There are some terms that are used in GSAS-II or in this book that are worth defining, as their use is not necessarily obvious.
Model: I prefer to use the term model rather than crystal structure when talking about the parameters that we fit in GSAS-II. I do this for a number of reasons. The parameters we fit include many things that are not properly part of the crystal structure, such as crystallite size or texture, but also because, in the end, what we call a crystal structure is indeed only a picture, or perhaps even better a cartoon, of what we believe to be present. There may be other models that explain the data even better. There are always aspects of the material (such as defects) that are not part of our model. We can only fit a model up to complexity supported by the data.
Histogram: This is used for any type of dataset in GSAS-II that can be used in a refinement, which includes all types of powder diffraction and single-crystal data, but not diffraction images. The history behind this odd name is that original TOF instrumentation used CAMAC-based histogramming memory units to record diffraction data. Each time a neutron was recorded, the electronics turned the time into an address in memory and that word in memory was incremented, so it was natural to think of a TOF diffraction pattern as a histogram.
Phase: Most commonly in GSAS-II, this will reference the full characterization model for a chemical element or compound that includes lattice parameters, atomic coordinates and displacement parameters and related details. Sometimes the term crystal structure is used for this. This information is commonly read from Crystallographic Information Framework (CIF) files to the point that, even though I would prefer otherwise, sometimes crystal structure and CIF are used interchangeably. Another less-common use for phase in GSAS-II is for the phase angle associated with a structure factor (\(F_{hkl}\)).
Constraint: As will be discussed in Chapter 24, with more complex structural models, one may not have enough well-determined independent observations to fit all independent parameters that are available. To decrease the complexity of the fitting model and increase the leverage (see below) of a parameter on the fit, parameters are grouped by constraining them to have the same value.
Restraint: These are also used for complex structural models, where the data may be insensitive to an aspect of the model. Restraints provide a target value for some aspect of the model, which may be a parameter value or may be a quantity derived from parameters (such as a bond distance derived from atomic coordinates). While they are used similarly to constraints, restraints are implemented in an entirely different manner from constraints, as will be discussed in Chapter 25.
\(\chi ^2\): This is the quantity that is minimized by GSAS-II when fit is performed. It is defined as \[\chi ^2 = \sum _{N} w_j(Y_{obs,j} - Y_{calc,j})^2\] where \(Y_{obs,j}\) are the observations, which will include restraint values, and \(Y_{calc,j}\) are the values predicted for these observations from the model and \(w_j\) are the weights applied to each observation. Statistical analysis predicts that the optimal model will be obtained when \(w_j = 1/\sigma _j^2\) where \(\sigma _j^2\) is the standard uncertainty for observation \(Y_{obs,j}\).
Reduced \(\chi ^2\): This is computed from
\[\chi ^2 = \sum _{N} w_j\frac {(Y_{obs,j} - Y_{calc,j})^2}{N-p}\] following the same definitions as above for \(\chi ^2\), but where \(N\) is the number of observations and \(p\) is the number of refined parameters. Note that if the weights are defined as \(w_j = 1/\sigma _j^2\) and the model provides an ideal fit to the observations, then the expectation value for \(|Y_{obs,j} - Y_{calc,j}|\) is \(\sigma _j\) so the Reduced \(\chi ^2\) would be expected to be 1 when \(N>>p\) (and one always wants \(N>>p\)). If the Reduced \(\chi ^2 < 1\) then this means that the weights must be overestimating the actual uncertainty in the data or that the model is being “overfit” where too many parameters are being fit.
Note that some authors do not distinguish between “\(\chi ^2\)” and “Reduced \(\chi ^2\)”, but for GSAS-II these mean different quantities. I have never seen \(\chi \) used in any statistical context, only \(\chi ^2\), but see the next entry for a usage of \(\sqrt {\chi ^2}\).
Goodness of Fit, GOF: This is commonly used in single-crystal fitting.
\[GOF = \sqrt {\sum _{N} w_j\frac {(Y_{obs,j} - Y_{calc,j})^2}{N-p}}\] so the \(GOF^2 = \chi ^2\).
Leverage: This is a measure of how much impact a parameter has on the overall fit. An example of parameters that has a high leverage would be the coordinates of Pb atom in a metal oxide in a refinement of x-ray data, while the position or \(\rm U_{iso}\) value of a hydrogen atom in the same material would have very low leverage, as one would not expect the scattering from one electron to significantly affect the overall pattern.
ADP: This is an abbreviation for atomic displacement parameter. Although other representations exist, in GSAS-II ADPs can be an isotropic \(\rm U_{iso}\) value (named in GSAS-II as Uiso) or an anisotropic \(\rm U_{ij}\) value (named in GSAS-II as U11, U22, U33, U12, U13, and U23). This describes the average displacement of the atom from its ideal location. These were traditionally referred to as “thermal parameters,” but that is a poor choice, as ADPs encompass both thermal and static disorder.
Standard Uncertainty: This represents the expected uncertainty for a quantity that is observed or fit, based on statistical expectations, as opposed to an actual standard deviation value, which requires multiple independent measurements. Older literature uses the name “estimated standard deviation” or “esd,” but standard uncertainty or s.u. is now preferred. Any parameter fitted in GSAS-II will have a s.u. value associated with that value.
Refinement flags: This term is used for a GUI entry that determines if a parameter or set of parameters will be refined. In most parts of the GSAS-II GUI a checkmark button is used and is often labeled “Refine” or “Ref”. In the case of some refinement flags, the button can control more than one parameter. For example, the “Refine unit cell” flag on each Phase’s General tab will refine all allowed elements in the reciprocal lattice tensor, so a, b and c are all refined in an orthorhombic cell when this is selected. Likewise, the “Refine?” flag for a powder histogram’s Background entry will refine all terms in the background function. There are a few places where a more compact notation is used. As an example, for atoms there are possible refinement flags of “X”, “U”, “F” and if magnetic “M” which refine the coordinates, ADPs, fractional occupancies and magnetic moment, respectively. If “X” is set all coordinates allowed by symmetry are refined. For “U” either \(\rm U_{iso}\) or if an atom has been set to use an anisotropic ADP, the symmetry-allowed \(\rm U_{ij}\) terms. These flags can be combined, so “XU” refines both coordinate and ADP values for that atom.
Peaks vs. Reflections. I will try to make a distinction between these two terms consistently in this book. Reflections are created by diffraction from the lattice, but we see peaks in the diffraction pattern. Peaks are really our observables, though I would also argue that the lack of a peak at a location where a reflection is present is also an observable. A peak will never arise from a single reflection. At a minimum, every powder diffraction peak will be a superposition of at least two reflections, \(hkl\) and its Friedel pair, \(\overline {h}\overline {k}\overline {l}\), although in monoclinic and higher symmetry cells, as many as 48 symmetry-related reflections may be superimposed. The number of symmetry-related reflections that occur at a specific Q is called the multiplicity. Note that overlap of non-symmetry related reflections is also possible. There are some reflections that will always occur at exactly the same Q value. For example, in a cubic system the 300 and 221 reflections will always be overlapped. Accidental overlap occurs when two reflections by chance have nearly the same Q value. On occasion, crystallographers will collect data at slightly different temperatures, which can cause lattice dimensions to change at different ratios, which will often be sufficient to change the accidental overlaps. If the temperature changes are small, then the expectation is that the structures at the two different temperatures are indistinguishable and a single model can be fit to both, but due to the changes in overlap conditions, with two different patterns more observations have now been made. I believe that Christian Baerlocher and Lynne McCusker were the first to exploit this.