Collision Times

It is conventional to define the collision time, $\tau_{ss'}$, associated with collisions of species-$s$ particles with species-$s'$ particles such that

$${\bf F}_{ss'} = - \frac{m_s\,n_s}{\tau_{ss'}}\,{\bf V}_{ss'}.$$ (3.184)

It follows from Equation (3.180) that

$$\tau_{ss'} = \frac{3\pi^{1/2}\,m_s\,\mu_{ss'}\,(v_{t\,s}^{2}+v_{t\,{s'}}^{2})^{3/2}}{8\,\gamma_{ss'}\,n_{s'}}.$$ (3.185)

Furthermore, when expressed in terms of the collision time, expression (3.184) for the collisional heating rate becomes

$$W_{ss'} = -\frac{3\,\mu_{ss'}\,n_s}{\tau_{ss'}\,m_{s'}}\,(T_s-T_{s'}).$$ (3.186)

According to the definition (3.185), the collision time, $\tau_{ss'}$, is the time required for collisions with species-$s'$ particles to decelerate species-$s$ particles to such an extent that the mean drift velocity of the latter particles with respect to the former is eliminated. At the individual particle level, the collision time is the mean time required for the direction of motion of an individual species-$s$ particle to deviate through approximately $90^\circ$ as a consequence of collisions with species-$s'$ particles.

Consider a quasi-neutral plasma consisting of electrons of mass $m_e$, charge $-e$, and number density $n_e$, and ions of mass $m_i$, charge $+e$, and number density $n_i=n_e$. Let the two species both have Maxwellian distributions characterized by a common temperature $T$, and a small relative drift velocity. It follows, from the previous analysis, that we can identify four different collision times. First, the electron-electron collision time,

$$\tau_{ee} =\frac{12\pi^{3/2}\,\epsilon_0^{2}\,m_e^{1/2}\,T^{3/2}}{\ln{\mit\Lambda}_c\,e^{4}\,n_e},$$ (3.187)

which is the mean time required for the direction of motion of an individual electron to deviate through approximately $90^\circ$ as a consequence of collisions with other electrons. Second, the electron-ion collision time,

$$\tau_{ei} =\frac{6\!\sqrt{2}\pi^{3/2}\,\epsilon_0^{2}\,m_e^{1/2}\,T^{3/2}}{\ln{\mit\Lambda}_c\,e^{4}\,n_e},$$ (3.188)

which is the mean time required for the direction of motion of an individual electron to deviate through approximately $90^\circ$ as a consequence of collisions with ions. (Here, we have made use of the fact that $m_e\ll m_i$.) Third, the ion-ion collision time,

$$\tau_{ii} =\frac{12\pi^{3/2}\,\epsilon_0^{2}\,m_i^{1/2}\,T^{3/2}}{\ln{\mit\Lambda}_c\,e^{4}\,n_e},$$ (3.189)

which is the mean time required for the direction of motion of an individual ion to deviate through approximately $90^\circ$ as a consequence of collisions with other ions. Finally, the ion-electron collision time,

$$\tau_{ie} =\frac{6\!\sqrt{2}\pi^{3/2}\,\epsilon_0^{2}\,m_i\,T^{3/2}}{\ln{\mit\Lambda}_c\,e^{4}\,n_e\,m_e^{1/2}},$$ (3.190)

which is the mean time required for the direction of motion of an individual ion to deviate through approximately $90^\circ$ as a consequence of collisions with electrons. Note that these collision times are not all of the same magnitude, as a consequence of the large difference between the electron and ion masses. In fact,

$$\tau_{ee}\sim \tau_{ei}\sim (m_e/m_i)^{1/2}\,\tau_{ii}\sim (m_i/m_e)\,\tau_{ie},$$ (3.191)

which implies that electrons scatter electrons (through $90^\circ$) at about the same rate that ions scatter electrons, but that ions scatter ions at a significantly lower rate than ions scatter electrons, and, finally, that electrons scatter ions at a significantly lower rate than ions scatter ions.

The collision frequency is simply the inverse of the collision time. Thus, the electron-electron collision frequency is written

$$\nu_{ee} \equiv \frac{1}{\tau_{ee}}= \frac{\ln{\mit\Lambda}_c\,e^{4}\,n_e}{12\,\pi^{3/2}\,\epsilon_0^{2}\,m_e^{1/2}\,T^{3/2}}.$$ (3.192)

Given that $\ln{\mit\Lambda}_c\sim \ln {\mit\Lambda}$ (see Section 3.10), where ${\mit\Lambda} =4\pi\,\epsilon_0^{3/2}\,T^{3/2}/(e^{3}\,n_e^{1/2})$ is the plasma parameter (see Section 1.6), we obtain the estimate (see Section 1.7)

$$\nu_{ee}\sim \frac{\ln{\mit\Lambda}}{{\mit\Lambda}}\,{\mit\Pi}_e$$ (3.193)

where ${\mit\Pi}_e=(n_e\,e^{2}/\epsilon_0\,m_e)^{1/2}$ is the electron plasma frequency (see Section 1.4). Likewise, the ion-ion collision frequency is such that

$$\nu_{ii}\equiv \frac{1}{\tau_{ii}}\sim \frac{\ln{\mit\Lambda}}{{\mit\Lambda}}\,{\mit\Pi}_i,$$ (3.194)

where ${\mit\Pi}_i=(n_i\,e^{2}/\epsilon_0\,m_i)^{1/2}$ is the ion plasma frequency.