Observing strain – the Stephens formalism

When a powder diffraction pattern is looked at, the peak linewidths may vary as a function of scattering angle 2\theta in a non-uniform fashion. This could arise from a given internal strain distribution, from anisotropic sample size broadening, or from a given defect pattern (for example, stacking faults). Here, I will concentrate on the broadening due to strain. The simplest method for dealing with this is to consider isotropic or uniform strain, in which case the full-width half-maxima (FWHMs) of the Bragg peaks will be proportional to tan \theta. If the strain is not uniform, the problem is a bit harder. Here, I will follow P. W. Stephens’ approach, outlined in an article in the Journal of Applied Crystallography, published in 1999.

Stephens starts by considering that there is a distribution of lattice parameters and angles in a given sample, centered around the nominal values. He then incorporates this into an expression for the d-spacing for the planes defined by the Miller indices hkl:

\frac{1}{d_{hkl}^2}= M_{hkl} = \alpha_1 h^2 + \alpha_2 k^2 + \alpha_3 l^2 + \alpha_4 kl + \alpha_5hl +\alpha_6hk

where \alpha_i have a particular set of values for each grain – implicitly, we are assuming that each grain is internally consistent. The parameters are then assumed to have a Gaussian distribution, and os we can construct a covariance matrix, ]C_{ij} reflecting the variance of \alpha_i. Since M_{hkl} is linear in \alpha_i, the variance is

\sigma^2(M_{hkl}) = \sum_{i,j} C_{ij} \frac{\partial M}{\partial \alpha_i} \frac{\partial M}{\partial \alpha_j} = \sum_{HKL} S_{HKL} h^H k^K l^L

where H + K + L = 4. The S_{HKL} format gives 15 independent parameters, an improvement on the 21 independent elements of C_{ij}. This reduction is due to the combination of terms like C_{23} and C_{44}, which have the same hkl dependence. This can then be used to give the contribution to the anisotropic broadening of the Bragg line at Bragg angle \theta, and for single-wavelength experiments, this will be

\Gamma_A = \sqrt{\sigma^2(M_{hkl})} \tan \theta / M_{hkl}

where the factor differentiating the rms and the FWHM of the Gaussian is wrapped into S_{HKL}.

However, lineshapes measured in powder diffraction are not typically Gaussian in shape, and are usually modelled as pseudo-Voigt functions, where the widths of the Gaussian and Lorentzian components will develop differently as a function of scattering angle (as discussed by Thompson, Cox and Hastings). The variance of the Lorentzian behaves quite differently to that of the Gaussian, which means that the generalization is not obvious. To deal with this, Stephens assumes without justification that the expressions for the variance as a function of S_{HKL} can be interpreted to give a FWHM for the Lorentzian anisotropic broadening term. With this assumption made, the experimental lineshape can then be convolved with the anisotropic strain broadening term.

For both the Gaussian and Lorentzian contributions to the lineshape, there are terms (U and X respectively) that may duplicate the effect of the S_{HKL} parameters. To avoid this, they should be set in any refinement by use of an unstrained reference standard. Once this is done, the S_{HKL} parameters can be refined. The number of these parameters may be reduced from 15 by considering constraints imposed by the underlying crystal lattice. For example, for an orthorhombic lattice, \alpha_{4,5,6} must be zero, and so the relevant terms are S_{400},S_{040},S_{004},S_{220},S_{202} and S_{022}. However, as Stephens notes, this does not permit the prediction of the actual values of the strain parameters for the material.

In the next post, I will look at a couple of examples using this formalism to look at orthorhombic systems.

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