Usually the first step in intensity fitting will be to refine the scale factor, often combined with the background fitting and in the next run to add phase fractions, if multiphase. In the worst cases, it may be best to find an initial reasonable value for the scale factor manually, until the cell and perhaps profiles can be improved, as will be described in §12.1.1.
As mentioned before, the diffraction intensities cannot be well-fit if the profiles are not well-fit. As has been discussed in the chapter on fitting profiles (Chapter 14) a Le Bail or Pawley fit (described further in the next section) may be helpful to obtain accurate peak profiles before attempting to fit peak intensities.
If there are multiple phases in the material, be sure to refine phase fractions before trying to optimize structural parameters for any of the phases. While one can optimize structural parameters for several phases at the same time, particularly if the phases have minimal overlapping peaks, until you are experienced with Rietveld, my recommendation is to work with one phase at a time and to tackle them in the order of greatest mass (weight) fraction to lowest.
It is common that atoms will be missing from the model, particularly with ab initio structure determination. While it may be necessary to work on structure completion at later stages in the fit, if it is clear that atoms are missing, it makes sense to view the structure visually (see Chapter 18) and perhaps examine a difference Fourier map (see §20.2) to look for positions of missing atoms at this stage, as the refinement will proceed much better with all atoms in the model.
Refinement of atom positions should only be attempted after the background, peak shapes and peak positions are well fit. There should be no significant unassigned peaks that would indicate the presence of an impurity phase or a lowering of symmetry for the phase (see Chapter 28 for a discussion of subgroups). While the isolated peaks from an impurity would be expected to have minimal effect on the model for other phases, the probability is significant that there are also peaks from the impurity superimposed on the other phases. Such unaccounted for intensity would throw off the fitting of the fitted phase.
Start the fitting by optimizing the coordinates for the most significant scatterers in the phase, as noted by their larger scattering power and higher multiplicity. To add the atomic coordinates into the refinement in GSAS-II, select the phase in the data tree and then select the “Atoms” tab on that data window. Double-click on the refine box for the atom to be optimized and a pull-down menu will offer choices for refinement flags, “F”, “X” and “U” and their combinations. Select “X” to refine coordinates. Note that GSAS-II will handle site symmetry constraints for you. If the atom is a special site, say (0,y,1/2), only the y value will be refined. If the coordinates are (x, 1/2-x, z) then the relationship between x and y will be maintained. The only exception for this is with polar space groups, where a coordinate must be fixed. This will be discussed separately in §3.16. As atoms are fit and the atom coordinates converge, you can add more weakly scattering scattering atoms into the fit.
It is wise to keep an eye on the bond distances for the structure to make sure that the model is not distorting in unreasonable ways. Bonding values are available on the Atom tab, using the Compute/“Show distances & angles” menu command. This can happen either because the model that you are using is not the correct one for the material, but it can also happen when the model is correct, but the data are not sufficiently sensitive to some of the atom locations to allow the full structure to be fit without constraints or restraints. The difference between these two cases is usually clear. If the structure is wrong, then nothing will produce a really good fit, but the quality of the fit improves significantly as the structure distorts. When the problem is that the data are not sufficient, the distortion of the structure is only producing very minor improvements in the fit. In this latter case, one must either remove structural parameters from the model (which can be done by introducing constraints (see Chapter 24), or simply by not varying parameters) or introduce restraints that “push” the fitting towards a reasonable structure, as described in Chapter 25, on restraints.
If missing atoms could not be located previously, once the atom positions have been fit, at least for the most significant scattering atoms and better for all atoms, it is worthwhile to look again for sites of missing atoms. As before, view the structure visually (see Chapter 18) and examine a difference Fourier map (see §20.2) to look for positions of missing atoms.
Refinement of ADPs with powder diffraction can be tricky, particularly with larger structures or the “lightest” atoms in the structure (with neutrons, these are the atoms with the smallest magnitude scattering lengths), and commonly I will group atoms either by element type or by chemical environment (this is discussed further in Chapter 24), but for an initial refinement it is worthwhile to “let it rip” and refine them all individually. You want to look for negative \(\rm U_{iso}\) as that can indicate sites where there is not enough scattering density. A single strongly negative \(\rm U_{iso}\) value could indicate that the element assignment for that site is wrong and should be changed to place more scattering density on that site. More commonly, I will see most or all the \(\rm U_{iso}\) values in a structure as very small, including a good fraction that are negative. This is often a symptom that the background needs attention. This is discussed further in §11.5. Although negative \(\rm U_{iso}\) values are often criticized by referees, I am personally more concerned with \(\rm U_{iso}\) parameters that refine to very large values. A very large \(\rm U_{iso,j}\) value (remember that \(\sqrt {\rm U_{iso}}\) is on the order of the root-mean-square distance in Å) could indicate that an atom site is disordered, is partially occupied, or even that there is no atom at that site. One of my “tricks” for evaluating if any atom is present at a site is to set the \(\rm U_{iso}\) value to something reasonable and then refine the occupancy. If the occupancy is very low, I will be have high doubts if that element is really present at that site. One can take a guess as to what element might be present, if at full occupancy, by looking at the ratio of the number of electrons or the scattering powers, but note that some elements have negative scattering lengths. The scattering length for Ti is -3.37 fm while 3.64 fm. So if one of these elements is placed at a site that is really populated by the other, one would expect an occupancy in the vicinity of -1.
You may ask, what then is a reasonable value for \(\rm U_{iso}\)? Alas, this depends on the type of material, the mass of the atom and temperature where data were collected. Commonly, for light atoms such as C and O at room temperature in organic or organometallic materials, a \(\rm U_{iso}\) value of 0.015 to 0.035 Å\({}^2\) is common, but for a heavy atom in a dense inorganic material a value of of 0.002 Å\({}^2\) could be reasonable, particularly at 20 K. You are best off looking at reported values for similar types of materials collected at similar temperatures, but give more credence to neutron studies as their wider Q range makes the values more reliable.
For a well-behaved structure, one can refine all \(\rm U_{iso}\) for all atoms with significant scattering power together after the coordinates have been refined. For less robust refinements, one needs to add \(\rm U_{iso}\) values for the strongest scatterers first. While I have recommended at least seeing the ranges of the refined values allowing all \(\rm U_{iso}\) values to refine, it is quite common that I will group them by atom type after that, particularly if there is a wide range in the fitted values and the \(\rm R_{wp}\) or other fit metrics only show a minor change when the values are changed.
What constitutes a significant scatterer depends on the probe and the material composition: H is almost always next to invisible for x-rays while V at natural abundance is next to invisible for neutrons. Likewise, with x-rays the O atoms are not major scatters in \(\rm U_3 O_8\). These atoms probably have \(\rm U_{iso}\) values that can be refined. Certainly, if an atom does not have sufficient scattering power for its position to be refined independently, the \(\rm U_{iso}\) value also will not be possible to refine either.
The first thing that I should point out is that site occupancies are determined on a relative basis. If you set all occupancies to 0.5, or 2 for that matter, the refinement will produce exactly the same result, once the scale factor compensates by changing, also by a factor of 2. Thus, when varying occupancies, one must assume that the occupancy for at least one atom is known and will be fixed. This is usually not a problem, because in materials that have vacancies usually only one type of atom will not be at full occupancy.
Likewise, when two elements share a site one might wish to refine the occupancy for each species as well as the amount of vacancies, but this also can’t be done with a single x-ray or neutron measurement. To understand why, consider that an x-ray diffraction measurement at best effectively measures the electron density at each site in the unit cell. (Neutrons measure the nuclear scattering.) If we know only the number of electrons in a site, except for the “edge case” where that number requires 100% of the higher-Z element, there are an infinite number of combinations of occupancies for the two species, combined with vacancies that will match that total. Hence, with one diffraction measurement, the number of vacancies cannot be determined. With two measurements where the elements each have different ratios of scattering power, for example with both an x-ray and neutron dataset, the occupancies can be varied. Likewise, a resonant scattering experiment (see §2.4) near the edge for one of the elements can provide some sensitivity for vacancies, but given that the change in scattering powder at edges is not that large, this will not usually be as good as the combination of neutrons and x-rays. Occupancy measurement for three elements sharing a site is the same problem as two elements and vacancies and also requires two differing types of datasets. Three elements and vacancies would need three datasets to fit.
Since vacancy and site sharing problems are frequently under-determined, what can one do? With two elements sharing a site, usually one must assume that the site is fully occupied and create a constraint that forces the sum of the occupancies to be 1 (see Chapter 24) or sometimes site sharing or vacancies arise due to valence constraints. One can design a constraint that forces the sum of charges to be a fixed value. The restraints implementation in GSAS-II offers compositional restraints (see Chapter 25 which will push the model towards the believed (or better, measured) composition. The compositional restraints can also be used for charge balance. Restraints and constraints are not ideal choices, as assumptions must be made, but when faced with a set of measurements that cannot fully answer a question, one must make assumptions.
Note that since we are measuring the number of electrons (or nuclear scattering power), we do not have much sensitivity for elements with similar scattering power. A classic problem that cannot be solved from powder diffraction is the assignment of Si and Al to sites in a zeolite. With x-rays, Al with 13 electrons and Si with 14 electrons have very similar scattering power. Powder diffraction is just not sensitive enough to distinguish this 7% difference. With neutrons, the difference is a bit larger, with scattering lengths of 3.45 fm and 4.15 fm, respectively, but even this 18% difference is close to the limit for the technique.
Refining site occupancies can be delicate. While, in general, I try to add more parameters to a fit as the refinement progresses, in the case of occupancies, sometimes I will initially only refine the occupancy for an atom without refining coordinates and \(\rm U_{iso}\) for that atom. The reason is that as the occupancy drops, the leverage for that atom drops and it becomes easy for that atom to become “lost” as should the occupancy overshoot and become overly small, then the location of the atom no longer has much effect on the fit and the coordinates may change in random ways. I will let the occupancy converge, then add the coordinates back into the fit and finally attempt to also fit the \(\rm U_{iso}\) as well. As noted previously, site occupancies and \(\rm U_{iso}\) can be quite highly correlated and sometimes only one can be refined. So, I will keep an eye on how those values change and if the \(\rm U_{iso}\) and/or occupancy values refine and will fix the \(\rm U_{iso}\) if it refines to a value that I don’t consider reasonable.
One important question to be considered is: how reliable is the conclusion that a site is partially vacant or has shared occupancy? Since the site occupancy and \(\rm U_{iso}\) value for a site may be quite highly correlated, the standard uncertainty for the occupancy value will increase when \(\rm U_{iso}\) and the occupancy are refined together, but this does not happen when one must fix \(\rm U_{iso}\) in order to refine the occupancy. Fixing the value means that in some manner one is choosing the \(\rm U_{iso}\) value. Since that number is a guess, I will be concerned with how much much this choice changes the occupancy. When trying to decide how much I trust an occupancy value, I will look to see how that value changes if I set the \(\rm U_{iso}\) significantly. For this I will take the \(\rm U_{iso}\), which I consider reasonable and perform two more refinements where I multiply the \(\rm U_{iso}\) value by 0.5 and 2 (or sometimes even larger numbers) to see how that affects the refined occupancy value. That gives me a feeling for how much I trust the occupancy value. I will consider the experimental uncertainty on occupancy to be the range of values that I see as I fix \(\rm U_{iso}\) at different values plus the standard uncertainty for the occupancy value. If changing \(\rm U_{iso}\) can bring the site to full occupancy, I will consider the conclusion that there are vacancies present as not having been proven by the refinement. Even when I can refine site occupancy and \(\rm U_{iso}\) values together, I may still perform this sensitivity analysis.
The final aspect of a model that I want to present that affects peak intensities is correction for texture. Before correcting for this, it is worthwhile asking oneself, how likely is it that the sample could have preferred orientation? If one knows from microscopy that the crystallites are needles or thin plates, then the presence of texture becomes more probable. Likewise, sample preparation can also be significant. For Bragg-Brentano diffraction, one often fills a sample mounting from the front side and then smooths the surface. This, alas, is a recipe for introducing preferred orientation. Better to have a sample mounting that is drilled through and fill the sample from the rear and then add a backing plate. Some people will even place a finely abrasive surface on the front side of the sample as they fill from the rear, as to not encourage the crystallites to align to a smooth surface. Capillaries are less likely to have texture, but needles will tend to align along the axis. The large size of neutron sample holders makes those measurements less likely to exhibit texture.
GSAS-II offers two modes for correcting for texture, as was discussed in §4.2, March-Dollase and spherical harmonics. Note that these corrections, like the phase fractions and sample broadening terms, are available for each phase and for each histogram and are thus “HAP” parameters (see 14.7). The March-Dollase model describes a simple case, where one crystallographic direction (and its symmetry replicants) is either over or under-represented. Use of March-Dollase allows only a single parameter to refine, but it does require supplying the direction for that axis of preferred orientation to the software. When that direction is known from crystallite morphology or is a fixed direction due to unit cell symmetry, then this might be the first thing to try, but that is not what I usually do. When I have a refinement in the final stages and the fit could still be improved, I will test if preferred orientation is present by selecting the spherical harmonics option in the HAP parameters (either on the data tab of the phase or in the Hist/Phase data tree entry). Once the “Preferred orientation model” is selected as “Spherical harmonics”, one can set the “Harmonic order” term. This dictates the number of terms that can be refined, but number of refinable parameters will depend on the Bravais lattice for the phase. My recommendation is to set the to the lowest Harmonic order term to the lowest value that provides any refinable terms (usually 2 but sometimes 4) and see if that provides a significantly lower \(\rm R_{wp}\) or \(\chi ^2\). If so, I will try increasing the Harmonic order term to see if a more complex preferred orientation model does an even better job of fitting the data. If I do not see significant improvement in the fit, I turn off texture corrections by setting the Harmonic order term to 0.
Note that somewhat confusingly, GSAS-II has a data tab on the phase data window labeled “Texture” which also offers access to spherical harmonic terms. That tab is used for the study of texture in samples and usually requires that multiple measurements have been made, either by repositioning the sample or by having multiple detectors placed around the sample. Unless you have collected data with set of measurements designed to characterize texture, you will not use the “Texture” tab.
I do remember having a conversation with Charles Prewitt where he corrected my pronunciation of March-Dollase and when I asked Charlie how he knew, he told me that W.A. Dollase had been his college roommate. Alas, I don’t remember what the correct pronunciation was.