# Peak shape

The peak width observed in x-ray scattering can be related to the grain size of the ordered structure giving rise to the scattering peak. More generally, the peak shape also encodes information about the sample order. Thus, peak shape analysis can be used to extract higher-order information.

One interpretation of peak shape is that encodes the average grain shape. That is, the peak shape function is the Fourier transform of the grain shape. Specific examples:

• Gaussian peak in reciprocal-space implies a Gaussian-like decorrelation in realspace (i.e. that the average of grains/correlation-volumes decays with a Gaussian profile).
• Lorentzian peak in reciprocal-space implies an exponential decorrelation in realspace.
• Sinc function in reciprocal-space implies a top-hat function in realspace (sharp domain boundaries, with little to no variation in size of domains).

Note also that instrumental resolution contributes to peak width, and also to peak shape. Scattering peaks are thus sometimes fit using functions that include two contributes (e.g. a Gaussian, representing material grain size, plus a Lorentzian, representing instrumental resolution).

## Generalized Peak Shape

A generalized peak shape can be computed using:

{\begin{alignedat}{2}L_{hkl}(q)&={\frac {2}{\pi \delta }}\prod _{n=0}^{\infty }{\left(1+{\frac {\gamma _{\nu }^{2}}{(n+\nu /2)^{2}}}{\frac {4q_{s}^{2}}{\pi ^{2}\delta ^{2}}}\right)^{-1}}\\&={\frac {2}{\pi \delta }}\left|{\frac {\Gamma \left[\nu /2+i\gamma _{\nu }(4q_{s}^{2}/\pi ^{2}\delta ^{2})^{2}\right]}{\Gamma \left[\nu /2\right]}}\right|^{2}\end{alignedat}} Where $q_{s}=(q-q_{hkl})$ , $\delta$ describes the peak width, and $\nu$ describes the peak shape. The parameter $\gamma _{\nu }$ is a ratio of gamma functions:

$\gamma _{\nu }={\sqrt {\pi }}{\frac {\Gamma \left[(\nu +1)/2\right]}{\Gamma \left[\nu +/2\right]}}$ The limiting cases for peak shape are:

$L_{hkl}(q_{s})=\left\{{\begin{array}{c l l}{\frac {\delta /2\pi }{q_{s}^{2}+(\delta /2)^{2}}}&\mathrm {for} \,\,\nu \to 0&\mathrm {(Lorentzian)} \\{\frac {2}{\pi \delta }}\exp \left[-{\frac {4q_{s}^{2}}{\pi \delta ^{2}}}\right]&\mathrm {for} \,\,\nu \to \infty &\mathrm {(Gaussian)} \\\end{array}}\right.$ Thus the parameter $\nu$ allows one to vary continuously between a Lorentzian peak shape and a Gaussian peak shape. Note that for Lorentzian, $\delta$ describes the full-width at half-maximum (FWHM):

$\delta _{\mathrm {lorentz} }=\mathrm {fwhm_{\mathrm {lorentz} }}$ The Gaussian form can be written a few different ways:

{\begin{alignedat}{2}L_{hkl,\mathrm {gauss} }(q_{s})&={\frac {2}{\pi \delta }}\exp \left[-{\frac {4q_{s}^{2}}{\pi \delta ^{2}}}\right]\\&={\frac {1}{{\sqrt {2\pi }}\sigma }}\exp \left[-{\frac {q_{s}^{2}}{2\sigma ^{2}}}\right]\\&={\sqrt {\frac {\ln {2}}{\pi }}}{\frac {1}{\mathrm {fwhm} }}\exp \left[-{\frac {4\ln {2}q_{s}^{2}}{\mathrm {fwhm} ^{2}}}\right]\\\end{alignedat}} where the width is described by:

$\delta _{\mathrm {gauss} }={\sqrt {\frac {8}{\pi }}}\sigma _{\mathrm {gauss} }={\frac {\mathrm {fwhm} _{\mathrm {gauss} }}{\sqrt {\pi \ln {2}}}}$ $\sigma _{\mathrm {gauss} }={\sqrt {\frac {\pi }{8}}}\delta _{\mathrm {gauss} }={\frac {\mathrm {fwhm} _{\mathrm {gauss} }}{2{\sqrt {2\ln {2}}}}}$ $\mathrm {fwhm} _{\mathrm {gauss} }=2{\sqrt {2\ln {2}}}\sigma _{\mathrm {gauss} }={\sqrt {\pi \ln {2}}}\delta _{\mathrm {gauss} }$ And note that $2{\sqrt {2\ln {2}}}\approx$ 2.35482004503...