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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. For information about how to create parameters from a GSAS parameter file see CreateBackToBackParametersGSAS.

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

Source

C++ header: Bk2BkExpConvPV.h

C++ source: Bk2BkExpConvPV.cpp