GSAS-II offers separate terms for crystallite broadening and for microstrain and those terms can be applied isotropically or two models are provided for anisotropic broadening for each. As was noted before, there are sample broadening terms associated with each histogram for each phase, so if there are \(m\) histograms and \(n\) phases, there will be \(m \times n\) sets of sample broadening terms.
For crystallite broadening, the GSAS-II offers three models, which will be discussed directly below.
The default broadening model is labeled as isotropic, which of course assumes a uniform crystallite size in all directions. This model has two refinable parameters: One is for the mean crystallite size and the second, labeled LGmix, determines if the broadening will be Lorentzian, Gaussian or a mix between the two.
The isotropic crystallite size parameter has units of mm (10-6 m). This is determined using the Scherrer equation with a K value of unity. While K=1 does not exactly match the most common crystallite morphologies that are encountered, we consider that crystallite broadening is not a rigorously defined quantity, so while relative differences between measurements on different specimens of the same sample may be determined quite accurately, no one should ever be concerned about a discrepancy of 10% between different quantification methods or as a difference from measurements on different materials, where the morphology may differ. Note that as the crystallite size parameter decreases, more broadening occurs, while larger values provide very minimal broadening. From a practical perspective, the instrumental resolution limits the maximum value of crystallite size broadening that can be discerned from a dataset. GSAS-II does not allow the crystallite size to refine below 0.01 \(\mu \)m. At this value peaks are so broad as to defeat Rietveld analysis. It the size parameter also is not allowed to refine above 10 \(\mu \)m, which would produce imperceptible broadening with any existing powder diffractometer. In practice, with a very high resolution instrument, values above perhaps 3 \(\mu \)m are meaningless. More typical for most diffractometers may be 1 \(\mu \)m. It is the user’s responsibility to review the refined value for the size parameter and to eliminate refinement for this parameter once the value is above the discernible range for the instrument that has been used.
The default for LGmix is 1, which means the peaks will be 100% Lorentzian. If LGmix is 0 then the crystallite broadening would be 100% Gaussian and values between 0 and 1 create a linear mix between the two. Physically, an LGmix value of 1 is what is expected, except from materials that have a very unusual distribution of crystallite sizes, such as a monodisperse size. While the parameter LGmix can be varied, values other than 1 seldom improves a fit. Unless a significant improvement is seen for other values, the value is best fixed at 1. Values less than 0 or greater than 1 represent peak shape functions that are highly unlikely, but I am not comfortable saying that they are always physically impossible. Seeing a value other than unity most likely indicates a problem with the background fitting, but perhaps could arise from something very unusual in crystallite size distributions. I will only change LGmix from unity if I have extreme problems in obtaining a good fit to the peak profile.
When there is evidence for anisotropic peak broadening and the broadening arises primarily from crystallite size, a more advanced model is that of uniaxial size broadening. This describes the average crystallite shape as a cylindrical, which in extreme cases extends to platy or needlelike morphologies. For this, one provides the crystallographic direction for the axis of the cylinder and this model provides two size parameters, axial and equatorial sizes, which again have \(\mu \)m units and are otherwise similar to the previous isotropic value except that the axial value is the average size value along the specified crystallographic direction and the equatorial value is the size perpendicular to that direction. There is also a single LGmix parameter to describe the Lorentzian vs. Gaussian character of the broadening, which again is rarely any value other than one.
Picking axes to select for broadening can be a trial-and-error process. For some Bravais lattice choices, there is only one sensible axis for broadening: For tetragonal, hexagonal or rhombohedral (hexagonal setting) the (001) axis is unique. [Note that for rhombohedral space groups in a rhombohedric setting, the (111) axis is equivalent to the hexagonal (001) axis and thus is unique.] For cubic cells, there are effectively three unique and non-perpendicular axes (001), (110) and (111). Highly anisotropic habits in cubic systems are unusual but uniaxial broadening nonetheless can be attempted to see if the fit improves. For orthorhombic, monoclinic and triclinic cells, there are no symmetry-based restrictions on preferred broadening directions, but analysis of peak widths may suggest a direction that has the broadest or narrowest widths or trial-and-error may suggest a preferred axis that improves the fit significantly.
The remaining crystallite size broadening model that GSAS-II offers is labeled ellipsoidal, as the average crystallite is assumed to have a shape described by a tensor that is analogous to the anisotropic displacement parameter (ADP) ellipsoid. The terms in the tensor are labeled by GSAS-II as Sij. The diagonal elements (i=j) of the tensor specify crystallite lengths along three orthogonal axes and the off-diagonal terms (i¹j) orient the ellipsoid relative to the cell axes. Thus, there are six terms to refine, though one may want to start with only the diagonal terms. Note that in higher symmetry Bravais lattices, just as for ADP’s for atoms on high-symmetry sites, symmetry may require constraints, but at present GSAS-II does not force this. Thus, use of this model is not recommended for cubic, hexagonal/rhombohedral or tetragonal symmetry, unless users understand the symmetry implications on the tensor terms.
For microstrain broadening, the GSAS-II default broadening model is also labeled as isotropic, and again assumes microstrain is uniform in all directions. As is the case for isotropic crystallite size broadening, this model has two parameters: One is for the mean microstrain and the second, labeled LGmix, determines if the broadening will be Lorentzian, Gaussian or a mix between the two.
The microstrain parameter is the unitless quantity \(\Delta Q/Q \times 10^6\). Note that this value has the opposite broadening effect from the crystallite size parameter. Smaller values produce less broadening. The default of value for microstrain is 1000, which can be considered a lower limit value for most diffractometers. (With 11-BM and other high-resolution diffractometers, one can see broadening with microstrain in the range of 300-500, perhaps). If the value decreases to a value that is insignificant, reset it and turn off the refinement of this parameter.
The default again for LGmix is 1, which means the peaks will be 100% Lorentzian. This parameter can be varied, but it seldom improves fits and unless a significant improvement is seen for other values, the value is best fixed at 1. I am unable to imagine what could give rise to an LGmix values less than 0 or greater than 1, but perhaps there might be physical reasons why this might be the case. Again, if I see a significant improvement in a fit when LGmix refines to a value outside the range of 0-1, I would look carefully to make sure the background fit is reasonable.
When there is evidence for anisotropic peak microstrain broadening, the next model to try is that of uniaxial microstrain broadening, which is very much analogous to the uniaxial crystallite size broadening model. Here, this describes a system where the material is either much more stiff or much softer in one direction, particularly compared to the perpendicular directions. Again, one provides the crystallographic direction for this unique direction and again there are now two microstrain parameters, axial and equatorial, which again are \(\Delta Q/Q \times 10^6\) (unitless). There is a single LGmix parameter to describe the Lorentzian vs. Gaussian character of the broadening, which again is rarely any value less than one. The same discussion for selecting a broadening axes from uniaxial crystallite size broadening applies here as well.
The third model for microstrain is labeled as “generalized” and is based on the work of Popa and Stephens to expand microstrain with symmetry-consistent Bessel function terms. The number of terms provided depends on the Laue class for the phase, but because these terms must be consistent with lattice symmetry, they usually refine with good stability. Unlike the elliptical case for crystallite size broadening, one can try fitting with the generalized microstrain model in a routine manner.