Profile Fitting II. Practice

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Practice

How is this expression for Δ evaluated in practice given that it contains three nested summations?

Although the summation is over all reflections hkl, it is not necessary to sum over all reflections for every point i in the diffraction pattern. This is because the tails of the Gaussian peak-shape function decrease rapidly to almost zero: for practical purposes the range can usually be limited to 3H. Thus any reflection whose calculated 2θ position is separated from the 2θ value of the ith profile point by, say 1.5× the peak width H, may be ignored in the summation. Thus a contribution from reflection hkl is only calculated when:

In order to make the calculation efficient, reflections are generated and sorted according to the value of their d spacing. In addition, the width of each reflection is also calculated. Thus for any point in the diffraction pattern, it is possible to calculate in advance which reflections contribute to which point. Since each reflection contributes to many profile points, the summation is optimized by calculating the structure factor and intensity of each reflection in advance, and this intensity is then distributed according to the peak-shape function in use.

Rietveld also took into account two other observations that effect how the profile is calculated in practice. At low scattering angles the peak shape was not perfectly Gaussian because the straight detectors scanned through a curved Debye-Scherrer cone of radiation. This leads to some peak asymmetry, which increases with decreasing scattering angle. Rietveld allowed for this by introducing an empirical asymmetry parameter, P, as a correction term. More recently, asymmetry parameters with a more theoretical basis have been used, the most notable being that coded by Finger, Cox, & Jephcoat. §

The second observation is that some samples (even with neutron diffractometry) exhibit preferred orientation due to the fact that plate-like crystals have a tendency to lie with the normals to the plane of the plate parallel to the normal of the specimen holder, even for cylindrical samples. The correction used by Rietveld is given by

where ζ is the acute angle between the scattering vector and the normal to the crystallites. It is not the best expression and alternatives have been introduced in various Rietveld codes. Generally, however, it is best to take care in mounting the sample so as to avoid preferred orientation effects far as possible. In my experience these can never be reliably corrected for.

There have been several approaches to the treatment of the background count: one method is to estimate the background level by graphical methods (e.g. by plotting the data) at several points where there are no diffraction peaks in the diffraction profile and to assume a linear function between them. This works well for angle-dispersive data sets for unit cells of a moderate size: for vary large unit cells, extensive peak overlap may make estimation of the background count difficult or even inaccurate.

The alternative method is to use a background function with refinable parameters, the most popular of which seems to be the polynomial function. The use of such functions may ultimately result in more accurate atomic displacement parameters, but are often unable to cope well with "lumps" in the background due to the presence of amorphous phases, e.g. glass scattering. They have the advantage (and disadvantage for the unwary user!) that they can be used in automatic fitting procedures in which the user does not have to view the profile fit. Refinable background functions are particularly recommended for wavelength-dispersive data where severe overlap of the peak due to small d spacings is common.

In the original Rietveld codes, the background was subtracted at an early stage from the observed profile so that a pre-background-subtracted profile, y_i′(obs) was fitted. Furthermore, regions between the calculated peak positions were simply set to zero with the danger for the unwary user that parts of the observed profile simply vanished. Modern codes tend to include all of the data as observed with the background function being treated as part of the calculated profile. This has repercussions when calculating agreement factors as discussed in the later page on R-factors.

In Rietveld's original paper, there is no mention of any weighting scheme for the profile points: this is equivalent to setting the weights w_i equal to unity in the above equations. However, if the error on each profile point is y_i(err), then correct statistical weights are given by:

When the error on the count is the square root of the count itself, then:

However, you should be aware that "counts" are often scaled during data processing, and so care must be taken to use correct weights. Beware of Rietveld programs that do not require either weights or statistical errors in the data to be included as part of the input information: the codes may be assuming square-root errors even when they are not applicable.

In the original paper by Rietveld, the atomic displacement parameters were assumed to be isotropic. The isotropic temperature factor term in the calculation of the structure factor therefore has the form:

It is quite common for the quality of powder neutron diffraction data to be such that an anisotropic temperature factor is refinable. This led to modification of the Rietveld code by Hewat, who introduced into the equation an expression for anisotropic atomic displacements. You should be aware that, in practice, several equivalent formats are used:

Internally, most computer programs will use the expression based on β (and not B or U) since this clearly provides the most efficient expression for calculation in terms of computer time. (Note that the order of the terms may vary according to version of the Rietveld code used, and that sometimes the factor of two is omitted from the off-diagonal terms.)

The parameters in the Rietveld method may therefore be summarized into two distinct groups as follows:

It is important to realize that the parameters in the left of the table affect mainly peak position and shape, while those on the right relate directly to the reflection intensity and hence to the structure factor itself.

X-rays

The above discussion focussed around the use of angle-dispersive neutron diffractometry to obtain refined structural parameters from powder data. It was soon realized that the same method could, in principle, be extended to the fitting of powder diffraction data obtained using both laboratory and synchrotron X-rays. All that was required was a few minor modifications to the basic code: firstly, X-ray form factors, f(2θ), must be used instead of the neutron scattering lengths, b, in the calculation of the structure factors; secondly, a polarisation correction, P(2θ), must be included in addition to the Lorentz factor for data collected with laboratory X-rays, together with an absorption correction, A(2θ), for data collected in capillary geometry; thirdly, a different (non-Gaussian) peak-shape function is required.

There has been much discussion on the latter subject with respect to laboratory data: the peak-shape function is highly dependent on the instrumental setup. For laboratory diffractometers with a narrow-bandpass primary-beam monochromators, and for most synchrotron diffractometers also, the pseudo-Voigt peak-shape function is empirically found to be most suitable. For diffractometers with only nickel filters or graphite monochromators, peak shapes based on less symmetric functions are preferred; alternatively, peak shapes based on so-called fundamental parameters have been heavily promoted by some code developers.

The applicability of the Rietveld method to the X-ray case has probably led to a much wider exploitation than that originally envisaged. More importantly, the extension to the X-ray case has resulted in better-quality, higher-resolution, X-ray data being collected, together with improvements in the design of X-ray powder diffractometers.

Refinement

The function Δ is very non-linear in terms of nearly all of the parameters. It is not possible to "invert" the function in order to obtain the parameters, so an iterative procedure is used. In order to appreciate how this procedure works in practice, the next page is used to explain some basic details concerning least-squares minimization.

§ L. W. Finger, D. E. Cox, A. P. Jephcoat. A correction for powder diffraction peak asymmetry due to axial divergence. Journal of Applied Crystallography, 1994, 27, 892-900

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