Before considering approaches for fitting the powder pattern intensities, it is worthwhile if I provide an overview of the factors that can change the intensities in the computed pattern, as these are the factors that will need to be optimized.
The intensity scale of an experimental powder diffraction pattern is determined by experimental conditions and the computed pattern, which is on an absolute basis must be matched to this. For this reason an arbitrary scaling parameter must be included to match the intensity scales of the two patterns. In most cases, a scaling constant will be needed for every powder diffraction pattern.
In GSAS-II there are two scaling parameters for every histogram,
In the case of a single-phase fit, either of the two can be fit, but not both. In the case of a multiphase fit, there are three choices:
The option to choose between these three is yours to make by personal preference. They will provide completely equivalent results. Note that in a multi-histogram fit, a scale factor and/or phase fraction(s) will need to be refined for every histogram.
The effect of this scaling is simply a multiplication of the pattern by a constant to obtain the best agreement between the computed and observed patterns; there will be no changes to any relative peak heights. When peaks are well aligned and the peak shape is well matched, the scale factor will refine in a single cycle to the optimum value, but when this is not the case, the scale factor fit will continue to be improve as the peak position and the profile fits are optimized.
While peak profiles do not change the integrated intensity of peaks, the appearance of the fit will change as profile shapes are modified. A peak that is overly broad will likely appear to be too low in height. The diffraction intensities cannot be well-fit if the profiles are not well-fit.
As we saw before, the diffraction intensities are determined by \(I_{hkl}\) which is determined from the square of the structure factors, \(F_{hkl}^2\), which is in turn determined by that atom positions in the unit cell:
\[F_{hkl}=\sum _j^{N^\prime } f_j m_j\exp [-2\pi i(hx_j + ky_j + lz_j)] \exp [-2\pi i(\vec {Q}\cdot \vec {U_j})]\] where we sum over the asymmetric unit (the unique atoms in the cell, as discussed in §3.13). Note that the atoms that will have the largest impact are the one that have the greatest values for \(f_j\) and \(m_j\), the scattering power and the multiplicity, respectively. The former, \(f_j\) is the number of electrons and thus the position of the heaviest atoms have the highest leverage for x-rays, unless their multiplicity is very low. The highest symmetry sites, with lower multiplicity relative to the general site, typically have few or no degrees of freedom for refinement of coordinates, however. For neutrons, the scattering length depends on the isotope and the leverage for the atom positions will be usually be very different from x-rays. Thus, we can expect that if the position of atoms within the unit cell change, then the intensities of the powder diffraction pattern will change, but the positions of some atoms may affect the intensities far greater than others. The effect of moving atoms will not change the total amount of scattering but will change distribution of intensities, possibly very significantly.
While it is occasionally possible to have data of sufficient quality and with simpler structures where it anisotropic ADPs can improve a structural model both in terms of fit and quality, powder diffraction almost always uses \(\rm U_{iso}\) values. This allows the second part of the structure factor equation to be simplified to \(\exp [-2\pi iQ{\rm U_{iso,j}}]\). This is a damping factor for each atom’s scattering contribution. At low values Q it is close to 1 and drops to smaller values as Q increases. The larger the value of \(\rm U_{iso,j}\) for an atom, the more rapidly with Q that the contribution for that atom is reduced in the summation and thus the powder diffraction pattern. Since the contribution of any one atom to the powder diffraction pattern can be subtle, the impact of changing \(\rm U_{iso}\) is even more subtle, but will manifest as a change in high Q intensities in comparison to those at low Q.
The scattering factor equation we have used so far does not include a term for the occupancy of each site, but how this is handled is to add a factor, \(o_j\) that simply multiplies \(f_j\), as discussed in §3.7. It should be noted that \(o_j\) will decrease the impact of that atom’s contribution to the \(F_{hkl}\) summation. It differs from the impact of \(\rm U_{iso,j}\) for that atom in that \(o_j\) will decrease the contribution for all reflections equally while \(\rm U_{iso,j}\) will have largest impact at high Q. Unless data have been collected over a wide range in Q, it can be difficult to discern the difference between these two factors. This leads to a high correlation between \(\rm U_{iso,j}\) and \(o_j\), meaning that the effect of lowering \(o_j\) can largely be offset by lowering \(\rm U_{iso,j}\) if the measurement range for Q is not large. Thus, experimentally it can be very difficult or even impossible to determine both values simultaneously. On the other hand, when there are a significant level of vacancies for an atom and the lowered occupancy of this site is not addressed in the model, then obviously \(o_j\) is higher than the correct value. The fit may compensate for this by increasing \(\rm U_{iso,j}\) to a larger than expected value. Likewise, if there is more scattering from an atom site than is in the model; this can happen when an atom type incorrectly assigned to one with less scattering power, then \(\rm U_{iso,j}\) will compensate by refining to a smaller number, including the possibility of refining to a negative value.
Texture in materials can be a very useful property, adding strength or other desired properties and many diffraction experiments are performed to characterize texture, but for crystallographic structure determination, texture is largely a pain. Use of sample preparation techniques that minimize it is to be encouraged. The affect of texture will be to lower the intensity for classes of reflections for crystallite orientation directions that have less abundance than would be expected, and move that intensity into the other classes of reflections for directions that are overrepresented. While simple and strong texture has a clear effect on diffraction intensities, and when untreated it can be visible in the “Rietveld plot”, where the reflections in a particular zone have too much or too little intensity, but in practice I find it is usually very hard for me to see that untreated texture is present, so I test by seeing if a refinement improves the fit.
As noted previously in §5.2, absorption does not affect intensities from Bragg-Brentano diffraction, but can significantly affect Debye-Scherrer diffraction intensities, though sample absorption does have the same general form as the Debye-Waller factor and thus correlates strongly with \(\rm U_{iso}\) values. GSAS-II provides a sample parameter that is an absorption correction for cylindrical samples. For CW measurements, the absorption correction value is \(\mu R\) (unit-less), using the actual density of the sample, which includes voids due to sample packing.
For TOF measurements, the correction varies with both angle and wavelength and the absorption correction value is \(\mu R/\lambda \) (units of \(\mathring {\rm A}^{-1}\)). Note that this correction can only be made when the data have only minor time-focusing corrections. The angle reported for the histogram should be close to the actual angle for all detectors included in that histogram and should ideally be the average Bragg angle across the included detection region. As far as I am aware, the only TOF instrument where this is not true is POWGEN. This wavelength-mixing at POWGEN also creates a problem for extinction corrections and adjusting for neutron resonance edges for isotopes where that occurs, since this also requires wavelength-dependent computations. Further, mixing together data with significantly different wavelengths degrades instrumental resolution because when high-resolution data are added to lower resolution data, the result is lower resolution data.
The absorption correction can be refined, but due to the correlation with \(\rm U_{iso}\) values, refinement of the absorption correction is usually only possible when the \(\rm U_{iso}\) values are not refined. So, for highly absorbing samples, you are strongly encouraged to measure the actual density of the sample used for diffraction so that the absorption correction value is a known quantity. I have not tried use of a general restraint on the average \(\rm U_{iso}\) value, but I wonder if that would allow refinement of \(\rm U_{iso}\) and sample absorption.