The forerunner to modern P-I functions was work by Caglioti et al. in 1958 (the year of my birth!) where he and co-workers characterized the instrumental resolution expected from mosaic neutron monochromators and found that broadening should be Gaussian and follow a curve where the resolution minimum is close to the monochromator takeoff angle. From this Hugo Rietveld developed a resolution curve where the Gaussian broadening function is written as \(\sigma ^2\ =\ U\ \tan ^2{\theta }\ +V\ \tan {\theta }+W\), and this expression is now known as the Caglioti equation. In this equation, \(\sigma =FWHM\ /\ \sqrt {8\ln {2}}\) (\(FWHM\) is the full-width at half maximum) and \(U\), \(V\) and \(W\) are terms determined for the instrumental configuration. This equation was not developed for x-rays, but in truth Caglioti equation is simply a polynomial and has the flexibility to fit instruments of all types. Ted Prince pointed out that the fitting process is much more stable if the equation is reformulated as \(\sigma ^2\ =\ U^\prime \ \tan ^2{\left (\theta -\theta _m\right )}\ +V^\prime \ \tan {(\theta }-\theta _m)+W^\prime \), where \(\theta _m \) is the monochromator “takeoff” angle. Ted was completely correct on this, but no one other than he ever implemented the modified equation in a Rietveld code. Whenever you find fitting the \(U\), \(V\) & \(W\) terms to be difficult, I invite you to curse the code developers (alas, I am on that list) who did not listen to Ted.
When Rietveld analysis was initially adapted to treat x-ray data, it became clear that the pure-Gaussian peak shape initially used was inadequate for treatment of x-ray line shapes. This is because the peaks had the longer tails seen in a Lorentzian function (this is also sometimes called a Cauchy function). This is mostly because the better resolution of x-ray instruments at low angles in comparison to the neutron instruments at that time was better able to show sample broadening effects, which are largely Lorentzian. After considerable discussion in the literature, the Voigt function, which is a convolution of a Gaussian and Lorentzian function, was accepted as a good descriptor for powder diffraction peak shapes. The Voigt function is relatively slow to compute, so a substitute for this is commonly used, where a Gaussian and Lorentzian function are multiplied rather than convoluted. This produces a peak shape that is quite close to that from a similarly parameterized Voigt function. The powder diffraction community calls this product a pseudo-Voigt and it is a good approximation for the Voigt. One of my early activities in Rietveld analysis was to take a Rietveld code that used a pseudo-Voigt peak shape and add an option to instead compute a Voigt peak shape function instead. The result was a refinement that was considerably slower than the original where improvement in the fit was nearly imperceptible. My option was never used after that.
The commonly used form for the Lorentzian contribution to the profile is \(FWHM\ =\ X\ /\ \cos {\theta }\ +Y\ \tan {\theta }\). Note that X and Y are the broadening contributions expected from crystallite size and microstrain broadening, respectively, and thus might not be the best choices for describing instrumental broadening, but these two terms together provide quite a bit of flexibility in fitting the broadening, particularly considering that Lorentzian contributions to instrumental broadening tend to be minor. As will be discussed further, below, GSAS-II introduces a third term to offer even more flexibility in the treatment of CW broadening, but it is not clear that this is ever needed.
One other significant improvement seen in peak shape functions was treatment for the asymmetry seen in low angle peaks due to axial divergence. The history here is that Hugo Rietveld introduced an ad hoc correction for low angle asymmetry that did not work very well. Several other unsatisfactory treatments were tried later. The physical cause of this low angle asymmetry was well understood quite early (and is well explained in Klug & Alexander’s 1974 book) but was first formulated in integral form in 1984 by Van Laar and Yelon, but that formalism was not conducive for peak shape computation. In 1986 Eddy, Cheetham and David published a paper demonstrating that low-angle asymmetry could be fit well using the Van Laar and Yelon approach, but did not make their code available. In 1994, Finger, Cox and Jephcoat, who were not then aware of the Eddy et al work, revisited the Van Laar and Yelon work, correcting some of the math. Larry Finger, who was a friend and prized role model to me, not only published this work, but also distributed their peak shape computation code and encouraged its employment in other software. The result is that this work is now known as the Finger-Cox-Jephcoat (FCJ) correction. While this does short-change the contribution from my very much respected friends Mike Eddy, Tony Cheetham and Bill David, I do use this naming as well; there is a moral somewhere in this about the value of putting one’s ideas into tools that others can use.
The FCJ correction requires three parameters, the diffractometer radius (L), the length of the sample (S) and the length of the detector slit (H), where these lengths are measured along the diffractometer axis of rotation (perpendicular to the diffraction plane). The radius is used as a ratio with each of the lengths so there are really only two free parameters. These parameters in theory are known and should not need to be fit, but indeed must be adjusted when x-ray or neutron focusing is used. The two lengths have very similar effects on asymmetry and thus usually cannot be refined together.
These three contributions, the \(U\), \(V\), \(W\) parameterized Gaussian function, the \(X\) and \(Y\) parameterized Lorentzian function, and the FCJ correction, form the backbone for P-I descriptions of peak shapes for constant-wavelength neutron and x-ray instruments in most Rietveld software, though some additional terms may be incorporated.
Things are quite different for TOF neutrons, which require a much more complex formulation to account for the performance of the neutron moderator, the neutron guides and path length of the instrument and the arrangement of detectors. Many different approaches have been used. The specific model used in GSAS-II was developed by my partner in GSAS-II development, Bob Von Dreele, based on his unique expertise in TOF instrumentation that goes back to the initial development of the technique. Note that the parameters used to fit the instrumental broadening for a TOF instrument are determined by the facility and instrument design, which is largely fixed, though perturbations may be offered through changes in collimation options and frame selection via chopper settings. Fitting the TOF profile terms can be a tricky process, since more than one combination of values can produce comparable fits. This fitting is best done periodically by an instrument scientist, who will supply profile values along with reduced data suitable for input to a refinement.