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, pp. 710711

The model of the structure is refined by leastsquares minimization of the residual is the intensity measured at a point i in the diffraction pattern corrected for the background intensity , is its weight, and 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 The summation in (8.6.1.1) 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 ) of the structural parameters (see Subsection 8.6.2.5), which are dependent on the number of linearly independent columns in the design matrix [equation (8.1.2.3) ]. 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) and a more rigorous statistical estimate has been described by Sivia (2000). The strategy for choosing the number of steps and apportioning the available counting time has been discussed by McCusker et al. (1999) and references therein. The relation between counting statistics and the s.u.'s has been discussed by Baharie & Pawley (1983) and by Scott (1983).
The calculated intensity is evaluated using the equation where s is a scale factor, is the multiplicity factor for the kth reflection, is the Lorentz–polarization factor, is the structure factor and is the `peakshape function' (PSF). The summation in (8.6.1.2) is over all nearby reflections, to , contributing to a given data point i.
A fundamental problem of the Rietveld method is the formulation of a suitable peakshape function. For Xrays, a mixture of Gaussian and Lorentzian components is sometimes used (see Subsection 8.6.2.2). 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 where is the full width at halfmaximum (FWHM) of the peak, is the scattering angle at the ith point, and is the Bragg angle for reflection k.
The angular dependence of the FWHM for a Gaussian peakshape function can be written in the form where U, V and W are halfwidth parameters independent of (Caglioti, Paoletti & Ricci, 1958). To allow for intrinsic sample broadening and instrumental resolution, U, V and W are treated as adjustable variables in the leastsquares refinement. The tails of a Gaussian peak decrease rapidly with distance from the maximum, and the intensity at oneandahalf 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 and is cut off outside this range. The FWHM for the Lorentzian (see Subsection 8.6.2.2) can be modelled by the relation The Lorentzian function extends over a much wider range than the Gaussian. A more flexible approach to this linebroadening problem is described by McCusker et al. (1999).
The leastsquares 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 leastsquares refinement of singlecrystal data. These include the halfwidth 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).
The following indicators are used to estimate the agreement with the model during the course of the refinement.
Profile R factor:Weighted profile R factor: Bragg R factor: Expected R factor: I_{k} is the integrated intensity of the kth reflection, is the number of independent observations, and P is the number of refined parameters. The most important indicators are and . The ratio is the socalled `goodnessoffit', χ^{2}: in a successful refinement χ^{2} 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.
References
Altomare, A., Cascarano, G., Giacovazzo, C., Guagliardi, A., Moliterni, A. G., Burla, M. C. & Polidori, G. (1995). On the number of statistically independent observations in powder diffraction. J. Appl. Cryst. 15, 361–374.Baharie, E. & Pawley, G. S. (1983). Counting statistics and powder diffraction scan refinements. J. Appl. Cryst. 16, 404–406.
Caglioti, G., Paoletti, A. & Ricci, F. P. (1958). Choice of collimators for a crystal spectrometer for neutron diffraction. Nucl. Instrum. Methods, 3, 223–228.
Hewat, A. W. (1986). Highresolution neutron and synchrotron powder diffraction. Chem. Scr. 26A, 119–130.
McCusker, L. B., Von Dreele, R. B., Cox, D. E., Louer, D. & Scardi, P. (1999). Rietveld refinement guidelines. J. Appl. Cryst. 32, 36–50.
Scott, H. G. (1983). The estimation of standard deviations in powder diffraction refinement. J. Appl. Cryst. 16, 589–610.
Sivia, D. S. (2000). The number of good reflections in a powder pattern. J. Appl. Cryst. 33, 1295–1301.