Introduction to Patterson Function and its Applications “Transmission Electron Microscopy and Diffractometry of Materials”, B. Fultz and J. Howe, Springer-Verlag Berlin 2002. 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:

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

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

System 2, 3 Reciprocal lattice Diffraction condition

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)

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:

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

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

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

1D example of Patterson function

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

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

Shape function RN(x): extended  to 

N = 9 shift 8a a triangle of twice the total width  -3a -a 2a 4a -4a 2a 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

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

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

F.T.

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

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

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

 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

12

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

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 !

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

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

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

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

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