Coulomb Logarithm

According to Equation (3.85), the Coulomb logarithm can be written

$$\ln {\mit\Lambda}_c =\int\frac{d\chi}{\chi},$$ (3.114)

where we have made use of the fact that scattering angle $\chi$ is small, Obviously, the integral appearing in the previous expression diverges at both large and small $\chi$.

The divergence of the integral on the right-hand side of the previous equation at large $\chi$ is a consequence of the breakdown of the small-angle approximation. The standard prescription for avoiding this divergence is to truncate the integral at some $\chi_{\rm max}$ above which the small-angle approximation becomes invalid. According to Equation (3.84), this truncation is equivalent to neglecting all collisions whose impact parameters fall below the value

$$b_{\rm min} \simeq \frac{e_s\,e_{s'}}{2\pi\,\epsilon_0\,\mu_{ss'}\,u_{ss'}^{2}}.$$ (3.115)

The ultimate justification for the truncation of the integral appearing in Equation (3.114) at large $\chi$ is the idea that Coulomb collisions are dominated by small-angle scattering events, and that the occasional large-angle scattering events have a negligible effect on the scattering statistics. Unfortunately, this is not quite true (if it were then the integral would converge at large $\chi$). However, the rare large-angle scattering events only make a relatively weak logarithmic contribution to the scattering statistics.

Making the estimate $(1/2)\,\mu_{ss'}\,u_{ss'}^{2}\simeq T$, where $T$ is the assumed common temperature of the two colliding species, we obtain

$$b_{\rm min} \simeq \frac{e_s\,e_{s'}}{4\pi\,\epsilon_0\,T}= r_c,$$ (3.116)

where $r_c$ is the classical distance of closest approach introduced in Section 1.6. However, as mentioned in Section 1.10, it is possible for the classical distance of closest approach to fall below the de Broglie wavelength of one or both of the colliding particles, even in the case of a weakly coupled plasma. In this situation, the most sensible thing to do is to approximate $b_{\rm min}$ as the larger de Broglie wavelength (Spitzer 1956; Braginskii 1965).

The divergence of the integral on the right-hand side of Equation (3.114) at small $\chi$ is a consequence of the infinite range of the Coulomb potential. The standard prescription for avoiding this divergence is to take the Debye shielding of the Coulomb potential into account. (See Section 1.5.) This is equivalent to neglecting all collisions whose impact parameters exceed the value

$$b_{\rm max} = \lambda_D,$$ (3.117)

where $\lambda_D$ is the Debye length. Of course, Debye shielding is a many-particle effect. Hence, the Landau collision operator can no longer be regarded as a pure two-body collision operator. Fortunately, however, many-particle effects only make a relatively weak logarithmic contribution to the operator.

According to Equations (3.86), (3.116), and (3.117),

$$\ln{\mit\Lambda}_c = \ln\left(\frac{b_{\rm max}}{b_{\rm min}}\right) = \ln\left(\frac{\lambda_D}{r_c}\right).$$ (3.118)

Thus, we deduce from Equation (1.20) that

$$\ln{\mit\Lambda}_c\simeq \ln {\mit\Lambda}.$$ (3.119)

In other words, the Coulomb logarithm is approximately equal to the natural logarithm of the plasma parameter. The fact that the plasma parameter is much larger than unity in a weakly coupled plasma implies that the Coulomb logarithm is large compared to unity in such a plasma. In fact, $\ln{\mit\Lambda}_c$ lies in the range $10$–$20$ for typical weakly coupled plasmas. It also follows that $b_{\rm max}\gg b_{\rm min}$ in a weakly coupled plasma, which means that there is a large range of impact parameters for which it is accurate to treat Coulomb collisions as small-angle two-body scattering events.

The conventional definition of the Coulomb logarithm is as follows (Richardson 2019). For a species-$s$ particle, with mass $m_s$, charge $e_s$, number density $n_s$, and temperature $T_s$, scattered by species-$s'$ particles, with mass $m_{s'}$, charge $e_{s'}$, number density $n_{s'}$, and temperature $T_{s'}$, the Coulomb logarithm is defined $\ln{\mit\Lambda}_c= \ln(b_{\rm max}/b_{\rm min})$. Here, $b_{\rm min}$ is the larger of $e_s\,e_{s'}/(4\pi\,\epsilon_0\,\mu_{ss'}\,u_{ss'}^{2})$ and $\hbar/(2\,\mu_{ss'}\,u_{ss'})$, averaged over both particle distributions, where $\mu_{ss'}=m_s\,m_{s'}/(m_s+m_{s'})$ and ${\bf u}_{ss'}={\bf v}_s-{\bf v}_{s'}$. Furthermore, $b_{\rm max} =(\sum_s n_s\,e_s^{2}/\epsilon_0\,T_s)^{-1/2}$, where the summation extends over all species, $s$, for which $\bar{u}_{ss'}^{2}\lesssim T_s/m_s$.

Consider a quasi-neutral plasma consisting of electrons of mass $m_e$, charge $-e$, number density $n_e$, and temperature $T_e$, and ions of mass $m_i$, charge $+e$, number density $n_i=n_e$, and temperature $T_i$. For thermal (i.e., Maxwellian) electron-electron collisions, we obtain (Richardson 2019)

$$\ln{\mit\Lambda}_c = 23.5 - \ln\left(n_e^{1/2}\,T_e^{-5/4}\right)-\left[10^{-5} + (\ln T_e-2)^2/16\right]^{1/2}.$$ (3.120)

Likewise, for thermal electron-ion collisions

$$\ln{\mit\Lambda}_c = 24 - \ln\left(n_e^{1/2}\,T_e^{-1}\right) 10\,{\rm eV} <T_e,$$

$$\ln{\mit\Lambda}_c =23 - \ln\left(n_e^{1/2}\,T_e^{-3/2}\right) T_e<10\,{\rm eV}.$$ (3.121)

Finally, for thermal ion-ion collisions,

$$\ln{\mit\Lambda}_c = 23 - \ln\left(n_e^{1/2}\,T_i^{-3/2}\right).$$ (3.122)

Here, $n_e$ is measured in units of ${\rm cm}^{-3}$, and all species temperatures are measured in units of electron-volts.

The standard approach in plasma physics is to treat the Coulomb logarithm as a constant, with a value determined by the ambient electron number density, and the ambient electron and ion temperatures, as has just been described. This approximation ensures that the Landau collision operator, $C_{ss'}(f_s,f_{s'})$, is strictly bilinear in its two arguments.