qspec.normal_chi2_convolved_ex_pdf  (  ex ,  t ,  scale_e ,  e0 = 0  )[source]

The probability density at the energy $E_x$, distributed according to a convolution of a normal and a $\chi^2_1$ distribution $$\begin{aligned} \rho(E_x) &= \frac{1}{\sqrt{2\pi^2 k_\mathrm{B}T}\sigma_\mathrm{e}}\int\limits_0^\infty \frac{1}{\sqrt{\varepsilon}} \exp\left[-\frac{1}{2}\left(\frac{E_x - E_0 - \varepsilon}{\sigma_\mathrm{e}}\right)^2 - \frac{\varepsilon}{k_\mathrm{B}T}\right]\mathrm{d}\varepsilon\\[1ex] &= N\times\begin{cases} \sqrt{-\frac{\mu}{2\pi}}\exp[-(x + \mu)] K_{1/4}(x), & \text{if } \mu < 0 \\[1.5ex] \sqrt{\frac{\varpi}{\sqrt{\pi}}\sigma}, & \text{if } \mu = 0 \\[1.5ex] \sqrt{\pi\mu}\exp[-(x + \mu)] \frac{1}{2}\left[ I_{1/4}(x) + I_{-1/4}(x) \right], & \text{if } \mu > 0 \end{cases}\\[2ex] x &= \left(\!\frac{\mu}{2\sigma}\!\right)^{\!2},\quad\mu = \frac{E_x - E_0}{k_\mathrm{B}T} - \sigma^2, \quad\sigma = \frac{\sigma_\mathrm{e}}{k_\mathrm{B}T}\\[2ex] N &= \frac{1}{\sqrt{2\pi}\sigma_\mathrm{e}}\exp\left(-\frac{\sigma^2}{2}\right), \end{aligned}$$ where $I_\alpha$, and $K_\alpha$ are the modified Bessel functions of first and second kind, and $\varpi \approx 2.6220575543$ is the lemniscate constant.

Parameters:

exarray_like

The energy quantiles $E_x$ (eV).

tarray_like

The temperature $T$ of the environment (K).

scale_earray_like

The standard deviation $\sigma_\mathrm{e}$ of the normal distribution (eV).

e0array_like

The mean energy $E_0$ of the normal distribution (eV).

Returns:

rho_exndarray

The probability density in thermal equilibrium at the energy ex (1/eV).