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International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C. ch. 8.6, p. 711
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The appropriate function to use varies with the nature of the experimental technique. In addition to the Gaussian PSF in (8.6.1.3)![[link]](/graphics/greenarr.gif)
, functions commonly used for angle-dispersive data are (Young & Wiles, 1982): ![[\eqalignno{ G_{ik}&= {2\over\pi H_{k}}\left [1+4\left ({\Delta2\theta _{ik} \over H_{k}}\right) ^{2}\right] ^{-1} &{\rm (Lorentzian)} \cr G_{ik} &= {2\eta \over\pi H_{k}}\left [1+4\left ({\Delta2\theta _{ik} \over H_{k}}\right) ^{2}\right] ^{-1} \cr &\quad+\left (1-\eta \right) {2\sqrt {\ln 2} \over \sqrt {\pi }H_{k}}\exp \left [-4\ln 2\left ({\Delta2\theta _{ik} \over H_{k}}\right) ^{2}\right] & \hbox{(pseudo-Voigt)} \cr G_{ik} &= {2\Gamma (n) \left (2^{1/n}-1\right)\over \pi H_k \Gamma (n-{1 \over 2}) }\left [1+4\left (2^{1/n}-1\right) \left ({\Delta2\theta _{ik} \over H_{k}}\right) ^{2}\right] ^{-n} \cr& & \hbox{(Pearson VII)}}]](/teximages/cbch8o6/cbch8o6fd11.svg)
where ![[\Delta2\theta _{ik}=2\theta _{i}-2\theta _{k}]](/teximages/cbch8o6/cbch8o6fi20.svg)
. η is a parameter that defines the fraction of Lorentzian character in the pseudo-Voigt profile. Γ(n) is the gamma function: when n = 1, Pearson VII becomes a Lorentzian, and when n = ∞, it becomes a Gaussian.
The tails of a Gaussian distribution fall off too rapidly to account for particle size broadening. The peak shape is then better described by a convolution of Gaussian and Lorentzian functions [i.e. Voigt function: see Ahtee, Unonius, Nurmela & Suortti (1984) and David & Matthewman (1985)]. A pulsed neutron source gives an asymmetrical line shape arising from the fast rise and slow decay of the neutron pulse: this shape can be approximated by a pair of exponential functions convoluted with a Gaussian (Albinati & Willis, 1982; Von Dreele, Jorgensen & Windsor, 1982).
The pattern from an X-ray powder diffractometer gives peak shapes that cannot be fitted by a simple analytical function. Will, Parrish & Huang (1983) use the sum of several Lorentzians to express the shape of each diffraction peak, while Hepp & Baerlocher (1988) describe a numerical method of determining the PSF. Pearson VII functions have also been successfully used for X-ray data (Immirzi, 1980). A modified Lorentzian function has been employed for interpreting data from a Guinier focusing camera (Malmros & Thomas, 1977). PSFs for instruments employing X-ray synchrotron radiation can be represented by a Gaussian (Parrish & Huang, 1980) or a pseudo-Voigt function (Hastings, Thomlinson & Cox, 1984).
References
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