International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 8.6, p. 711

Section 8.6.2.2. Peak-shape function (PSF)

A. Albinatia and B. T. M. Willisb

aIstituto Chimica Farmaceutica, Università di Milano, Viale Abruzzi 42, Milano 20131, Italy, and bChemical Crystallography Laboratory, University of Oxford, 9 Parks Road, Oxford OX1 3PD, England

8.6.2.2. Peak-shape function (PSF)

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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], functions commonly used for angle-dispersive data are (Young & Wiles, 1982[link]): [\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)}}]where [\Delta2\theta _{ik}=2\theta _{i}-2\theta _{k}]. 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[link]) and David & Matthewman (1985[link])]. 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[link]; Von Dreele, Jorgensen & Windsor, 1982[link]).

The pattern from an X-ray powder diffractometer gives peak shapes that cannot be fitted by a simple analytical function. Will, Parrish & Huang (1983[link]) use the sum of several Lorentzians to express the shape of each diffraction peak, while Hepp & Baerlocher (1988[link]) describe a numerical method of determining the PSF. Pearson VII functions have also been successfully used for X-ray data (Immirzi, 1980[link]). A modified Lorentzian function has been employed for interpreting data from a Guinier focusing camera (Malmros & Thomas, 1977[link]). PSFs for instruments employing X-ray synchrotron radiation can be represented by a Gaussian (Parrish & Huang, 1980[link]) or a pseudo-Voigt function (Hastings, Thomlinson & Cox, 1984[link]).

References

Ahtee, M., Unonius, L., Nurmela, M. & Suortti, P. (1984). A Voigtian as profile shape function in Rietveld refinement. J. Appl. Cryst. 17, 352–357.
Albinati, A. & Willis, B. T. M. (1982). The Rietveld method in neutron and X-ray powder diffraction. J. Appl. Cryst. 15, 361–374.
David, W. I. F. & Matthewman, J. C. (1985). Profile refinement of powder diffraction patterns using the Voigt function. J. Appl. Cryst. 18, 461–466.
Hastings, J. B., Thomlinson, W. & Cox, D. E. (1984). Synchrotron X-ray powder diffraction. J. Appl. Cryst. 17, 85–95.
Hepp, A. & Baerlocher, Ch. (1988). Learned peak-shape functions for powder diffraction data. Aust. J. Phys. 41, 229–236.
Immirzi, A. (1980). Constrained powder profile refinement based on generalized coordinates. Application to X-ray data of isotactic polypropylene. Acta Cryst. B36, 2378–2385.
Malmros, G. & Thomas, J. O. (1977). Least-squares structure refinement based on profile analysis of powder film intensity data measured on an automatic microdensitometer. J. Appl. Cryst. 10, 7–11.
Parrish, W. & Huang, T. C. (1980). Accuracy of the profile fitting method for X-ray polycrystalline diffractometry. Natl Bur. Stand. (US) Spec. Publ. No. 567, pp. 95–110.
Von Dreele, R. B., Jorgensen, J. D. & Windsor, C. G. (1982). Rietveld refinement with spallation neutron powder diffraction data. J. Appl Cryst. 15, 581–589.
Will, G., Parrish, W. & Huang, T. C. (1983). Crystal structure refinement by profile fitting and least-squares analysis of powder diffractometer data. J. Appl Cryst. 16, 611–622.
Young, R. A. & Wiles, D. B. (1982). Profile shape functions in Rietveld refinements. J. Appl. Cryst. 15, 430–438.








































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