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# Bk2BkExpConvPV¶

## Description¶

A back-to-back exponential convoluted pseudo-voigt function is defined as:

$F(X) = I \cdot \Omega(x)$

where $$\Omega$$ is defined to be

$\Omega(x) = (1-\eta)N\left\{e^u\mathit{erfc}(y)+e^v\mathit{erfc}(z)\right\} - \frac{2N\eta}{\pi}\left\{ \Im(e^p\mathit{E}_1(p))+ \Im(e^q\mathit{E}_1(q)) \right \},$

given that

$u=\frac{1}{2}\alpha\left( \alpha\sigma^{2}+2(x-X0) \right),$
$y=\frac{1}{\sqrt{2\sigma^{2}}}(\alpha\sigma^{2}+x-X0),$
$v=\frac{1}{2}\beta\left( \beta\sigma^{2}-2(x-X0) \right),$
$z=\frac{1}{\sqrt{2\sigma^{2}}}(\beta\sigma^{2}-x+X0),$
$p=\alpha(x-X0)+\frac{\alpha H}{2}i,$
$q=-\beta(x-X0)+\frac{\beta H}{2}i,$
$N = \frac{\alpha\beta}{2(\alpha+\beta)}.$

$$\eta$$ is approximated by

$\eta = 1.36603\frac{\gamma}{H} - 0.47719\left(\frac{\gamma}{H}\right)^2 + 0.11116\left(\frac{\gamma}{H}\right)^3,$

where,

$H = \gamma^5+0.07842\gamma^4H_G+4.47163\gamma^3H_G^2+2.42843\gamma^2H_G^3+2.69269\gamma H_G^4+H_G^5,$
$H_G=\sqrt{8\sigma^2\log(2)}.$

$$\mathit{erfc}$$ is the complementary error function and $$\mathit{E}_1$$ is the exponential integral with complex argument given by

$\mathit{erfc}(x) = 1 - \text{erf}(x) = 1 - \frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-u^{2}}du = \frac{2}{\sqrt{\pi}}\int_{x}^{\infty}e^{-u^{2}}du,$
$\mathit{E}_1(z) = \int_{z}^{\infty} \frac{e^{-t}}{t}dt.$

The parameters $$A$$ and $$B$$ represent the absolute value of the exponential rise and decay constants (modelling the neutron pulse coming from the moderator) and $$S$$ represent the standard deviation of the gaussian. The parameter $$X0$$ is the location of the peak; more specifically it represent the point where the exponentially modelled neutron pulse goes from being exponentially rising to exponentially decaying. $$I$$ is the integrated intensity.

For information about how to convert Fullprof back-to-back exponential parameters into those used for this function see CreateBackToBackParameters.

## Properties (fitting parameters)¶

Name

Default

Description

X0

-0.0

Location of the peak

Intensity

0.0

Integrated intensity

Alpha

0.04

Exponential rise

Beta

0.02

Exponential decay

Sigma2

1.0

Sigma squared

Gamma

0.0

Categories: FitFunctions | Peak