The goal for a background fitting function is that it should produce a smoothly varying intensity curve over the range of \(2\theta \) or TOF values and that the complexity should be something that you can select as needed for the fit. GSAS-II provides four different approaches for computing background scattering. In theory, all four could be used for fitting background in a single powder diffraction pattern, but I suspect no one will ever do that. It is fairly common, however, to use two of the approaches together.
GSAS-II provides a number of functions that can be used to provide smoothly-varying curves:
* “chebyschev”, * “chebyschev-1”, * “cosine”, * “lin interpolate”, * “inv interpolate”, * “log interpolate”.
In addition to selecting one of the above functions, You able to choose the number of terms and if the terms will be refined; all terms are refined when the refine flag is selected. The more terms that are allowed the more “wiggly” the functions can be. The problem with using too many terms is that by allowing the function to be too flexible, the baseline may be refined to cut through a peak, or more commonly a set of peaks, that compute with more intensity than was observed or the opposite, to add extra intensity to a set of peaks by placing the background too low if the opposite occurs. The problem goes away if the crystallographic model fits the peaks well, but a poor background fit can prevent you from obtaining a good crystallographic fit.
You are recommended to choose the chebyshev-1 function. The original implementation of the Chebyshev function (labeled chebyshev) does not refine as well, so that is not as good a choice, but that older function has been retained in the software so that older refinements will be unchanged if rerun. We may remove the older function at some point.
Chebyshev polynomials provide smooth spline-line curves and are chosen because they refine well – the terms are all orthogonal, but it could be possible that for a particular instrument or sample, another function could perform better. One good way to investigate this is to use fixed background points (see §11.3). Once a set of fixed background points have been defined, one can fit with each of the functions quickly and see if any produce a better match to the desired background shape.
This option uses the Debye diffuse scattering equation to compute a multi-peaked broad background function that is commonly seen from scattering by amorphous materials. This can be quite useful for materials where the background has a large contribution from one (or more) amorphous phases.
The Debye diffuse scattering equation is:
\[ B = \sum _{i} A_i \frac {\sin {QR_i}} {QR_i} \exp {(-U_iQ^2)}\]
where \(B\) is the background, \(Q = 4\pi /d\) and there are \(i\) sets of adjustable terms \(A_i\), \(R_i\) and \(U_i\), which can be individually selected for refinement, as desired. Note that the \(A_i\) are multipliers that set the overall background intensity, \(R_i\) corresponds to an interatomic distance of sorts, in Å and the \(U_i\) dictates how quickly the intensity of the Debye function falls off with Q. An example of the background computed by this function is shown in Fig. 11.1.
More complex background curves are possible by adding more than one term in the Debye equation, as seen in Fig. 11.2. Note that in both figures, the baseline has values well below zero – which is not physically possible. Thus, the Debye scattering equation must always be used with another function to offset it to be positive everywhere. The chebyshev-1 function with only a single term is sufficient for this, but I would likely use either 2 or 4 terms.
These background corrections provide a set of individual background peaks that can be placed where needed. When a broad peak is seen in the background, it is much better to fit it with this option rather than to try to introduce enough terms into a Chebyshev polynomial (etc.). The background peaks allow one or more pseudo-Voigt profile peaks to be included in the baseline. The parameters for each peak are “pos” (peak position), “int” (integrated intensity), “sig” (Gaussian width) and “gam” (Lorentzian width). The definitions for “sig” and “gam” are the same as for “sigma\({}^2\)” and “gamma” in peak fitting (see §17.1). Note that GSAS-II allows both “sig” and “gam” to be selected for refinement together, but it is quite uncommon to fit both. I usually use sig only. The default values for sig and gam are 0.10, which generate very sharp peaks. For typical background use, values of 100 to 10,000 for sig are more common. You may want to manually adjust them accordingly to the kind of peak you are trying to fit before trying to refine them. It is common to combine background peaks with the ’chebyschev-1’ function. Addition of a small number background peak(s) to account for broad “lumps” in the pattern, such as what arises from Kapton scattering or with other sample container peaks, allows the chebyschev-1 function to fit a smooth, slowly-varying baseline with a small number of terms.
Spurious sharp peak(s) will occasionally occur in diffraction data from a sample container or furnace, etc. It can be easiest to model these as an impurity phase, possibly using Le Bail fitting if the material is highly textured, but if the scattering for these peaks arises from a location other than the center of the Bragg circle, the peak positions may be shifted and treating them as a phase will not work. Measurement of a fixed background scan (see next section) may allow the peak to be removed, but if the peak needs to be modeled, use of background peaks is the only other choice; none of the smooth background functions can fit a sharp feature.
Background peaks need to be refined with care. I will usually place them into the pattern and may adjust the values for int, pos and sig manually to get them into the right range. Since the background is updated any time new values are entered into this window, it can be very instructive to change the values to see what a background peak is contributes. I will sometimes boost the “int” to make sure that I can see where a background peak is located and confirm that the width and position are on the order of what I want before doing any refinement. I will usually refine the int value, then add the sig value and finally add the pos value to the refinement later, looking to make sure that the numbers remain reasonable, but there are times when refining pos is not possible. Use of fixed background points (see §11.3) can be very helpful with background peaks, as the refinements are instantaneous.
Another way to treat background is to experimentally measure the background scattering from the instrument without a sample and then import that measurement as an additional histogram. This can be a very useful thing, when working with “background-rich” data collection environments, common with operando or in situ experiments, where ancillary equipment (furnaces, refrigerators/cryostats, reaction cells, etc.) can produce significant amounts of scattering that can have sharp features that are hard to model. One might think that simply subtracting the observed background from empty ancillary equipment should account for all background, but there are several issues that this will not address. The first is that the absorption and scatter from the sample reduces the amount of radiation that can be scattered from the ancillary equipment. This means that the measured background will usually be a slight overestimate for what is actually observed from that source. Also, this measurement will not treat the additional background that arises from the sample. For this reason, it is not a good idea to directly subtract the measured background from the diffraction signal as part of the measurement process. (Note that background subtraction would also require increasing the s.u. values on each data point.) A scaling factor for the background histogram is included in the GSAS-II fixed background treatment that can help treat the difference between the measured background. This value can be refined. Since the fixed background will not include sample background scattering and since the sample can subtly affect the instrumental background in ways other than attenuation, one will usually need to also refine a Chebyshev polynomial with at least a few terms.
Note that the background histogram will be subtracted from your observed pattern, point by point. This means that the two patterns must have the same point spacing and the same start and ending values in \(2\theta \) or TOF. It is not necessary that the counting times be the same, as the background pattern will be scaled.
GSAS-II does provide another way to generate a fixed background histogram, using the auto-background feature, (see §11.4).
