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, pp. 710-712
https://doi.org/10.1107/97809553602060000614

Chapter 8.6. The Rietveld method

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

An overview of the Rietveld method of analysing powder diffraction data, in which the crystal structure is refined by fitting the entire profile of the diffraction pattern to a calculated profile, is provided. The basic theory of Rietveld refinement is described and some problems that may lead to failure of the method are discussed.

In the Rietveld method of analysing powder diffraction data, the crystal structure is refined by fitting the entire profile of the diffraction pattern to a calculated profile. There is no intermediate step of extracting structure factors, and so patterns containing many overlapping Bragg peaks can be analysed. The method was applied originally by Rietveld (1967[link], 1969[link]) to the refinement of neutron intensities recorded at a fixed wavelength. Subsequently, it has been used successfully for analysing powder data from all four categories of experimental technique, with neutrons or X-rays as the primary radiation and with scattered intensities measured at a fixed wavelength (and variable scattering angle) or at a fixed scattering angle (and variable wavelength).

Powder refinements are usually less satisfactory than those on single-crystal data, as the three-dimensional information of the reciprocal lattice is compressed into one dimension in the powder pattern. Nevertheless, a large number of successful refinements by the Rietveld method has been reported; reviews have been given by Taylor (1985)[link], Hewat (1986)[link], Cheetham & Wilkinson (1992)[link], Young (1993)[link], Harris & Tremayne (1996)[link], Masciocchi & Sironi (1997)[link], Harris et al. (2001)[link] and David et al. (2002)[link]. Here we shall discuss only the basic principles of the refinement procedure.

8.6.1. Basic theory

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The model of the structure is refined by least-squares minimization of the residual [ M=\textstyle\sum\limits _{i=1}^{N}w_{i}\left [y_{i}\left ({\rm obs.}\right) -y_{i}\left ({\rm calc.}\right) \right] ^{2}.  \eqno (8.6.1.1)][y_{i}\left ({\rm obs.}\right) ] is the intensity measured at a point i in the diffraction pattern corrected for the background intensity [b_i], [w_{i}] is its weight, and [y_{i}\left ({\rm calc.}\right) ] is the calculated intensity. If the background at each point is assumed to be zero, and if the only source of error in measuring the intensities is that from counting statistics, the weight is given by [ w_{i}=\left [y_{i}\left ({\rm obs.}\right) \right] ^{-1}. ]The summation in (8.6.1.1)[link] runs over all N data points. The number of data points can be arbitrarily increased by reducing the interval between adjacent steps. However, this does not necessarily imply an improvement in the standard uncertainties (s.u.'s; see Section 8.1.2[link] ) of the structural parameters (see Subsection 8.6.2.5[link]), which are dependent on the number of linearly independent columns in the design matrix [equation (8.1.2.3)[link] ]. The number of independent observations in a powder pattern is determined by the extent of overlapping of adjacent reflections. An intuitive argument for estimating this number has been proposed by Altomare et al. (1995[link]) and a more rigorous statistical estimate has been described by Sivia (2000[link]). The strategy for choosing the number of steps and apportioning the available counting time has been discussed by McCusker et al. (1999[link]) and references therein. The relation between counting statistics and the s.u.'s has been discussed by Baharie & Pawley (1983[link]) and by Scott (1983[link]).

The calculated intensity is evaluated using the equation [ y_i({\rm calc.})=s\textstyle\sum\limits _{k=k_1}^{k_2}m_kL_k\left | F_k\right | ^2G_{ik}, \eqno (8.6.1.2)]where s is a scale factor, [m_k] is the multiplicity factor for the kth reflection, [L_k] is the Lorentz–polarization factor, [F_k] is the structure factor and [G_{ik}] is the `peak-shape function' (PSF). The summation in (8.6.1.2)[link] is over all nearby reflections, [k_1] to [k_2], contributing to a given data point i.

A fundamental problem of the Rietveld method is the formulation of a suitable peak-shape function. For X-rays, a mixture of Gaussian and Lorentzian components is sometimes used (see Subsection 8.6.2.2[link]). For neutrons, it is easier to find a suitable analytical function, and this is, perhaps, the main reason for the initial success of neutron Rietveld analysis. For a neutron diffractometer operating at a fixed wavelength and moderate resolution, the PSF is approximately a Gaussian of the form [ G_{ik}= {2 \over H_{k}}\sqrt { {\ln 2\over \pi }}\exp \left [-{4\ln 2\,(2\theta _{i}-2\theta _{k})^{2} \over H_{k}^{2}}\right] ,  \eqno (8.6.1.3)]where [H_{k}] is the full width at half-maximum (FWHM) of the peak, [2\theta _{i}] is the scattering angle at the ith point, and [\theta _{k}] is the Bragg angle for reflection k.

The angular dependence of the FWHM for a Gaussian peak-shape function can be written in the form [ H_k^2=U\tan ^2\theta _k+V\tan \theta _k+W, ]where U, V and W are half-width parameters independent of [\theta _k] (Caglioti, Paoletti & Ricci, 1958[link]). To allow for intrinsic sample broadening and instrumental resolution, U, V and W are treated as adjustable variables in the least-squares refinement. The tails of a Gaussian peak decrease rapidly with distance from the maximum, and the intensity at one-and-a-half times the FWHM from the peak is only about 0.2% of the intensity at the peak. Thus, no large error is introduced by assuming that the peak extends over a range of approximately [\pm 1.5H_k] and is cut off outside this range. The FWHM for the Lorentzian (see Subsection 8.6.2.2[link]) can be modelled by the relation [H_k=X\tan\theta_k+Y/\cos\theta_k.]The Lorentzian function extends over a much wider range than the Gaussian. A more flexible approach to this line-broadening problem is described by McCusker et al. (1999[link]).

The least-squares parameters are of two types. The first contains the usual structural parameters: for example, fractional coordinates of each atom in the asymmetric unit and the corresponding isotropic or anisotropic displacement parameters. The second type represents `profile parameters' which are not encountered in a least-squares refinement of single-crystal data. These include the half-width parameters and the dimensions of the unit cell. Further parameters may be added to both groups allowing for the modelling of the background and for the asymmetry of the reflections. The maximum number of parameters that can be safely included in a Rietveld refinement is largely determined by the quality of the diffraction pattern, but intrinsic line broadening will set an upper limit to this number (Hewat, 1986[link]).

The following indicators are used to estimate the agreement with the model during the course of the refinement.

Profile R factor:[R_{P}= {\sum \limits_{i} | y_{i}({\rm obs.}) -y_{i}({\rm calc.})| \over \sum \limits_{i} y_{i}({\rm obs.}) }.]Weighted profile R factor: [ R_{wP}=\left [{\sum \limits _{i}w_{i}\left | y_{i}\left ({\rm obs.}\right) -y_{i}\left ({\rm calc.}\right) \right | ^{2} \over \sum \limits _{i}w_{i}\,y_{i}^{2}\left ({\rm obs.}\right) }\right] ^{1/2}.]Bragg R factor: [ R_{I}= {\sum \limits _{k}\left | I_{k}\left ({\rm obs.}\right) -I_{k}\left ({\rm calc.}\right) \right | \over \sum \limits _{k}I_{k}\left ({\rm obs.}\right) }. ]Expected R factor: [ R_{E}=\left [{{\cal N}-P \over \sum \limits _{i}w_{i}\,y_{i}^{2}\left ({\rm obs.}\right) }\right] ^{1/2}. ]Ik is the integrated intensity of the kth reflection, [{\cal N}] is the number of independent observations, and P is the number of refined parameters. The most important indicators are [R_{wP}] and [R_E]. The ratio [R_{wP}/R_E] is the so-called `goodness-of-fit', χ2: in a successful refinement χ2 should approach unity. The Bragg R factor is useful, since it depends on the fit of the structural parameters and not on the profile parameters.

8.6.2. Problems with the Rietveld method

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One should be aware of certain problems that may give rise to failure in a Rietveld refinement.

8.6.2.1. Indexing

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The first step in refinement is the indexing of the pattern. As the Rietveld method is often applied to the refinement of data for which the unit-cell parameters and space group are already known, there is then little difficulty in indexing the pattern, provided that there are a few well resolved lines. Without this knowledge, the indexing requires, as a starting point, the measurement of the d values of low-angle diffraction lines to high accuracy. According to Shirley (1980[link]): `Powder indexing works beautifully on good data, but with poor data it usually will not work at all'. The indexing of powder patterns and associated problems are discussed by Shirley (1980[link]), Pawley (1981[link]), Cheetham (1993[link]) and Werner (2002[link]).

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]).

8.6.2.3. Background

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The background may be determined by measuring regions of the pattern that are free from Bragg peaks. This procedure assumes that the background varies smoothly with sinθ /λ, whereas this is not the case in the presence of disorder or thermal diffuse scattering (TDS), which rises to a maximum at the Bragg positions. An alternative approach is to include a background function in the refinement model (Richardson, 1993[link]). If the background is not accounted for satisfactorily, the temperature factors may be incorrect or even negative. The various procedures for estimating the background for X-ray, synchrotron, constant-wavelength and TOF neutron powder patterns are reviewed by McCusker et al. (1999[link]).

In neutron diffraction, the main contribution to the background from hydrogen-containing samples is due to incoherent scattering. Deuterating the sample is essential in order to substantially reduce this background.

8.6.2.4. Preferred orientation and texture

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Preferred orientation is a formidable problem which can drastically affect the measured intensities. A simple correction formula for plate-like morphology was given by Rietveld (1969[link]). Ahtee, Nurmela, Suortti & Järvinen (1989[link]) have shown how the effects of preferred orientation can be included in the refinement by expanding the orientation distribution in spherical harmonics. Quantitative texture analysis based on spherical harmonics has been implemented in the Rietveld refinement code by Von Dreele (1997[link]). A general model of the texture has also been described by Popa (1992[link]). It may be possible to remove or reduce the effect of preferred orientation by mixing the sample with a suitable diluent.

An additional problem is caused by particle size and strain broadening, which are not smooth functions of the diffraction angle. These effects can be taken into account by phenomenological models (e.g. Dinnebier et al., 1999[link]; Pratapa, O'Connor & Hunter, 2002[link]) or by an analytical approach such as that of Popa & Balzar (2002[link]).

The determination of the elastic stresses and strains in polycrystals can be determined from diffraction line shifts using Rietveld refinement (Popa & Balzar, 2001[link]).

8.6.2.5. Statistical validity

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Sakata & Cooper (1979[link]) criticized the Rietveld method on the grounds that different residuals [ y_{i}\left ({\rm obs.}\right) -y_{i}\left ({\rm calc.}\right) ]related to the same Bragg peak are correlated with one another, and they asserted that this correlation leads to an uncertainty in the standard uncertainties of the structural parameters. Prince (1981[link]) has challenged this conclusion and stated that the s.u.'s given by the Rietveld procedure are correct if the crystallographic model adequately fits the data. However, even if the s.u.'s are correct, they are measures of precision rather than accuracy, and attempts to assess accuracy are hampered by lack of information concerning correlations between systematic errors (Prince, 1985[link], 1993[link]).

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