 2( ) ( )

Ti T

T T

T

F e F

I I

π

α α π

⋅= ⋅

= ⋅ + ⋅

= ⋅

h th R hh R h h th R h

171

For a 21

axis along b

2 2 ( )

1 0 0 00 1 0 and 1/ 2

00 0 1therefore

( ) ( )[ ]T Ti i

realhalfthecell

F e e dvπ πρ ⋅ ⋅ +

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟= =⎜ ⎟ ⎜ ⎟

⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

= +∫ h r h R r t

R t

h r

172

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

+ 0

and hT·t = k/2 This simplifies:

2(0 0) [1 ] ( )ik ikyreal

halfthecell

F k e e dvπ πρ= + ∫ r

If k

is odd, F(0k0) = 0

173

For C-centering

( ) 2

1/ 21/ 2 , so / 2 / 2

0and

( ) [1 ] ( )T

T

i h k ireal

halfthecell

h k

F e e dvπ πρ+ ⋅

⎛ ⎞⎜ ⎟= ⋅ = +⎜ ⎟⎜ ⎟⎝ ⎠

= + ∫ h r

t h t

h r

174

The Patterson Function

2

2 2

Let ( ) ( )1( ) ( ) cos(2 )

or equivalently (no anomalous dispersion)1( ) ( ) i

P uvw P

P FV

P F eV

π

π

⋅

=

⎛ ⎞= ⋅⎜ ⎟⎝ ⎠

⎛ ⎞= ⎜ ⎟⎝ ⎠

∑

∑

h

h u

h

u

u h h u

u h

175

What does the Patterson function mean?

Consider an alternate expression:

( ) ( ) ( ) realP dvρ ρ= +∫r

u r r u

176

More about Structure Factors

2

1

1 1

( )

For noncentrosymmetric structures, it is useful:

( ) cos(2 ) sin(2 )

( ) ( )

jn

ij

j

n n

j j j jj j

f e

f i f

A iB

π

π π

⋅

=

= =

=

= ⋅ + ⋅

= +

∑

∑ ∑

r SF S

F S r S r S

S S

177

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

•

The falloff in f

(greater interference)•

Static/dynamic disorder (thermal motion)

2 20 (sin ) /

2 2where 8

Bi if f e

B u

θ λ

π

−=

=

178

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

scale. For large numbers of well-distributed S

2 2

2 2

2 sin /2 0 2

( ) ( , )

but

( ) i

ii

Bi i

F I abs f

f f e θ λ−

= =

=

∑S S

179

The Wilson Plot2 22 sin / 0 2

0 2 2 2

( ) ( , ) ( )

ln[ ( ) / ( ) ] ln 2 sin /

Bi

i

ii

I K I abs Ke f

I f K B

θ λ

θ λ

−= =

= −

∑

∑

S S

S

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

180

Normalized Structure Factors

2 2

1/ 2

2

1/ 2

sin / 0 2

( ) ( ) /

( ) ( )

jj

Bj

j

E F f

F e fθ λ

⎛ ⎞= ⎜ ⎟

⎝ ⎠

⎡ ⎤= ⎢ ⎥

⎣ ⎦

∑

∑

S S

S

181

Least Squares Refinement

•

Assume observations are uncorrelated•

IfError distributions are GaussianObservations are weighted by 1/σ2

LS gives maximum likelihood estimates of xj

•

Restraints count as observations (weights?)

E. Prince, Mathematical Techniques in Crystallography and Materials Science, Springer-Verlag

(1994)

182

Accuracy vs.

Precision

•

The standard uncertainty (esd) is a measure of precision

•

LS yields minimum variances•

Correlations

•

Systematic errors•

Accuracy?

International Tables for Crystallography, Volume F, Section 18.5

183

The Macromolecular CIF Dictionary (mmCIF)

International Tables for

Crystallography, Volume G Section 4.5

pages 295-443 (!)

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