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Strain Fitting Options

Principles

Relating stress and x-ray measured lattice strains in materials deforming under high pressure is a tricky business. Several versions of elastic theories have been developed, but they are known to show strong limitations because they do not account for plastic deformation and stress heterogeneities within the polycristal.

Those limitations can be lifted if one uses numerical models such as Elasto-Plastic Self-Consistent (EPSC) models, but this remains an active field of research.

Here is a list of papers that you could read

• Elastic theory of lattice strains
  • A. K. Singh, C. Balasingh, H. K. Mao, R. J. Hemley and J. Shu, Analysis of lattice strains measured under non-hydrostatic pressure, J. Appl. Phys., 1998, 83, 7567-7575 , doi: 10.1063/1.367872
  • S. Matthies, H. G. Priesmeyer and M. R. Daymond, On the diffractive determination of single-crystal elastic constants using polycrystalline samples, J. Appl. Cryst., 2001, 34, 585-601 , doi: 10.1107/S0021889801010482
• Limitations of the elastic model
  • D. J. Weidner, L. Li, M. Davis and J. Chen Effect of plasticity on elastic modulus measurements, Geophys. Res. Lett., 2004, 31, L06621 , doi: 10.1029/2003GL019090
  • S. Merkel, N. Miyajima, D. Antonangeli, G. Fiquet and T. Yagi, Lattice preferred orientation and stress in polycrystalline hcp-Co plastically deformed under high pressure, J. Appl. Phys., 2006, 100, 023510 , doi: 10.1063/1.2214224
• Application of EPSC models to account for effects of plasticity
  • L. Li, D. J. Weidner, J. Chen, M. T. Vaughan, M. Davis and W. B. Durham, X-ray strain analysis at high pressure: Effect of plastic deformation in Mg O, J. Appl. Phys., 2004, 95, 8357-8365 , doi: 10.1063/1.1738532
  • P. C. Burnley and D. Zhang, Interpreting in situ x-ray diffraction data from high pressure deformation experiments using elastic–plastic self-consistent models: an example using quartz, J. Phys.: Condens. Matter, 2008, 20, 285201 , doi: 10.1088/0953-8984/20/28/285201
  • S. Merkel, C. Tomé and H.-R. Wenk, A modeling analysis of the influence of plasticity on high pressure deformation of hcp-Co, Phys. Rev. B, 2009, 79, 064110 , doi: 10.1103/PhysRevB.79.064110

General equations

Elastic theories, as described by Singh et al, predict that measured d-spacings should vary according to

where

• d_m is the measured d-spacing for the hkl line at the angle psi
• d_p is the d-spacing for the hkl line under hydrostatic pressure
• ψ is the angle between the diffracting plane normal and the deformation direction. In compression, planes with ψ=0 are perpendicular to the compression, planes with ψ=90 are parallel with the compression,
• Q is the lattice strain parameter.

If the compression direction is at the azimuth δ_c on the detector, ψ can be calculated from the azimuth direction using

where θ is the diffraction angle.

The assumptions behind this expression are not correct. However, it has been shown to be fairly consistent with high pressure deformation data, at least to the first order.

The idea in Polydefix, is to fit this equation to the experimental data and extract

• d-spacings under hydrostatic pressure d_p
• lattice strain parameters Q

for each measured hkl line and each diffraction image in the experiment.

D-spacings under hydrostatic pressure d_p will be used for Pressure and unit cell optimizations.

Lattice strain parameters Q can be used to study stress distribution by

• comparing experimental values to results of EPSC calculations,
• using an elastic stress model.

Corrections

Polydefix offers two types of correction to the lattice strain equation above:

• automatic adjustment of the axial symmetry (deformation) direction δ_c,
• adjustment for drifting beam centers.

Both are incompatible and they can not be used at the same time.

Adjustment of the axial symmetry (deformation) direction

In some experiment, the axial symmetry direction is not always well know. Polydefix can adjust it automatically. In this case, for each image, we fit

• the azimuth of the deformation direction δ_c,
• lattice strain parameters Q(hkl) and d-spacings under hydrostatic pressure d_p(hkl) for each measured diffraction line.

The starting value should not be too far off as Polydefix can confuse the axial symmetry direction and the one located at 90 degrees. As a check, in compression, lattice strain parameters Q(hkl) should be positive. In extension, they are negative.

Adjustment for drifting beam centers

In other experiment, the stand for the detector can be moving and the beam center of the image plate will be drifting. Even a few microns drift of a CCD 20 cm away from the sample can be seen and will be significant. Polydefix can correct for this. Equations are a bit more complex, but they can be solved.

In this case, Polydefix will adjust (for each image)

• the relative position of the beam center x/D and y/D, where D is the sample-to-detector distance,
• lattice strain parameters Q(hkl) and d-spacings under hydrostatic pressure d_p(hkl) for each measured diffraction line.

Again, in compression, lattice strain parameters Q(hkl) should be positive. In extension, they are negative.