The Stephens formalism in practice

Here, I am going to look at a couple of exams that use the Stephens formalism in their consideration of data collected from strained samples. The examples will all be orthorhombic, so that the relevant Stephens parameters are S_{400}​, S_{040}​, S_{004}​, S_{220}​, S_{202}​ and S_{022}​.


I will start with a study of the microstructural strain energy in \alpha-uranium, by Manley et al., published in Physical Review B. This is orthorhombic with space group Cmcm. The study was carried out on polycrystalline samples, collecting neutron powder diffraction patterns from 77 K to 300 K. The authors extract the Stephens parameters from their fits, and later give 3D representations of what they call the microstrain, generated from the Stephens parameters – an example is shown below.

Microstrain representation obtained from work by Manley et al.  This image is the top right panel of Figure 4.  This figure is copyright of the American Physical Society; rights and permissions information is available at

Unfortunately, the actual parameters extracted from the fits are not given in the paper. They also found that they the measured parameters were different in different detector banks, pointing to textural changes in the sample.

They then compared the directionality of the strain broadening with both the Young’s modulus data availables and the hydrostatic compressibility, finding that the pattern is closer to the Young’s modulus state, indicating that the observations more closely mimic a uniaxial state rather than a hydrostatic state.

There then follows an interesting discussion on how to evaluate the strain energy stored in the microstructure. As noted previously, it is not possible to extract values for the stiffness constants directly from the Stephens parameters (note that an orthorhombic lattice has 9 independent stiffness constants, three pure normal stresses, three pure shear stresses, and three coupled terms). They then consider two limits. In the first, the strains are statistically isotropic, such that there are no correlations and the three coupled terms are assumed to be zero. The second limit considers the cases where the stresses are statistically isotropic. In this case, there will be contributions to the strain in one direction, and stress in another through the Poisson effect, and characterized by the Poisson ratio. In reality, the result will lie somewhere between these two limits.


This material was being investigated as a colossal magnetoresistive manganite, by Yaicle and co-authors, in a paper published in the Journal of Solid State Chemistry. Some synchrotron X-ray powder diffraction was done, where there was significant broadening of some Bragg peaks, and so the Stephens formalism was used to characterise this. Of the six parameters, four of them were quite large, indicating that the problem could not be simplified to a consideration of only one dimension. However, changes in the strain parameters appeared to correlate with the onset of a charge ordered/orbital ordered state. We can make the same visualisation plots as Manley et al. used to see how the response changes.

This makes clear the changes in the strain parameters between 300 K and 10 K, where the two phases have quite different levels of strain – in the Phase I case, the scale is 5 times larger than the other two, indicating the dramatic difference between Phase I and Phase II.

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.

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