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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

Source

C++ header: Bk2BkExpConvPV.h

C++ source: Bk2BkExpConvPV.cpp