The creation of GSAS-II provided us with the option to rethink how profiles are handled in Rietveld. In earlier programs, such as GSAS/EXPGUI, a single set of profile terms are provided to describe both the instrumental and sample broadening effects. Even when broadening for an instrument was well-characterized, as is common for user instruments at neutron sources, the mixing of instrumental and sample contributions in the refinement stage made it difficult to discern which was which.
This was not a serious problem with the initial role for Rietveld fitting, where a diffraction dataset was collected (perhaps with less care than desirable for the reproducibility in the instrumental settings,) from a material that consisted primarily of a single phase and where the goal was a crystal structure determination for that primary phase. Fitting of broadening was a necessary chore, and learning the profile parameters was not typically a goal. We now use Rietveld for so many more types of measurements. When considering a more common contemporary use-case for Rietveld, we are likely to have data from a well-characterized diffractometer on a multiphase material. Rietveld is commonly employed for materials where the atomic structures are well understood, but we want to learn about materials effects, such as microstrain or phase composition or details for site substitutions or lattice constants or perhaps how the material evolves with time or the application of temperature, an external electric field, pressure, etc. From this perspective we realize that the instrumental contributions for broadening should be the same for all datasets from that instrumental configuration. Further, in the ideal case those effects will be known and will not require fitting. On the other hand, crystallite size and microstrain will not be known in advance, will be different for every phase and may change over the course of a parametric experiment. Further, more complex models for anisotropic sample broadening may be needed for some phases, but not others. From this perspective it becomes clear that one wants the ability to separate instrumental and sample broadening contributions.
For this reason, GSAS-II associates instrumental broadening terms with each PWDR (powder) histogram, where the initial values for the parameters are read from the instrument parameter file used when the dataset histogram is initially imported. The sample broadening terms are considered both a property of the phase (as there is no reason that two different phases would be expected to have the same crystallite sizes or microstrain) but also a property of the histogram, since the specimen used for each measurement might have differing sizes or morphologies. Thus, there are independent values for sample broadening for every histogram in every phase, so if there are \(m\) histograms and \(n\) phases, there will be \(n \times m\) sets of sample broadening terms. Bob and I call such parameters HAP parameters (histogram-and-phase) in a name that dates back to the original GSAS program. Note that where one wishes to force sample broadening values to be the same, for example where the same specimen is used in multiple histograms, constraints can be used to group the values. These HAP parameters are placed in a location of the GSAS-II data structure associated with each phase, but there is some choice on how they appear in the GUI. This will be discussed further below.
GSAS-II places the instrumental broadening terms into the “Instrument Parameters” section of each PWDR (powder) histogram. Different parameters are displayed depending on the instrument type, which is determined by the instrument parameters file contents that was read when the histogram was input. It should be noted that there are terms here that are not going to be needed under normal circumstances; a common beginner mistake is to assume that “if it is there I should refine it”. For example, the Zero term is retained for CW work largely for historical purposes. There are exceptional cases where this might still be used, when particular care has been used to measure it as part of a calibration, but I have only seen this done in standards development. Likewise, the Z term is present for completeness in those correction equations, should the need for them arise in some future instrument. You are quite unlikely to improve a fit by using it. The difB term was found to be needed to properly calibrate the unique SNS POWGEN instrument, but has not been needed anywhere else to date.
The instrumental parameters for sample broadening for constant wavelength x-ray and neutron histograms depend on radiation source. For monochromatic instruments one set of parameters is used, as shown in Table 15.6, but for dual wavelength instruments (typically CuK\(\alpha _1\)-CuK\(\alpha _2\) instruments, a second set is used, as shown in Table 15.7. Note that not all these parameters affect the peak profile; the wavelength and zero values determine peak positions and the polarization term affects peak intensities.
As noted earlier, time of flight diffraction has asymmetric peaks due to the way that neutrons pulses are generated. The parameters used for TOF peak shapes are tabulated in Table 15.8. These terms are somewhat complex but their use is easy, since they are determined using calibration standards by an instrument scientist who is an expert on that diffractometer. Instrument users will read their values from the instrument parameter file and simply use them modified. The values will provide a good fit to the instrument peak shape ”out of the box.” The refinement flags for these parameters are only to used during calibration. (Note that rarely, and typically only with highly absorbing samples, difC may be varied by users, when needed to provide an improved fit to data from highly absorbing samples.)
The TOF peak shape is generated from convolution of double exponential and pseudo-Voigt. The exponential parameters are split between the “rise side” (shorter TOF, or equivalently high Q) and the “decay side” (longer TOF or small Q). On the low TOF side of the peak, there is one parameter a, which is multiplied by d*. On the high TOF side of the peak, there are three terms, beta0, beta1 and betaq, which are scaled by 1, d*4 and d*2, respectively. The pseudo-Voigt is parameterized where the variance for the Gaussian contribution to the peak, \(\sigma \), (\(\sigma =FWHM\ /\ \sqrt {8\ln {2}}\)) is determined from \[\sigma \ =\ \sigma _0\ +\ \frac {\sigma _1}{d^{\ast 2}}\ +\ \frac {\sigma _2}{d^{\ast 4}}\ +\ \frac {\sigma _q}{d^\ast }\] and the Lorentzian contribution, \(\Gamma \), (\(\Gamma \) = FWHM) is determined from \[\Gamma =\frac {X}{d^\ast }+\frac {Y}{d^{\ast 2}}+Z\] There are theoretical reasons for the choice of these functions, but the parameterization is largely determined from experience with a variety of TOF instruments. Most instruments do not require all these terms for a good fit, but each of these terms has been needed for a good fit for at least one instrument, with the exception of Z, which is there “for future need” should that arise.