21.1 What is a sequential fit?

In a sequential fit is one where a fit is performed independently for a set of phases using one dataset after another. This is in contrast to a combined fit, where all phases are fit to all datasets. In a combined fit, if more than one histogram is present and each is linked to one or more phases, all linked phases will be fit to all linked and “Used”² histograms. Phases that are not linked to histograms are ignored. Likewise, histograms where the “Use flag” is turned off or that are not linked to any phases will also be ignored. The difference between the two is in how the models are parameterized. In a combined fit, there are separate histogram parameters for each histogram and separate HAP parameters (HAP parameters are explained in §14.7) for each phase and histogram, but only one set of phase parameters must fit all histograms. Thus, when several histograms are fit, the atom positions and ADP values will represent a compromise that best-fits all histograms simultaneously. This it is some sort of average. This is often optimal. In a sequential fit, each dataset will have individually-fit parameters, so atoms positions and ADP values can evolve as the datasets change parametrically, for example, as the temperature is changed. Note that, as will be discussed later in §21.1.2, lattice parameters can be fit separately in either refinement type so lattice parameters are not forced to be the same for every dataset in either type of fit. In both cases, it is possible to designate which phases are associated with individual histograms, so it is possible to model cases where phases appear or disappear. Sequential refinements offer one additional result: a table is prepared with the fit parameters showing how they change, as will be discussed in §21.4. There are a number of nice tools to follow the progress of the fits. A good resource for learning about sequential refinements can be found in the “Sequential refinement of multiple datasets – CuCr2O4 from 7K to 300K” tutorial, https://advancedphotonsource.github.io/GSAS-II-tutorials/SeqRefine/SequentialTutorial.htm. A second tutorial, “Parametric Fitting and Pseudo Variables for Sequential Fits”, https://advancedphotonsource.github.io/GSAS-II-tutorials/SeqParametric/ParametricFitting.htm, shows some of the tools that GSAS-II provides from the results of a sequential fit. The goal of this chapter is not to duplicate those tutorials, but to explain how sequential fitting works and discuss different ways that sequential fitting can be approached.

There are a few important concepts that are worthwhile to discuss before considering the mechanics of performing a sequential fit.

21.1.1 Switching to a sequential fit

By default in GSAS-II, a combined fit is performed. To establish that a sequential fit will be performed, you need to select the Controls data tree entry and in the middle of the data window, as seen in Fig. 21.1 click on the Sequential Settings “Select datasets” button. This brings up a window where you can select histograms. If you select one or more histograms, sequential mode is enabled. Why one might want to perform a sequential fit with only a single histogram will be made more clear later, in §21.3.

Once sequential mode is entered, this section of the data window changes, as seen in Fig. 21.2. Also, the “Refine”command in the Calculate menu is replaced with “Sequential refine.”

The new controls that are available are:

• “Reselect datasets” button: You can change which datasets are included in the sequential fit with this. If for some reason you would want to turn off sequential mode, use this, but do not select any datasets before pressing OK.
• “Reverse order?”: When this is unselected, the selected datasets are processed in the order they appear in the data tree (top down). When this is selected the order is reversed (bottom to top).
• “Copy results to next histogram?”: When this is selected, the fitted parameters at the end of one fit are used as the starting point for the next histogram. After a sequential fit is performed, this is always turned off. The use of this is discussed in more depth in §21.1.3.
• “Clear previous seq. results” button: Results from a sequential fit are placed into a data tree entry. This button deletes that data tree entry from the GSAS-II project. This is discussed in more depth in §21.4.

Note that when a sequential fit is established, only the histograms included in the list of selected dataset are used in a refinement. All others are ignored. Thus, a sequential fit can be performed in stages, where datasets are added in stages. This is different from the combined fit process, where all histograms that are linked to one or more phases will be used in the fit unless their “Use” flag is set to off.

21.1.2 Lattice parameter offsets

A key change between combined and sequential fit has to do with how lattice parameters are treated in GSAS-II. Note that a phase has only a single set of lattice constants, so these constants alone will not treat the case where a phase description is being used to fit multiple datasets where the data were collected under different conditions as the lattice parameters are likely to not be identical at the great sensitivity that Rietveld refinement offers.

As was introduced in §3.4.1, a hydrostatic strain tensor, \(D_{ij}\), allows for delta values to be applied to lattice constants. These \(D_{ij}\) values are HAP values, so there is a set of them for every histogram associated with a phase. The effect of these terms is that lattice parameters are being treated as \(a = a_0 + \delta a_j\) where \(a_0\) is the lattice parameter associated with the phase, and there is an offset \(\delta a_j\) associated with each histogram, except the \(D_{ij}\) terms are actually applied to the reciprocal lattice tensor. I will leave deriving an expression for \(\delta a_j\) from the \(D_{ij}\) terms in the general triclinic case “as an exercise for the reader.” (No partial credit for the cubic case.) An example showing how the lattice parameters and the \(D_{ij}\) terms are displayed in the GUI is shown in Figs. 21.3 and 21.4, respectively.

These hydrostatic strain tensor (\(D_{ij}\)) terms are essential for sequential fits. Should you attempt to refine the lattice parameters in a sequential fit, a warning message will be displayed and the software will offer to turn off the lattice parameter flag and turn on refinement of the \(D_{ij}\) terms. The software will not allow a sequential refinement where lattice parameters are refined directly.

21.1.3 Use of “Copy results to next histogram?”

. When performing a parametric refinement, a model that fits the first dataset may not be a good starting point for a data set collected much later in the series. For example, the lattice parameters may have changed sufficiently, that the peak positions no longer are close enough to the computed reflection positions that refinement can progress.³ This “Copy results to next histogram” option will take the results from the fit of histogram \(n\) and copy those parameters as the starting point for histogram \(n+1\). Even if the starting point parameters in a parametric fit would be close enough for a fit to progress, if the datasets are showing a orderly progression in changes, starting from an adjacent set of parameters will cause a more rapid convergence.

Note, however, that you will only want to use this option for the first time that a set of histograms are fit. Once the a refinement has been run on a set of histograms, the initial values have been optimized and use of this option would reset the parameters, which is probably not what you want. For this reason, the “Copy results to next histogram?” setting is automatically cleared after a sequential fit is performed. However, if you are performing a sequential fit in stages, you may want to turn this back on when you select a new set of histograms to use in a sequential fit.