.. index:: Bk2BkExpConvPV
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.
.. attributes::
.. properties::
.. categories::
.. sourcelink::