Pseudo Voigt

Someone models instrument response function as voigt profile. Since convolution with voigt profile is hard, I approximate voigt profile to linear combination of cauchy and gaussian distribution. Such approximated function is usually called pseudo voigt profile.

\[\begin{equation*} {IRF}(t, {fwhm}, {eta}) = \eta C(t, {fwhm}) + (1-\eta)G(t, {fwhm}) \end{equation*}\]

where, \({fwhm}\) is unifrom full width at half maximum parameter and \(\eta\) is the mixing parameter.

However, the paramter (\({fwhm}\), \(\eta\)) in pseudo voigt profile shown in above is different from the parameter (\({fwhm}_G\), \({fwhm}_L\)) in voigt profile function.

To resolve such paramter mismatch, another type of pseudo voigt profile function is proposed.

\[\begin{align*} {pv}(t, {fwhm}_G, {fwhm}_L) &= \eta({fwhm}_G, {fwhm}_L) C(t, {fwhm}({fwhm}_G, {fwhm}_L)) \\ &+ (1-\eta({fwhm}_G, {fwhm}_L)) G(t, {fwhm}({fwhm}_G, {fwhm}_L)) \end{align*}\]

\[\begin{align*} {fwhm} &= ({fwhm}_G^5+2.69269{fwhm}_G^4{fwhm}_L+2.42843{fwhm}_G^3{fwhm}_L^2 \\ &+ 4.47163{fwhm}_G^2{fwhm}_L^3+0.07842{fwhm}_G{fwhm}_L^4 \\ &+ {fwhm}_L^5)^{1/5} \\ \eta &= 1.36603({fwhm}_L/f)-0.47719({fwhm}_L/f)^2+0.11116({fwhm}_L/f)^3 \end{align*}\]

Such \({fwhm}\) and \(\eta\) function are taken from J. Appl. Cryst. (2000). 33, 1311-1316.