Chapter 13 Diffraction

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Chapter 13 Diffraction

Reading Assignment:

1. D. Sherwood, Crystals, X-rays, and

Proteins-chapter 9

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Contents

Electron Density Function Electron Density Function

Atomic Scattering Factor Atomic Scattering Factor

Structure Factor Structure Factor 3

1 2

4

Scattering of X-ray by electron Scattering of X-ray by electron

5 Systematic Absences Systematic Absences

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Contents of the Unit Cell

- diffraction pattern of a crystal- in terms of reciprocal lattice

( ) ( ) [ ( ) * ( )]

- motif (contents of the unit cell) envolop that of the finite cryst

Tf crystal = Tf motif i Tf infinite lattice Tf shape function al lattice

intensity determined by the diffraction pattern of motif - intensities of all the diffraction maxima

inverse Fourier transform

parameters pertaining molecules within unit cell scatt

→

→ ering of the incident X-rays by electrons → distribution of electron within the unit cell

(4)

Scattering of X-ray by a Single Electron

- interaction of X-rays with an electron

Thomson scattering vs. Compton scattering - Thomson scattering (coherent scattering) assumption- free electron, passive response incident X-ray with electr

2 2

2 0

ic field

accelerate electron of charge and mass

act as a source of radiation, scattered wave 1 cos 2

, well defined phase relationship

4 2

in

scat scat

in

E

e m

a e E m

E

E e

E rmc

θ πε

→

=

→

= +

(5)

Scattering of X-ray by a Single Electron

- Compton scattering

a definite set of waves with wavelength rather longer than that of the incident wave

collision between an electron and a photon particle-particle collision

no well def

No discrete diffraction effects result from Com ined phase relationship

incoherent scattering

(overall backgrou

pton nd sc

Scatt atter

ering ing)

(6)

Scattering of X-ray by a Distribution of Electrons

2 2

2 0

- X-ray diffraction by Thomson scattering 1 cos 2

(2 )

4 2

: electronic scattring factor, (2 ): polarization factor

scat

e in

e

E e

rmc f p E

f p

θ θ

πε

θ

= + =

D. Sherwood, Crystals, X-rays, and Proteins

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- ( )

( )

- principle of superposition

( ) ( ) ( )

( )

- from an array of electrons

( )

ⁿ F( k) ( )

scat A e in

i

scat B e in

scat total scat A scat B

scat total i

e e

in

scat total

e in n

i i k

E f E

E f E e

E E E

E f f e

E

E f e f r e

E

φ

φ Δ

=

⇔ +

= +

=

∑

Δ =

all r

rd r

∫

ⁱ

(8)

- electron density function ( )

any element of volume, centered on the point , which represented by

average number of electrons within the volume is given by ( )

scattered wa

r

r d r

d r r d r

ρ

ve amplitude ( )

- F( k) ( )

scat e

e all r

i k r

dE f r d r f r e d r

ρ

ρ ^Δ

=

Δ =

∫

ⁱ

continuous distribution

(9)

Diffraction Pattern of the Motif

- within a single unit cell

electron density function ( ) position of atoms from the electron density maxima bond lengths, bond angles

- ( ) _e ( ) ( ) _e ( )

all r

i k r i k

r

F k f r e d r F k f r e

ρ

ρ ^Δ ρ ^Δ

→

Δ =

∫

ⁱ ⇒ Δ =

- ( ) / : relative scattering ability of the contents of the unit cell compared to a single electron

- ( ) ( )

unit cell e

rel

unit cell

r

i k r

d r

F k f

F k ρ r e ^Δ d r

Δ

Δ =

∫

i

(10)

Diffraction Pattern of the Motif

- triclinic system, a a, b b, c c

- ( ) ( )

- any point within the unit cell

a b c 0 , 0 , 0

/ , / , /

0 1, 0

rel

unit cell

i k r

a b c

F k r e d r

r X Y Z X a Y b Z c

x X a y Y b z Z c

x y

ρ ^Δ

= = =

Δ =

= + + ≤ ≤ ≤ ≤ ≤ ≤

= = =

≤ ≤ ≤ ≤

∫

ⁱ

1, 0 1

a b c

- volume element

z

r x y z

d r dXdYdZ a b c dxdydza b c

Vdxdydz

≤ ≤

⇒ = + +

= ×

=

i i

(11)

1 1 1

)

0 0 0

( a b c

- amplitude of the diffraction pattern of the contents of the unit cell

( ) ( , , )

envolop the diffraction pattern of the crystal lat

x y z

rel

x y z

i k x y z

F k V ⁼ ⁼ ⁼ ρ x y z e dxdydz

= = =

Δ + +

Δ =

⇒

∫ ∫ ∫

ⁱ

tice restrict the values of the scattering vector to those which correspond crystal lattice diffraction maxima - 2

2 ( * * *) ( )

2 (

hkl

k

k G

k r ha kb lc xa yb zc

hx π

π π

⇒ Δ

Δ =

Δ = + + + +

= +

i i

) ky lz+

Diffraction Pattern of the Motif

(12)

1 1 1

0 0 0

2 ( )

- for a single particular maximum corresponding to the reciprocal lattice point

[ ( )] ( , , )

amplitude of the diffraction pattern at the

x y z

rel hkl

x y z

i hx ky lz

hkl

F k V ⁼ ⁼ ⁼ ρ x y z e ^π dxdydz

= = =

Δ =

∫ ∫ ∫

+ +

1 1 1

0 0 0

2

2 ( )

point indexed as - structure factor

( , , )

- in general , complex quantity

-

hkl

x y z

hkl

x y z

hkl hkl

i hx ky lz

i

hkl

F V x y z e dxdydz

F F e

I

F

π

α

ρ

= = =

= + +

=

∫ ∫ ∫

Diffraction Pattern of the Motif

(13)

Calculation of the Electron Density Function

1 1 1

0 0 0

2 ( )

( )

2 ( )

- ( ) ( , , )

- ( , , ) 1 ( ) ( )

- ( , , ) 1 ( )

- diffraction pattern- discr

x y z

rel

x y z

rel all k

hkl all k

i hx ky lz

i k xa yb z c

i hx ky lz

F k V x y z e dxdydz

x y z F k e d k

V

x y z F e d k

V

π

ρ ρ

ρ

= = =

Δ

+ +

− Δ + +

− + +

Δ =

= Δ Δ

= Δ

∫ ∫ ∫

∫

i

2 ( )

ete, not continuous

( , , ) 1 _hkl

h k l

i hx ky lz

x y z F e

V

ρ =

∑∑∑

⁻ π ^{+ +}

(14)

Fourier Synthesis

electron density map of protein myoglobin

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Electron Density Projections

1

0 0

1

0

2 ( )

2

- determination of relatively small molecules- 2D function ( , )

- ( , ) ( , , ) ( , , )

( , , ) 1

( , ) 1

c

hkl

h k l

hkl

h k l

hkl

i hx ky lz

i

x y x y x y z dZ c x y z dz

x y z F e

V

x y c F e dz

V A F e

π

ρ

ρ ρ ρ

ρ ρ

− + +

−

= =

=

∫ ∫

∑∑∑

∫ ∑∑∑

1 1

0 0

0

( ) 2 2

2 ( )

( )

1

h k l

hk

h k

hx ky ilz ilz

i hx ky

e dz e dz l

A F e

π π

π

+ − − δ

− +

=

∑∑∑ ∫ ∫

∑∑

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Electron Density Projections

nickel phthalocyanine

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Calculation of Structure Factor

1 1 1

0 0 0

2 ( )

- ( , , ) - ( , , ) 1

- only infer atomic position from the maxima in the electron density function

- alternatively,

x y z

hkl

x y z

hkl

h k l

i hx ky lz

F V x y z e dxdydz

x y z F e

V

r xa y

π

ρ ρ

= = =

+ +

− + +

=

= +

∫ ∫ ∫

∑∑∑

( )

2

-

hkl

unit cell

j j

iG r

b zc

F r e d r

r r R

ρ

π

+

=

= +

∫

ⁱ

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atomic electron density function ρj(r − r ^j)

( ) j( j)

j

r r r

ρ =

∑

ρ −

Calculation of Structure Factor

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Calculation of Structure Factor

2

)

2

2 (

2

- ( )

, positions of nuclei are constant

- ( )

( )

j j

j

hkl

hkl j j

unit cell j

j j

j

j R j

hkl j

unit cell j

j j j

iG r

F r r e d r

r r R

d r d R

F R e d R

R e e

^π

π

ρ

ρ ρ

+

= −

= +

=

∫ ∑

∑

i

2 2

^j ( )

i Ghkl R j

h i Ghk R j

l l

k

j uni

j j

j atom t cell

j

iG r

d R

e

^π

ρ

R e ^π d R

=

∫

∑ ∫

i

i i

(20)

Calculation of Structure Factor

2

2 (

- atomic scattering factor ( )

-

- th atom

( * * *) ( ) ( )

-

j

i Ghkl R j

hkl

j j

atom

hkl j

j

j j j j

hkl j j j j

hk

j j

j l

j

iG r

i

f R e d R

F f e

j r x a y b z c

G r ha kb lc x a y b

F f e

z c hx ky lz

π

π π

ρ

=

= + +

=

= + + + + = + +

∫

∑

i

i i

1 1 1

0 0 0

)

2 ( )

- ( , , )

j j j

x y z

hkl

x z

j

y

i h hx ky lz

x ky lz

F V

⁼ ⁼ ⁼

ρ x y z e

^π

dxdydz

= = =

+

+ + +

=

∫

∑

∫ ∫

(21)

Atomic Scattering Factor

2

2 0

2

- ( )

scattering ability of an atom as compared to that of a single electron ( )

- spherically symmetric

sin(4 sin )

2 ( )

(2 sin ) 4 ( ) sin

i Ghkl R j

j j

atom

e

j j

j

f R e d R

f

R

f R R dR

R R R s

ρ π

π θ

π ρ πλ θ

λ π ρ

∞

=

∫

i

0

4 sin s=

RdR sR

π θ λ

∞

∫

(22)

B. D. Cullity, Elements of X-ray Diffraction

(23)

Crystal Symmetry and X-ray Diffraction

)

2 (

2 ( 2 (

- symmetry of crystal symmetry of diffraction pattern 230 space group

-

- ex) mirror parallel to plane ( , , ) ( , , - )

( ) [

^A ^A

j j j

A A A

hkl j

j

A A A A A A

hkl A A

i hx ky lz

i hx ky lz i hx ky

F f e

ab

x y z x y z

F f e e

π

π π

+ +

+ + + −

↔

=

= +

∑

)

) )

2 ( 2 (

]

( ) [ ]

- mirror plane, rotation axis

A A

A

A A A A

hkl hk l

A A

hk l

lz

i hx ky lz i hx ky lz

F f e e

I I

π + − π + +

= +

=

(24)

Friedel’s Law

)

*

* 2

2 (

-

- is real, (complex conjugate)

- magnitude of the structure factor of centrosymmetr

j j j

hkl j

j hkl j

j

j hkl hkl

hkl hkl hkl hkl hkl

i hx ky lz

F f e

f F F

F F I I F

π π

+ +

− + +

=

= = =

∑

ically related reciprocal lattice points are equal

- intensity of the diffraction maxima corresponding to

centrosymmetrically related reciprocal lattice points are equal (inversion center)

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- Friedel's law was proved on a perfectly general basis and its significance is that the X-ray diffraction picture in

general are centrosymmetric whether or not the crystal has an inversion center

- observation of the symmetry of the diffraction pattern in general allows to identify mirror planes and proper rotation axes, but inversion centers and improper rotation axes.

(26)

Systematic Absence

1 1 1 2 2 2)

) )

)

1 1 1

2 (

2 ( 0 0 0 2 2 2

(

- body centered cubic (000,

[1 ] 2 2

0 2 1 - systematic absenc

hkl j j

j j

i h k l i h k l

i h k l

F f e f e

f e f if h k l n

if h k l n

π π π

+ + + +

+ +

= +

= + = + + =

+ + = +

1 2

) )

)

2 [ ( 1

2 ( 2

2 ( 4

e or extinction - a glide ( (001))

,

[1 ]

0 0, 2 1 0

hkl j j

j

hkl

i h x ky lz i hx ky lz

i hx ky lz i h ilz

xyz x y z

F f e f e

f e e e

l hk h n F

π π

π π π

+ + − + +

+ + −

+

= +

= ⇒ = + ⇒ =

(27)

Reciprocal lattice

- cubic I a = =b c

90^o

α β γ

= = =

C. Hammond, The Basics of Crystallography and Diffraction

(28)

Systematic Absence

1 1 1 1 1 1

, ,

2 2 2 2 2 2

) )

)

) ) )

1 1 1 1

2 ( 0 2 ( 0

2 ( 0 0 0 2 2 2 2

1 1

2 ( 0

2 2

( ( (

- face centered cubic (000, 0 0 0)

[1 ]

4 ,

hkl j j j

j

j j

i h k l i h k l

i h k l

i k l i h l i h k

F f e f e f e

f e

f e e e

f if h k

π π

π π π

+ + + +

+ +

+ + +

= + +

+

= + + +

= , unmixed

0 , , mixed

100 110 111 200 210 220 300(221) 310 311 x x o o x o x x o

l

if h k l

(29)

(30)

Systematic Absence

1 1 1 1 1 1 1 1 1 1 3 3 3 1 3 3 3 1

, , , , ,

2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4

) ) )

)

) )

1 1 1 1 1 1

2 ( 0 2 ( 0 2 ( 0

2 ( 0 0 0 2 2 2 2 2 2

1 1 1 1 3 3

2 ( 2 (

4 4 4 4 4 4

- diamond structure (000,0 0 0, )

hkl j j j j

j j

i h k l i h k l i h k l

i h k l

i h k l i h k l

F f e f e f e f e

f e f e

π π π

π

π π

+ + + + + +

+ +

+ + + +

= + + +

+ + ⁾ ⁾

) ) ) ) / 2

3 1 3 3 3 1

2 ( 2 (

4 4 4 4 4 4

( ( ( (

[1 ][1 ]

, , mixed, 0

, , unmixed 4 (1 i) , , are odd

j j

j

hkl

hkl j

i h k l i h k l

i k l i h l i h k i h k l

f e f e

f e e e e

if h k l F

if h k l F f if h k l

π π

π π π π

+ + + +

+ + + + +

+ +

= + + + +

=

= ±

8 , , are even multiple of 2 0 , , are odd multiple of 2

100 110 111 200 210 220 300(221) 310 311 222 x x

hkl j

hkl

F f if h k l

F if h k l

=

o x x oo x x o x

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2000 TMS Fall Meeting: Special Tutorial

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Determination of Crystal Symmetry

- 230 space group

- M.J. Buerger: 122 different diffraction pattern types

- what symmetry information can be derived from the symmetry of an X-ray photograph?

- systematic absence lattice type

→

- 11 out of 32 point group-centrosymmetric (Laue group)

- as many as necessary for the systematic absence (glide, screw) (diffraction symbol)

- 58 uniquely define a single space group 64 ambiguity as to space group

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Determination of Crystal Symmetry

- of all the symmetry operations, the only one about which we can derive no information is the presence or absence of an inversion center

(i) breakdown of Friedel's law (ii) piezoelectric effect

(iii) pyroelectric effect (iv) optical activity