[Physics] How are lattice parameters determined from reciprocal space maps


It seems that the papers speak of reciprocal space maps with very high praise because of its ability to study strain in epitaxial films and determine the amount of relaxation. Also one can determine the in-plane and out-of-plane lattice parameters independent of each other (which is nice because it prevents one from propagating uncertainties). However, I cannot find how the lattice parameters are determined from these reciprocal space maps.

Every paper is the same, shows me the RSM, tells me the lattice parameters they found, but no one tells me how they found it. Review papers and pdfs I've found online plus 3 x-ray diffraction/diffractometry/scattering textbooks go over RSM in the same superficial way: go over the principles and the math but no one offers examples. It's hard to believe that I cannot find a book that can teach me how to index even a regular diffraction pattern.

What's the point of having this whole literature of results if no one can tell me exactly how they were found/calculated?

The only vestige of an explanation I've found is that the value of Q_z where a peak is, is proportional to the inverse of the out-of-plane lattice parameter. Likewise, the value of Q_x where the peak is, is proportional to the inverse of the in-plane lattice parameter. But that's it. It's amazing how little is published in terms of how experimental techniques are carried out. Everything is too result-oriented and without publishing the methodology not only does it make me a frustrated, under-slept grad student but no one can check other people's work.

Best Answer

From an asymmetrical peak one can determine both a- and c- lattice constant.

$$\begin{align}a=\frac{\lambda\cdot h}{(q_\text{par} \cdot \sqrt3)} \\c=\frac{\lambda\cdot l}{(q_\text{ort} \cdot 2)}\end{align}$$

Where $\lambda$ is the wave length of your X-rays, $q_\text{par}$ (q_parallel) is the peak position alongside the in-plane direction ($q_x$ or $q_y$) in reciprocal lattice units. $q_\text{ort}$ (q_orthogonal) is the peak position alongside the out of plane direction, also in reciprocal lattice units. $h$ and $l$ are the miller indices of the peak ($hkl$).

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