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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Distorted hexagonal lattices

When a vortex lattice forms in a Type-II superconductor, it usually forms a hexagonal lattice, although this can be distorted, for example by some sort of coupling to the underlying crystal lattice, or by the directions of any nodes in the superconducting gap.  In some cases this can give rise to a square lattice.  Here, we are going to focus on small distortions of the hexagonal latiice.  In the perfect hexagonal case, the triangles that make up the hexagon are equilateral.  The easiest distortion to consider is that these equilateral triangles become isosceles triangles.  If they become scalene triangles, as in niobium, this is a bit trickier, and we will leave that for later.

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Scientists in Conference

I was recently asked to sit on the organizing committees for a couple of scientific meetings, and, almost at the same time, stumbled across Scientists in Conference: The Congress Organizer’s Handbook, by Volker Neuhoff, in my departmental library, and rapidly came to appreciate its charms.  It is a comprehensive aide for the conference organizer with a lot of good general tips although the specifics are now rather outdated.  Some comments on the secrets of good chairmainship are also hidden inside, and the author emphasises the importance of the social aspects of conferences.  There is a lovely note in the ‘Postface’ on issues with the translation from the original German, and the joint work between the author and the Australian chemist Robert Schoenefeld on this really shines.

In addition, it also provided reference to an article published in the British Medical Journal on Christmas Eve of 1983, called ‘Dreaming during scientific papers: effects of added extrinsic material‘, by Harvey, Schullinger, Stassinopoulos and Winkle, investigating sleeping and dreaming during scientific talks.  Like the book above, this is, in many respects, of its time, but the apparent prevalence of sleeping during medical conferences in 1983 is quite impressive.

Cricket in the Jungle

The name of this blog is a quote taken from the work of John Ziman, a theoretical physicist who wrote a series of books on solid state physics that I enjoyed discovering as an undergraduate.  One of them, Electrons and Phonons, has an excellent first paragraph that neatly explains the apparent problems with understanding things like electrical conductivity in metals.

The quote from the figure is:

It is, at first sight, remarkable that any influence can travel through a solid body.  We can imagine the passage of fast projectiles such as energetic neutrons, tearing their way through the crystal lattice, or of electromagnetic waves whose transport is primarily through an imponderable, all-pervading ether.  But besides such processes, impelled by forces much stronger than those binding the particles of the solid there exist the transport phenomena, in which heat, electricity, and matter itself are carried through the structure, under the gentle influence of a gradient of temperature, of electric field potential, or of atomic concentration.  If we insist on a particulate, electronic theory of electricity, the high conductivity of metals such as copper and silver is exceedingly difficult to explain.  The electrons must penetrate through closely packed arrays of atoms as though these scarcely existed.  It is as if one could play cricket in the jungle.

John Ziman, Electrons and Phonons, Oxford University Press (1960).