Part 15. Patterson Function and Diffuse Scattering

1
Part 15. Patterson Function and Diffuse
Scattering (From Transmission Electron
Microscopy and Diffractometry of Materials, B.
Fultz and J. Howe, Springer-Verlag Berlin 2002.
Chapter 9)
? Introduction ? We have briefly introduced
Patterson function in part 9. Here we will
pick up the subject again and give a more
detailed look at its application. ? The
Patterson function is a function in real space
with variable r. One could use the
Patterson function to explain diffraction
phenomena involving displacement of atoms
off periodic positions due to temperature or
atomic size. ? diffuse scattering from
chemical disorder, the radial distribution
function to describe amorphous materials,
small angle scattering phenomena. ? Phase
factor instead of
Definition of reciprocal lattice
2
? In the Fourier transform, the prefactors of
are neglected when the absolute
value of the diffraction intensity in
unimportant.
? The Patterson function ? Atom centers at
Points in Space Assuming there are N
scatterers, each is a point, located at
rj. The total diffracted waves is
It is convenient to consider a distribution of
scatterers, f(r), with a continuous variable r.
f(r) is zero over most of the space, but at atom
centers such as , is a Dirac
delta function times a constant
3
Property of the Dirac delta function
Later, the actual shape of the atomic form
factor, , will be included.
? Definition of the Patterson function
Patterson function is defined as
It is a little bit different from convolution
and called autoconvolution (the function is
not inverted).
Convolution
Autocorrelation
4
?
Fourier transform of the Patterson function is
the diffracted intensity in kinematical theorem.
Define , and change variables
.
Inverse transform
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The Fourier transform of the scattering
factor distribution, f(r), is the diffracted
wave, ?(?k).
and
i.e.
? Properties of Patterson function
Peaks in Patterson function is
broader than those in the scattering
factor distribution function (f ).
Periodicity for the figure on the
right is a.
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Patterson function has a higher symmetry
than the f(r). Primary peaks occur when
shift equal to the periodicity. For the
above case, secondary peaks occurs at ?0.3a.
Friedela law diffraction experiments cannot
distinguish between an atom arrangement
and the atom arrangement when it is
inverted. (phase problem) For simple
structure, knowing the Patterson function
presents no difficulties in studies of simple
crystals of pure elements. For complex
crystals, experimental techniques has been
developed to alter the scattering strengths
of different atoms isomorphous substitutions
of atoms at known sites of the unit cell,
isotopic substitutions in the case of
neutron diffraction, altering the atomic
form factor by choosing different X-ray
wavelength (anomalous scattering)
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? Perfect Crystals It is much easier to
handle f(r) The scattering factor
distribution for an entire crystal, f(r) is
the convolution of the form factor of one atom
with a sum of delta functions
The Patterson function of a one dimensional
perfect crystal,
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Convolutions are commutative and
associative ?
Use a shape function RN(x) so that the
summation of ?- function can be extended to
an infinite series.
?
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Now there is the function RN(x). If the
shift of the two chains is larger than 2N,
they do not overlap anywhere
N 9
-3a
2a
4a
0
-a
a
3a
-2a
-4a
shift 8a
-3a
2a
4a
0
-a
a
3a
-2a
-4a
a triangle of twice the total width
-a
-3a
-7a
-5a
-9a
2a
4a
0
6a
8a
a
3a
-2a
-4a
5a
7a
9a
-6a
-8a
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Or one can consider the convolution of two
identical rectanglar shape functions. ? a
triangle shape function of twice the total
width.
The Fourier transform of P0(x) ? I(?k)
Using convolution theorem two
convolutions and one multiplication after
Fourier transform ? two multiplications and
one convolution
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If ?ka ? 2?. The sum will be zero. The sum
will have a nonzero value when
and each term is 1.
N number of terms in the sum
1 D reciprocal lattice
Another term is
shape factor intensity
In 1D this function is
F.T.
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A familiar result in a new form. ? -function ?
center of Bragg peaks These peaks are broadened
by convolution with the shape factor intensity
(depends on d?k, deviation parameter Bragg
peak of Large ?k are attenuated by the atomic
form factor intensity.
For convenience, assume that the crystal is
infinite in length ? shape factor offers no
broadening
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? Patterson Functions for homogeneous disorder
and atomic displacement diffuse scattering
? Deviation from periodicity In many cases of
interest, f(r) can be expressed as
Deviation function
Perfect periodic function provide sharp Bragg
peaks
Look at the second term
Mean value for deviation is zero
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The same argument for the third term ? 0
1st term Patterson function from the average
crystal, 2nd term Patterson function from the
deviation crystal.
Sharp diffraction peaks from the average crystal
The second term is often a broad, diffuse
intensity! Two important source of ?f(r)
(1)atomic displacement disorder and (2) chemical
disorder.
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? Uncorrelated Displacements Types of
displacement (1) atomic size differences in an
alloy ? static displacement, (2) thermal
vibrations ? dynamic displacement Consider
a simple type of displacement disorder
each atom has a small, random shift, ?, off
its site of a periodic lattice
Consider the overlap of the atom center
distribution with itself after a shift of
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No correlation in ? ? probability of overlap of
two atom centers is the same for all
shift except n 0
When n 0, perfect overlap at ? 0, at ? ? 0
no overlap




The same number of atom- atom overlap
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constant deviation
FPdevs1(x) increasingly dominates over
FPdevs2(x) at larger k.
The diffuse scattering increases with ?k !
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? Correlated Displacements Atomic size effects
Different atoms have different sizes. Atoms
around the big atoms are pushed away from the
lattice site and atoms further away are
gradually relaxed back to lattice site. The
peak shapes (P(x)) are skewed to larger x.
? Diffracted intensity is therefore shifted
from the higher ?k sides of the peaks to the
lower ?k sides. Overall effect causes an
asymmetry in the shape of the Bragg peaks.
a big atoms locate
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? Diffuse Scattering from chemical disorder ?
Randomness uncorrelated chemical disorder
Assume a statistically-random occupancy of A-
and B-atoms on each site in the alloy.
f(x) comprises delta functions each
weighted by the scattering strength of the
individual atom. Concentration of A-atoms
cA Concentration of B-atoms cB.
Assume cA gt cB ?
Deviations in the scattering factor distribution,
?f(x). The average value of ?f is zero
The Patterson function for the deviation crystals
is largest for a zero shift. At shifts equal to
lattice vectors, the positive and negative peaks
overlap randomly.
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When the product is summed over x?.
More positive than negative ones, but most of the
positive ones are small. The Patterson function
of the deviation crystal is zero, except for
shifts of zero.
Lets calculate Pdevs(0) cAN peaks of cBN peaks
of
cA
cB
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Just like the case of perfect crystal
Total diffracted intensity
The diffuse scattering part is also called Laue
monotonic scattering. The intensity of the
diffuse scattering is the difference between the
total intensity from all atoms and the intensity
in the Bragg peaks
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? Short range order (SRO) parameter
Assume a binary alloy (A-B), total number of
atoms is N, concentration of A-atoms is cA,
concentration of B-atoms is cB, the
scattering factor for A atom at crystal site m
is fA(m) the scattering factor for B atom
at crystal site m is fB(m). Short
range order describe correlations between
neighboring pairs of atoms, typically separated
by a few atomic distance.
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Stronger correlations at shorter distance and
weaker correlations at larger pair separations
(n). When atoms are separated by very large
distances, we should expect no correlation.
Mathematically, two atoms at sites m and m?,
separated by a large nth neighbor distance
, the lack of correlation is a
statement of statistical independence
The relation will be
used for no correlation.
convolution is evaluated at the interatomic
separation rn.
Define the two conditional pair probability
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In the limiting case of complete disorder and
statistical independence between atom positions
Define Warren-Cowley SRO parameters, ?(n)
? total 0-2
0-1
0-1
If an alloy is random,
For alloy with chemical order
for at least some value of n
at those n
For alloy with clustering tendency
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? Patterson function for chemical SRO
The Patterson function for the average crystal
The Patterson function for the entire crystal
consists of two terms. (1) correlations around
A-atoms there are NcA with scattering
strength fA. About these A-atoms, the
probability of finding a B-atom at a distance n
is and that of finding an
A-atom at a distance n is (2) similarly
for the second term which finds the
correlations around B-atoms.
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Deviation
Look at
Look at
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Look at
Look at
The terms having
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The terms having
True random
Just like previous calculation
SRO diffuse intensity
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For chemically centro-symmetric alloys
sum vanished
In a random solid solution,
? Amorphous Materials ? One dimensional
model Amorphous materials usually contains
two or more elements position
correlations of atoms, local chemical
correlations. Consider one dimensional amorphous
materials with single element ? provide
qualitative features of the diffracted
intensity arising from position disorder
only. The average separation between atoms
is a.
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The separation between each adjacent atom is a
random Variable (statistically independent) with
a Gaussian distribution. Atoms separated by
larger distances have a greater uncertainty in
their separation. The scattering factor
distribution is
For a perfect crystal
The diffracted intensity is
First calculate the convolution
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The probability distribution for the separation
of the first -nearest-neighbor (1nn) planes is
A Gaussian distribution centered at a with a
width of ?
a
The probability distribution for the separation
of the second-nearest-neighbor (2nn) planes the
probability distribution of a sum of independent
random variables is the convolution of
the probability distributions of the variable.
2a
centered at 2a
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Similarly,
centered at 3a
.
Next get the Fourier transform of the above
function
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1
The Fourier transform of a Gaussian function is
a Gaussian function
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Neglecting the common prefector
x100
? ? ? perfect lattice
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? Radial Distribution Function Consider
the reference structure is not a periodic solid,
but a homogeneous distribution of atoms with
an average density ?0. All density
distributions are described as sums of the
constant density plus spatial variations in
density. The scattering factor distribution
For a perfect crystal one can use ?0 0 and
??(r) is a set of delta functions at lattice
sites. Here ?0 is the average bulk density, so
that
In form of integral
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The first term is needed to account for the
perfect overlap at zero shift . The
system have N scatterers and a volume V. ? ?0V
N.
V N/?0.
0
Define a new function R(r)
A correlation function of the density
heterogeneities in the material
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The diffracted intensity is
The first term is a structureless background in
the diffraction pattern (from the sharp
self-correlation). The second term the forward
scattering from a large homogeneous objects
(forward direction only).
Assume the density-density correlations in the
material are spatially isotropic, i.e. R(r)
depends on r only.
Look at part 8
This equation can be used to obtain R(r).
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First identify the contribution to the
measured diffraction data from the term ?(?k),
located near the transmitted beam, and we strip
this component from the data. Define normalized
intensity ( )
To get R(r), multiply both sides of the above
equation by and integrate
over
A forward Fourier transform of R(r)r into
k-space followed by a back transform into real
space, nonzero only when
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?/2 coming from the normalization factor of the
two sine transform