According to Toraya (1990), the following functional forms can be used for diffraction peak functions:
– Symmetric Pseudo Voigt: a mixture of Gaussian and Lorentzian functions
– Split Pseudo Voigt: an asymmetric extension of Symmetric Pseudo Voigt
– Symmetric Pearson VII: a type of probability density distribution function
– Split Pearson VII: an asymmetric extension of Pearson VII
The specific forms are as follows.
Symmetric Pseudo Voigt:
\(
f(x, \eta, H_k) = \frac{2}{H_k \pi} \left(
\frac{\eta}{1+ \left(\frac{2x}{H_k}\right)^2} +
(1-\eta) \sqrt{\pi\ln2} \cdot 2^{-\left(\frac{2x}{H_k}\right)^2}
\right)
\)
f(x, \eta, H_k) = \frac{2}{H_k \pi} \left(
\frac{\eta}{1+ \left(\frac{2x}{H_k}\right)^2} +
(1-\eta) \sqrt{\pi\ln2} \cdot 2^{-\left(\frac{2x}{H_k}\right)^2}
\right)
\)
Split Pseudo Voigt:
\(
f(x, \eta_l,\eta_h, A, H_k) =
\frac{2}{ H_k \pi + \frac{H_k \pi (\eta_h-\eta_l)(\sqrt{\pi\ln2} – 1) }{(1+e^A)(\eta_h+ \sqrt{\pi\ln2} (1-\eta_h))}}
\left(
\frac{\eta_l}{1+\left(\frac{e^A+1}{e^A}\cdot\frac{x}{H_k}\right)^2} + (1-\eta_l)\sqrt{\pi\ln2} \cdot 2^{-\left(\frac{e^A+1}{e^A}\cdot\frac{x}{H_k}\right)^2}
\right)
\)
f(x, \eta_l,\eta_h, A, H_k) =
\frac{2}{ H_k \pi + \frac{H_k \pi (\eta_h-\eta_l)(\sqrt{\pi\ln2} – 1) }{(1+e^A)(\eta_h+ \sqrt{\pi\ln2} (1-\eta_h))}}
\left(
\frac{\eta_l}{1+\left(\frac{e^A+1}{e^A}\cdot\frac{x}{H_k}\right)^2} + (1-\eta_l)\sqrt{\pi\ln2} \cdot 2^{-\left(\frac{e^A+1}{e^A}\cdot\frac{x}{H_k}\right)^2}
\right)
\)
The substitution \(A \rightarrow e^{A}\) has been made from the original form. The above expression is for \(X<0\). For \(X>0\), replace \(\eta_h\Leftrightarrow\eta_l, A\Leftrightarrow -A\).
Symmetric Pearson VII:
\(
f(x, R, H_k) =
\frac{2\sqrt{2^{1/R}-1}\cdot \Gamma(R)}{H_k\sqrt\pi \cdot \Gamma(R-1/2)}
\left(
1 + (2^{1/R}-1) \left(\frac{2x}{H_k}\right)^2
\right)^{-R}
\)
f(x, R, H_k) =
\frac{2\sqrt{2^{1/R}-1}\cdot \Gamma(R)}{H_k\sqrt\pi \cdot \Gamma(R-1/2)}
\left(
1 + (2^{1/R}-1) \left(\frac{2x}{H_k}\right)^2
\right)^{-R}
\)
Split Pearson VII:
\(
f(x, R_l, R_h, A, H_k) =
\frac{2(1+e^A)}{ H_k \sqrt\pi}
\left(
e^A \frac{\Gamma(R_l-1/2)}{\sqrt{2^{1/R_l}-1}\cdot\Gamma(R_l)} + \frac{\Gamma(R_h-1/2)}{\sqrt{2^{1/R_h}-1}\cdot\Gamma(R_h)}
\right)^{-1}
\left(
1+(2^{1/R_l}-1)\left(\frac{e^A+1}{e^A}\cdot\frac{x}{H_k}\right)^2
\right)^{-R_l}
\)
f(x, R_l, R_h, A, H_k) =
\frac{2(1+e^A)}{ H_k \sqrt\pi}
\left(
e^A \frac{\Gamma(R_l-1/2)}{\sqrt{2^{1/R_l}-1}\cdot\Gamma(R_l)} + \frac{\Gamma(R_h-1/2)}{\sqrt{2^{1/R_h}-1}\cdot\Gamma(R_h)}
\right)^{-1}
\left(
1+(2^{1/R_l}-1)\left(\frac{e^A+1}{e^A}\cdot\frac{x}{H_k}\right)^2
\right)^{-R_l}
\)
The above is for \(X<0\). \(\Gamma()\) is the gamma function. For \(X>0\), replace \(R_h\Leftrightarrow R_l, A\Leftrightarrow -A\).