Integral of Ion Distribution Function in Derivation of Dreicer Field
Disclaimer: I am including a fair bit of the physics background, but I believe this can be solved by someone with a strong calculus understanding
I am following the derivation of the critical electric field strength (Dreicer Field) that leads towards runaway electrons in a plasma exposed to a strong external electric field.
Source: https://journals-aps-org.stanford.idm.oclc.org/pr/pdf/10.1103/PhysRev.115.238
As part of the process they define the dynamical friction between two species (ions (i) and electrons (e)) using Fokker-Planck with Rosenbluth potentials.
The following definitions of the Rosenbluth potentials are given:
$H_{e,i} = \frac{m_e+m_i}{m_i}\Gamma_e\int{\frac{F_i(r,c',t)}{w}d^3c'}$
$w = |c-c'|$
$\Gamma_e= 4\pi\left(\frac{c^2}{4\pi \epsilon_0 m_e}\right)^2\log{\frac{\lambda}{\rho_0}}$
where:
* r: position space
* c: electron velocity space
* c': ion velocity space
* $F_i(r,c',t)$: 7 dimensional ion distribution function
everything else listed is assumed to be constant including $\log{\frac{\lambda}{\rho_0}}$
At a later point in the paper they define $F(r,c,t)$ in the general sense as a displaced Maxwellian distribution given as:
$F_\alpha(r,c,v(t)) = n(r)\left(\frac{\beta_\alpha(r)}{\pi}\right)^{3/2}\exp{\left(-\beta_\alpha(r)|c-v_\alpha(t)|^2\right)}$
$\beta_\alpha(r) = \frac{m_\alpha}{2kT_\alpha(r)}$
where $v_\alpha(t)$ is the bulk species velocity, and $T_\alpha(r)$ is the species temperature.
With this general definition of the species distribution function the author claims to take this definition and apply it to the previously listed definition of the Rosenbluth potentials to generate the outcome:
$H_{e,i} = \frac{m_e+m_i}{m_i}\Gamma_e\frac{\xi(\beta_i^{1/2}q)}{q}$
$\xi(x) = \frac{2}{\sqrt(\pi)}\int_{0}^{x}\exp{(-t^2)}dt$
$q = |c-v_i(t)|$
My question is how do the previous two definitions combine to produce this output? In my mind substituting and integrating does not produce this result, but my Math could be wrong. Does anyone with a better mind for Calculus have any thoughts on this? 402