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In real crystals, the atoms may not always sit at their ideal lattice sites. How to quantify this lattice distortion?

Based on my understanding from X-ray diffraction books, the Radial Distribution Function/Pair Distribution Function can qualitatively give information about the lattice distortion. Is there a way to give a number to this lattice distortion?

I want to know if there's a method/technique which can predict that the binary alloy $\ce{AB}$ has lattice parameter of XÅ with lattice distortion of ±YÅ.

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First of all it must be clarified that a lattice is a collection of geometrical point all characterized by a same surrounding (that is, each lattice point has the same number of nearest neighbour lattice points arranged with the same point distances and angles).

Hence, widespread definitions such as “lattice distortion”, “lattice dynamics”, “lattice energy” and so on are misleading, but unfortunately commonly used; in this case the term “lattice” should be substituted by “structure”. A crystal structure is obtained by the convolution of a lattice with a basis (i.e. an atom of a group of atoms).

Having clarified this, structural distortion can be evaluated by diffraction. Distortion can be dynamic or static. Pair distribution function analysis provides a better estimate of the Debye-Waller parameters, whereas the widths of the peaks give the bond distance distribution of the considered atomic pair. But this last information characterizes the local scale, not the structural parameters.

Diffraction line (peak) broadening can be used for evaluating the structural strain, that is the fluctuation of the inter-planar distances along the different crystallographic directions. Simplifying, the components of the diffraction line profile have a $\tan(2\theta)$ dependence on the scattering angle $\theta$ by means of which strain along the different directions can be valuated.

The qualitative analysis of the strain distribution is rather easy from high quality diffraction data, but an accurate quantitative evaluation is not. This is not a predictive method (neither is PDF analysis), but gives you qualitative information about the structural distortion.

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  • $\begingroup$ Is there way to quantify these distortions? There are numerous literature available on the PDFs. The peak broadening has been well studied. However, I am more intrested in giving a number to this peak broadening and correlating it to the structural distortion $\endgroup$
    – Sufyan
    Mar 20 at 11:50
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    $\begingroup$ Peak broadening in PDF is related to the distribution of the bond lengths for a specific pair, it is not always related to strain. Imagine an octahedron that by symmetry is somehow distorted. You will have a broadened PDF peak because you will have different, although similar, bond distances, but no structural strain. In other cases the broadening of the PDF peak could be related to structural strain and you can actually detect this strain by diffraction line broadening analysis. $\endgroup$
    – gryphys
    Mar 21 at 12:24
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    $\begingroup$ You can find details about diffraction line analysis for examplein this book "Diffraction Analysis of the Microstructure of Materials" by Mittemeijer and Scardi $\endgroup$
    – gryphys
    Mar 21 at 12:24
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In my group we develop continuous symmetry and chirality measures that can be used to quantify the distortion of a structure, given the 3D coordinates.

Please see: https://csm.ouproj.org.il/

See also a related method to calculate continuous shape measures by S. Alvarez and coworkers: http://www.ee.ub.edu/index.php?option=com_content&view=article&id=72&Itemid=469

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