1. Introduction to Patterson Function and its Applications
“Transmission Electron Microscopy and Diffractometry of
Materials”, B. Fultz and J. Howe, Springer-Verlag Berlin Chapter 9)
The Patterson function:
explain diffraction phenomena involving displacement
of atoms off periodic positions (due to temperature or
atomic size)  diffuse scattering
Phase factor: instead of
Fourier transform prefactor ignored:

2. Supplement: Definitions in diffraction
Fourier transform and inverse Fourier transform
System 1
System 4
System 2
System 5
System 3
System 6

3. Relationship among Fourier transform, reciprocal
lattice, and diffraction condition
System 1
Reciprocal lattice
Diffraction condition

4. System 2, 3
Reciprocal lattice
Diffraction condition

5. Patterson function Atom centers at Points in Space:
Assuming: N scatterers (points), located at rj.
The total diffracted waves is
The discrete distribution of scatterers  f(r)

6. f(r): zero over most of the space, but at atom centers
such as , is a Dirac delta function times a
constant
Property of the Dirac delta function:

7. Definition of the Patterson function:
Slightly different from convolution
called “autoconvolution” (the function is not inverted).
Convolution:
Autocorrelation:

8. Fourier transform of the Patterson function =
the diffracted intensity in kinematical theorem.
Define 
Inverse transform

9. The Fourier transform of the scattering factor
distribution, f(r)  (k)
and
i.e.

10. 1D example of Patterson function

11. Properties of Patterson function comparing to f(r):
1. Broader Peaks
2. Same periodicity
3. higher symmetry

12. Case I: Perfect Crystals
much easier to handle f(r); the convolution of the atomic
form factor of one atom with a sum of delta functions

13. Shape function RN(x): extended  to 

14. N = 9 shift 8a a triangle of twice the total width  -3a -a 2a 4a -4a2a 4a
-4a
-2a a 3a
shift 8a
-3a -a 2a 4a
-4a
-2a a 3a
a triangle
of twice
the total
width
-9a
-7a
-5a
-3a -a 2a 4a 6a 8a
-8a
-6a
-4a
-2a a 3a 5a 7a 9a

15. F(P0(x))  I(k)
Convolution theorem: a*b  F(a)F(b); ab  F(a)*F(b)

16. If ka  2, the sum will be zero. The sum will have a
nonzero value when ka = 2 and each term is 1.
N: number of
terms in the sum
1 D reciprocal lattice

17. F.T.

18. A familiar result in a new form.
  -function  center of Bragg peaks
 Peaks broadened by convolution with the
shape factor intensity
 Bragg peak of Large k are attenuated by the
atomic form factor intensity

19. Patterson Functions for homogeneous disorder
and atomic displacement diffuse scattering
Deviation from periodicity:
Deviation function
Perfect periodic function: provide sharp Bragg peaks
Look at the second term
Mean value for
deviation is zero

20. 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
often a broad
diffuse intensity

21.  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

22. 12

23. 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

24. The diffuse scattering increases with k !
constant
deviation
F[Pdevs1(x)]
increasingly
dominates over
F[Pdevs2(x)] at
larger k.
The diffuse scattering increases with k !

25. Correlated Displacements: Atomic size effects
a big atoms locate
Overall effect: causes an asymmetry in the shape of the Bragg peaks.

26. Diffuse Scattering from chemical disorder:
Concentration of A-atoms: cA;
Concentration of B-atoms: cB.
Assume cA > cB 
When the product
is summed over x.
# positive > # negative
H positive < H ones negative
Pdevs(x  0) = 0; Pdevs(0)  0

27. Let’s calculate Pdevs(0): cAN peaks of cBN peaks of

28. Total diffracted intensity
Just like the case
of perfect crystal
Total diffracted intensity

29. The diffuse scattering part is: the difference
between the total intensity from all atoms and the
intensity in the Bragg peaks