Date: 2005-02-14
Some supplementary information for
B. J. Braams and C. F. F. Karney,
Conductivity of a relativistic plasma,
Phys. Fluids 1B(7), 1355–1368 (1989).
doi:10.1063/1.858966
arXiv:plasm-ph/9502001
PPPL–2598
Equation (37) can be generalized to give the collision term for a Maxwellian colliding off an arbritrary azimuthally symmetric spherical harmonic: \[ \begin{align} &\hspace{-2.5em} \frac{C^{s/s'}(f_{sm}(u),f_{s'l}(u)P_l(\cos\theta))}{f_{sm}(u)P_l(\cos\theta)}\\ &=\frac{4\pi\Gamma^{s/s'}}{n_{s'}}\biggl\{ \frac{m_s}{m_{s'}} \biggl[\frac1\gamma \psi_{{s'}l[0]{}} - \frac u{u_{ts}^2} \frac{d\psi_{{s'}l[1]{1}}}{du} - \frac2{c^2 \gamma} \psi_{{s'}l[1]{1}} + \frac{2u}{c^2u_{ts}^2} \frac{d\psi_{{s'}l[2]{11}}}{du}\biggr] + \frac u{u_{ts}^2} \frac{d\psi_{{s'}l[1]{0}}}{du} \\&\qquad{} - \biggl(\frac{u^2}{\gamma u_{ts}^4} - \frac1{u_{ts}^2}\biggr) \psi_{{s'}l[1]{0}} +\biggl(\frac{2\gamma u}{u_{ts}^4} - \frac {2u}{c^2u_{ts}^2}\biggr) \frac{d\psi_{{s'}l[2]{02}}}{du} \\&\qquad{} -\biggl(\frac{l(l+1)}{\gamma u_{ts}^4}-\frac2{c^2u_{ts}^2}\biggr) \psi_{{s'}l[2]{02}} -\frac {8\gamma u}{c^2u_{ts}^4} \frac{d\psi{{s'}l[3]{022}}}{du} +\frac{4(l+2)(l-1) + 8\gamma^2}{\gamma c^2u_{ts}^4}\psi_{{s'}l[3]{022}}\biggr\}. \end{align} \]
There is a sign error in the exponential term in Equation (41). It should read: \[ \bar\sigma=\frac1{3\Theta^{7/2}K_2(\Theta^{-1})} \biggl(\frac{E_1(\Theta^{-1})}\Theta-(1-\Theta+2\Theta^2-6\Theta^3-24\Theta^4-24\Theta^5) \exp(-\Theta^{-1})\biggr). \]