Classical & Statistical Thermodynamics - open.ac.uk · PDF fileFor a classical harmonic...
Transcript of Classical & Statistical Thermodynamics - open.ac.uk · PDF fileFor a classical harmonic...
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