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Structural Scattering

Jon Goff

Outline of lecture

Topics - Fundamentals of scattering

- Diffraction

- Fourier transforms

- Correlation functions

- Diffuse scattering

Fundamentals

Definition of Bravais Lattice

Infinite array of discrete points with

an arrangement and orientation that

looks exactly the same, from

whichever of the points the array is

viewed

Lattice * Basis → Structure

We wish to consider the behaviour

of plane waves in such a lattice

cnbnanR321

Definition of Reciprocal Lattice

The set of all wave vectors

that yield plane waves with the

periodicity of a given Bravais

lattice

K

R

rKirRKiee

..

1.

RKi

e

Fundamentals

How is that related to diffraction?

0k

kkd ˆˆ.0

d

k

Path difference =

Condition for constructive interference

nkkd ˆˆ.0

nkkd 2.0

For a Bravais lattice

Hence, for diffraction

must be a vector of the

reciprocal lattice

nkkR 2.0

1

.0 Rkki

e

kkQ 0

Q

The scattered wave is spherical since the size of the nucleus is much smaller than the wavelength of the neutron.

Incoming

plane wave Outgoing spherical

wave in phase

r

bee

n

r

ikz1

Scattering from bound nucleus

Diffraction grating

Consider a linear array of

2N+1 nuclei arranged

transverse to the beam.

Their position vectors are

The observation point is a

long way away at

ynarn

ˆ

Nn ,...2,1,0

r

Diffraction grating

Consider a linear array of

2N+1 nuclei arranged

transverse to the beam.

Their position vectors are

The observation point is a

long way away at

ynarn

ˆ

Nn ,...2,1,0

r

2/cossin

2/12cossin

ka

Nka

r

ebr

ikr

sc

Diffraction grating

||||n

rr

cosnar

The scattered wave is

Assuming

N

Nn n

rrik

sc

rr

ebr

n

||

||

nnn

rrrrrr .||

where θ is the angle between the

diffracted beam and the grating

Hence the approximate expression

at large distances is

This Geometric Progression gives

N

Nn

nika

ikr

sce

r

ebr

cos

Diffraction grating

The maximum amplitude

obtained when

is given by

Peak width ~ 1/(2N+1) ~ 1/size

12 Nr

ebr

ikr

sc

2/cossin

2/12cossin

ka

Nka

r

ebr

ikr

sc

nka 2/cos

and the scattering can be

expressed as

Diffraction grating

In the limit N→∞

becomes a delta function, δ(x),

such that

2/cossin

2/12cossin

ka

Nka

r

ebr

ikr

sc

1dxx

0fdxxfx

afdxxfax

bfdxxfbx /0

m

msc

kar

Nbr

2

2

2 coscos122||

Delta functions

Cross sections

How much of the incident beam

is scattered, in total, and in any

particular direction?

Incident flux =

Total scattered flux through a

sphere of radius r

Thus, the total cross section is

m

kn

2

2

4 rm

k

r

bn

24 b

The differential cross section of

neutrons scattered at polar angles θ

and φ within spread dΩ is

Number scattered into dΩ

unit time*incident flux*dΩ

So for the diffraction grating

d

d

m

m

ka

Nb

d

d coscos1222

kMka

Nb

124

2

Three dimensional lattices

First generalise to the simple cubic

lattice. The lattice points are

defined by integer vectors

where

In the far field the scattered wave is

The sum may be written as the

product of three sums over h, k, l

amrm lkhm ,,

m

amQi

ikr

sce

r

ebr

.

From the definition of the δ

function

where we have used a delta

function with a vector argument

m

maQNbd

d

2122

2

haQmaQx

22

kaQy

2

laQz

2

Three dimensional lattices

Thus we find that the allowed

momentum transfers lie on the

reciprocal lattice

The Miller indices for the (h,k,l)

plane allow us to calculate spacing

between planes

The direction of is perpendicular

to the (h,k,l) plane

amGQ

2

For general lattice vectors

, , etc.

222,,

lkh

ad

lkh

lkhlkhdQ

,,,,/2||

Q

cnbnanR321

***

clbkahQ

cbv

a

0

* 2

acv

b

0

* 2

bav

c

0

* 2

cbav .0

0.*

ba2.*

aa

Three dimensional lattices

Or, more simply, each atom reradiates a

spherical wave of amplitude

proportional to the incoming plane wave

At the observation point r amplitude is

Drop the term in t and rearrange

Sum over all atoms in the far field limit

trkiiAe

'.

'.'.'

rrkitrki fi eeA

'..

'rkkirki

fif

eeA

atomsall

r

rkkirkifif

eeA

'

'..

'

O

ki kf

r’ r

Three dimensional lattices

The intensity is given by

For a Bravais lattice R , ,

where d is the displacement within the

unit cell.

Only need to sum over the unit cell

2

'

'.

atomsall

r

rkkifi

eI

O

ki kf

r’ r

dRr '

2

.

cellunitin

atomsall

d

dkkifi

eI

Consider fluctuations about equilibrium

position

Assume atoms fluctuate independently

about equilibrium position (Einstein

model) thermal average contains terms

Series expansion of the exponential is

Debye-Waller Factor

2

.

rQi

eQI

turtr

uQirQi

ee..

....2

1.1

2.

uQuQieuQi

But for random motion

and averaging over 3D

The following function has

the same series expansion

Hence

0. uQ

222

3

1. QuuQ

...6

11

226

1 22

QueQu

22

3

1

0

Qu

eII

The Debye-Waller factor

decreases the intensity of reflections at

higher Q, but does not alter peak widths

For a classical harmonic oscillator

and the temperature factor is

Debye-Waller Factor

Temperature dependence of

Debye-Waller factor for Cu

22

3

1

2Qu

Wee

kTU2

1

2

23

0

Q

M

kT

eII

The Debye-Waller factor

decreases the intensity of reflections at

higher Q, but does not alter peak widths

For a classical harmonic oscillator

and the temperature factor is

Debye-Waller Factor

Q dependence of Debye-Waller

factor for 3He at low temperature

Zero-point motion

22

3

1

2Qu

Wee

kTU2

1

2

23

0

Q

M

kT

eII

Fourier Transforms

Consider the scattered wave from a

continuous density of scattering

centres

The sum over scattering centres

becomes an integral

The integral is the Fourier transform

(F.T.) of

'r

'r

O ×

'''.

rderbr

er

rQi

spaceall

ikr

sc

'r

Fourier Transforms

Examples

ρ(x) S(Q)

F.T.

x Q

Delta function at the origin Scattering is completely flat

Array of delta functions

Fourier Transforms

Examples

ρ(x) S(Q)

F.T.

x Q

Another array of delta functions

Fourier Transforms

Examples

ρ(x) S(Q)

F.T.

x Q

2

2

2

x

ex

22

22

2

Q

eQS

Gaussian Another Gaussian

If a function is spread out in real space it is compact in reciprocal space

Fourier Transforms

Fourier transform of the convolution

'''* xgxxfdxxgf

The convolution of two functions

f (x) and g(x) is defined by

The F.T. of f *g is the product of

the F.T.s of f and g times 2π

F.T.(f *g) = 2π × F.T.(f ) × F.T.(g)

Convolving a lattice with a

Gaussian allows us to

account for a form factor or

a Debye-Waller factor

Convolving a lattice with a

basis shows how to calculate

intensities from enveloping

functions, and the missing

orders, etc. are vital for

structure determination

Fourier Transforms

Examples

ρ(x)

x

S(Q)

Q

Lattice

Reciprocal Lattice

*

×

=

=

Structure

Diffraction

Gaussian

Form/D-W Factor

Fourier Transforms

Examples

ρ(x)

x

S(Q)

Q

*

×

=

=

Lattice Basis

Reciprocal Lattice Enveloping Function

Structure

Diffraction

Correlation Functions

Disordered Glassy Material How do we deal with the situation

where we do not have long-range

order, such as for a glass?

The number density of particles is

The correlations between and

are given by

The density-density correlation

function is

n

nrrr '

r sr

srr

rdsrrsG

counts the nuclei separated

by

The F.T. of is given by

The amplitude of the scattered wave

is

m

mn

n

srrsG

Correlation Functions

sG

s

sG

m

rrQi

n

mn

eQG.

2

1

n

rQi

ikr

sc n

er

ebr

.

The differential cross section is

and this can be expressed in

terms of the F.T. of

Neutrons measure correlation

functions

m

rrQi

n

mn

ebd

d .22,

sG

QGbd

d

2

4,

Correlation Functions

Disordered Glassy Material

Diffuse Scattering from Defects

Simulate over many unit cells

Away from Bragg points

where pm is the probability of

occupation of the mth site, cm is the

ideal occupation number (0 or 1)

and Tm is the individual temperature

factor

j

D

j

D

cohQF

NQS

2||

1

mrQi

m

m

mmm

D

jeQTbcpQF

.

j

riQ

j

D

coh

j

ebN

QS2.

||1

Oxygen vacancies in Y2Ti2O7

Diffuse Scattering from Defects

Simulate over many unit cells

Away from Bragg points

where pm is the probability of

occupation of the mth site, cm is the

ideal occupation number (0 or 1)

and Tm is the individual temperature

factor

j

D

j

D

cohQF

NQS

2||

1

mrQi

m

m

mmm

D

jeQTbcpQF

.

j

riQ

j

D

coh

j

ebN

QS2.

||1

Oxygen vacancies in Y2Ti2O7

Small Angle Neutron Scattering

General expression for SANS

Two-component system

Small Angle Neutron Scattering

General expression for SANS

Two-component system

Babinet’s Principle

Scattering the same

Small Angle Neutron Scattering

General expression for SANS

Multi-component system

Contrast matching

E.g. core-shell system

Small Angle Neutron Scattering

General expression for SANS

Multi-component system

Contrast matching

E.g. core-shell system

Small Angle Neutron Scattering

General expression for SANS

Multi-component system

Contrast matching

E.g. core-shell system

Small Angle Neutron Scattering

Porod Scattering

At large Q,

I(Q) ~ Q – 4

Specific surface area = S/V

Guinier Scattering

At small Q, where

Small Angle Neutron Scattering

Small Angle Neutron Scattering

Small Angle Neutron Scattering

Small Angle Neutron Scattering

Small Angle Neutron Scattering

Rewriting the cross section as

We take the ensemble average

(i.e. calculate the average cross

section over many samples with

the same nuclear positions) and

since the sites are uncorrelated

if n m

if n = m

The differential cross section is

where depends upon

element

isotope

spin of nucleus

We can calculate this quantity if we

assume that the distribution of

isotopes and spin states is random

Coherent & Incoherent Scattering

n

rQi

n

n

ebd

d 2.

||

nb

m

rrQi

mn

n

mn

ebbd

d .

mnmnbbbb

2

nmnbbb

Therefore

Coherent scattering gives correlations between

same and different nuclei (structure, collective

dynamics) whereas incoherent gives information

on single particles (dynamics but not structure)

Coherent & Incoherent Scattering

mn

n

m

rrQi

mn

mn

bebbd

dmn 2.

mn

nn

m

rrQi

mn

n

bbebbd

dmn

22.

Coherent Incoherent

Cross sections

Examples (in barns)

H 1.8 80.2

C 5.6 0

V 0 5

2

4 bcoh

22

4 bbinc

coh inc

Neutrons or X-rays?

Structural cross sections

Neutrons interact with

nuclei and scattering length

varies irregularly with Z

X-rays interact with

electrons and form factor

varies as Z 2

Neutrons good for light

elements (H, Li, O, Na, etc.)

Neutrons discriminate

between nearby elements in

periodic table (but

anomalous x-ray diffraction

useful too!)

Hydrogen storage

High storage densities

Rapid charge and

discharge at acceptable

temperatures

Synchrotron x-rays to

search compositions

Neutron diffraction

determines location of

hydrogen

In-situ studies

Neutrons or X-rays?

Hydrogen storage

NH2 & BH4 groups

isoelectronic

Neutrons can

distinguish between them

LiNH2 high storage

density & reversible, but

gives off harmful

ammonia

Structurally similar

Li4BN3H10 desorbs H2

rather than ammonia, has

a high storage density,

but is not as reversible

The search goes on…

Li4BN3H10 LiNH2

Neutrons or X-rays?

Diffraction Summary

where

2.

|~| n

rQi

n

n

ebd

dFor general lattice vectors

, , etc.

'rRr cnbnanR

321

***

clbkahG

cbv

a

0

* 2

acv

b

0

* 2

bav

c

0

* 2

cbav .0

0.*

ba2.*

aa

2'..

|~| j

rQi

j

l

RQi jl

ebed

d

zwyvxur ˆˆˆ'

222'.2||||~

hkl

j

rGi

jFNebN

d

dj

j

lwkvhui

jhklebF

2