Historically, there have been four different approaches used for computing powder diffraction peak profiles numerically to fit powder diffraction so that patterns may be fit.
The ad hoc approach for fitting is less than ideal because it can allow inclusion of effects that are not physically possible. As an example for why this is the case, the asymmetric peak broadening in low angle diffraction peaks due to axial divergence not only causes the peak to have the low-Q and high-Q sides of peaks to have different widths, but it shifts the peak positions as well. Since these shifts will not be accounted for in the ad hoc equations, the fitting software will not compute the lattice parameters optimally. Use of learned peak shapes can handle aberrations that introduce irregular aspects to the peak shape, but it fails to handle anisotropic or Q dependent broaden effects. The greater use of machine learning techniques in recent years may mean that this approach may yet see a future renaissance. The physically-inspired approach was used in Hugo Rietveld’s initial codes and has been augmented over the decades as the basic physics description has improved. It is available in all of widely used Rietveld codes. Despite this, the FPA method has several nice features. Only a small number of codes implement FPA-generated peak shapes, but those codes are seeing wide use. If an instrument is well described, then no fitting parameters are needed to model a pattern’s peak profiles, other than for sample-related effects. The FPA method has the potential to reproduce minor deviations of the profile from the shapes produced from even the most advanced P-I functions. Creating the FPA model for a novel instrument is not a simple task, even for an FPA expert, but once done FPA fitting is unmatched for proving that an instrument is performing at the level expected for how it was designed.
However, my personal opinion is that the improvements in peak shape fitting from FPA for modern commercial instruments are of negligible value to improve the accuracy for the parameters derived from Rietveld analysis. For this reason, our choice in GSAS-II has been to use a physically-inspired peak shape model. It fits the profiles from laboratory instruments, as well as very specialized synchrotron and neutron instruments, well enough that the sample can be very accurately modeled and characterized; this is the goal for Rietveld analysis.
The one valid criticism of the GSAS-II model is that fundamental parameters profile computations have the ability to describe instrumental peak shape functions that are not well-fit by the pseudo-Voigt function that GSAS-II uses. However, modern diffraction instruments rarely have significant deviations from a pseudo-Voigt function peak shape and even where that can be noted, the contribution of the slight misfit to the model fitting results will be negligible. However, in theory with FPA one could develop sample broadening models that account for more complex treatment of sample broadening (for example, a bimodal crystallite size distribution or where defects create asymmetric broadening), but as far as I am aware no one has developed a generalized methodology for this with FPA.