Abstract
Although diffraction can be understood as wave interference involving wave scattering from individual atoms, in kinematical theory the diffraction intensity can also be understood as contributions from pairs of atoms, weighted by phase factors. The Patterson function is defined, and used to describe disorder from atom displacements and chemical disorder. The diffuse scattering from these types of disorder is explained. Local correlations in atomic displacements and short-range order are used to explain modulations in the diffuse scattering. This chapter explains how to obtain the pair distribution function by Fourier transformation of the diffraction pattern. Diffraction methods are extended for measuring structural features of amorphous materials and measuring structures of larger objects by small-angle scattering. The basic methods for analyzing small-angle scattering experiments are presented.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The work with convolutions in this chapter will help the reader through Chaps. 11–13.
- 2.
The actual shape of the atom is included later by convolution, and does not change the key results obtained with point atoms.
- 3.
- 4.
You can obtain the same P(r) by taking the mirror image of the f(r) in Fig. 10.1a (with the small peak to the immediate left of the large peak), and repeating the construction.
- 5.
For our N-atom chains, the number of shifts that allow overlaps is 2N, although the average number of overlaps in a shift is half this number, i.e., N. We therefore recover the factor of N in (10.30) for very large crystals.
- 6.
This does not restrict generality because any non-zero mean could have been transferred into 〈f(r)〉 in (10.38).
- 7.
For example, we assume that if one atom is displaced to the left, its neighbor to the right is equally likely to be displaced to the left or to the right.
- 8.
The total coherent cross-section remains constant.
- 9.
One argument for a Gaussian function is provided in Appendix A.9. Another is for a harmonic solid, where the potential energy U=1/2kx 2 for each independent atomic displacement, x. The probability of a displacement is p(x)=exp[−U/(k B T)]=exp[−(k/2k B T)x 2], which is a Gaussian in x, with a characteristic width \(\sigma \propto \sqrt{T}\).
- 10.
- 11.
Never forget that the total cross-section for coherent scattering is constant.
- 12.
Note that 〈x 2〉 is along the direction of Δk. In an isotropic material 〈x 2〉 would equal 1/3 of the mean-squared atomic displacement (cf., (10.170)).
- 13.
Another aspect of the problem is that a crystal has fewer long-wavelength than short-wavelength vibrational modes. However, the lower energy of the long-wavelength modes means that their occupancy is higher at all temperatures, especially low temperatures.
- 14.
The change in average displacement is accounted for by a simple change in lattice parameter. When the change in lattice parameter is linear in impurity concentration, the alloy is said to obey “Vegard’s law.” This is not the atomic size effect of interest here, however.
- 15.
This is a handy result. A shift by a constant, b, in real space, x′=x−b, amounts to a multiplication by the factor exp(−iΔkb) in k-space.
- 16.
In a perfect crystal, R(r) includes a series of δ-functions separated by crystal translations.
- 17.
This is analogous to the scattering from atoms in an ideal gas, where the lack of atom-atom positional coherence provides the atomic form factor intensity.
- 18.
Fortunately, we need not worry about the actual value of r 0 because it provides a constant phase that is eliminated when we calculate the intensity in (10.175).
- 19.
and other compact 3-dimensional objects
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Fultz, B., Howe, J. (2013). Patterson Functions and Diffuse Scattering. In: Transmission Electron Microscopy and Diffractometry of Materials. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29761-8_10
Download citation
DOI: https://doi.org/10.1007/978-3-642-29761-8_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-29760-1
Online ISBN: 978-3-642-29761-8
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)