Collision Times

Consider collisions between particles of type $1$ (with number density $n_1$ and mass $m_1$ ), possessing the Maxwellian distribution function

$$f_1({\bf v})=n_1\left(\frac{m_1}{2\pi\,T}\right)^{3/2}\exp\left[-\frac{m_1\,({\bf v}-{\bf V})^2}{2\,T}\right],$$ (3.167)

and particles of type $2$ (with number density $n_2$ and mass $m_2$ ), possessing the Maxwellian distribution function

$$f_2({\bf v})=n_2\left(\frac{m_2}{2\pi\,T}\right)^{3/2}\exp\left(-\frac{m_2\,v^2}{2\,T}\right).$$ (3.168)

Here, $T$ is the common temperature of the two species, and ${\bf V}$ is the mean drift velocity of species $1$ relative to species $2$ . As we saw in the previous section, collisions with particles of type $2$ give rise to a velocity-dependent force acting on individual particles of type $1$ . This force takes the form [see Equation (3.165)]

$${\bf R}_{12} =- \frac{\gamma_{12}\,n_2}{m_2}\left[-\frac{F_2(\zet... ...3}\,{\bf V} + \frac{3\,F_3(\zeta)}{v^5}\,({\bf v}\cdot{\bf V})\,{\bf v}\right],$$ (3.169)

where $\zeta=v/v_{t\,2}$ and $v_{t\,2} = \sqrt{2\,T/m_2}$ . The net force per unit volume acting on type $1$ particles due to collisions with type $2$ particles is thus

$${\bf F}_{12} = \int m_1\,{\bf v}\,C_{12}\,d^3{\bf v} = \int {\bf R}_{12}\,f_1\,d^3{\bf v}.$$ (3.170)

Suppose that the drift velocity, ${\bf V}$ , is much smaller than the thermal velocity, $v_{t\,1}=\sqrt{2\,T/m_1}$ , of type $1$ particles. In this case, we can write

$$f_1({\bf v})\simeq \frac{n_1}{\pi^{3/2}\,v_{t\,1}^{\,3}}\,\exp\le... ...c{v^2}{v_{t\,1}^{\,2}}\right)\left(1-\frac{2\,{\bf v}\cdot{\bf V}}{v^2}\right).$$ (3.171)

Hence, Equations (3.169) and (3.170) yield

$$({\bf F}_{12})_i = -\frac{\gamma_{12}\,n_1\,n_2}{\pi^{3/2}\,m_2\,v_{t\,1}^{\,3}}\i... ...(\zeta)}{v^3}\,V_i + \frac{3\,F_3(\zeta)}{v^5}\,v_k\,V_k\,v_i\right]d^3{\bf v}.$$ (3.172)

However, it follows from symmetry that

$$\int H(v)\,v_i\,d^3{\bf v} =0,$$ (3.173)

$$\int H(v)\,v_i\,v_j\,d^3{\bf v} = \frac{\delta_{ij}}{3}\int H(v)\,v^2\,d^3{\bf v},$$ (3.174)

$$\int H(v)\,v_i\,v_j\,v_k\,d^3{\bf v} =0,$$ (3.175)

where $H(v)$ is a general function. Hence, Equation (3.172) reduces to

$${\bf F}_{12} = -\frac{\gamma_{12}\,n_1\,n_2\,{\bf V}}{\pi^{3/2}\,... ...v_{t\,1}^{\,2}}\right)\left[\frac{F_3(\zeta)-F_2(\zeta)}{v^3}\right]d^3{\bf v}.$$ (3.176)

It follows from Equations (3.151) and (3.152) that

$${\bf F}_{12} = -\frac{16\,\gamma_{12}\,n_1\,n_2\,{\bf V}}{3\,\pi^... ...a)\,\exp\left(-\frac{v_{t\,2}^{\,2}}{v_{t\,1}^{\,2}}\,\zeta^{\,2}\right)d\zeta.$$ (3.177)

Integration by parts gives

$${\bf F}_{12} = -\frac{16\,\gamma_{12}\,n_1\,n_2\,{\bf V}}{3\,\pi\... ...-\left(1+\frac{v_{t\,2}^{\,2}}{v_{t\,1}^{\,2}}\right) \zeta^{\,2}\right]d\zeta,$$ (3.178)

which reduces to

$${\bf F}_{12} = -\left[\frac{8\,\gamma_{12}\,n_1\,n_2}{3\,\pi^{1/2}\,m_2\,v_{t\,2}^{\,2}\,(v_{t\,1}^{\,2}+v_{t\,2}^{\,2})^{1/2}}\right]{\bf V}.$$ (3.179)

The collision time, $\tau_{12}$ , associated with collisions of particles of type $1$ with particles of type $2$ , is conventionally defined via the following equation,

$${\bf F}_{12} = - \frac{m_1\,n_1}{\tau_{12}}\,{\bf V}.$$ (3.180)

According to this definition, the collision time is the time required for collisions with particles of type $2$ to decelerate particles of type $1$ 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 type $1$ particle to deviate through approximately $90^\circ$ as a consequence of collisions with particles of type $2$ . According to Equations (3.112) and (3.179), we can write

$$\tau_{12} =\frac{3\pi^{1/2}\,m_1\,T^{3/2}}{2\sqrt{2}\,\mu_{12}^{\... ...1\,T^{\,3/2}}{\ln{\mit\Lambda}_c\,\mu_{12}^{\,1/2}\,e_1^{\,2}\,e_2^{\,2}\,n_2}.$$ (3.181)

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,

$$\tau_{ee} =\frac{12\sqrt{2}\,\pi^{3/2}\,\epsilon_0^{\,2}\,m_e^{\,1/2}\,T^{3/2}}{\ln{\mit\Lambda}_c\,e^4\,n_e},$$ (3.182)

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,

$$\tau_{ei} =\frac{6\sqrt{2}\,\pi^{3/2}\,\epsilon_0^{\,2}\,m_e^{\,1/2}\,T^{3/2}}{\ln{\mit\Lambda}_c\,e^4\,n_e},$$ (3.183)

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. Third, the ion-ion collision time,

$$\tau_{ii} =\frac{12\sqrt{2}\,\pi^{3/2}\,\epsilon_0^{\,2}\,m_i^{\,1/2}\,T^{3/2}}{\ln{\mit\Lambda}_c\,e^4\,n_e},$$ (3.184)

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,

$$\tau_{ie} =\frac{6\sqrt{2}\,\pi^{3/2}\,\epsilon_0^{\,2}\,m_i\,T^{3/2}}{\ln{\mit\Lambda}_c\,e^4\,n_e\,m_e^{\,1/2}},$$ (3.185)

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,

$$\tau_{ee}\sim \tau_{ei}\sim (m_e/m_i)^{1/2}\,\tau_{ii}\sim (m_i/m_e)\,\tau_{ie},$$ (3.186)

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

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

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)

$$\nu_{ee}\sim \frac{\ln{\mit\Lambda}}{{\mit\Lambda}}\,{\mit\Pi}_e$$ (3.188)

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

$$\nu_{ii}\equiv \frac{1}{\tau_{ii}}\sim \frac{\ln{\mit\Lambda}}{{\mit\Lambda}}\,{\mit\Pi}_i,$$ (3.189)

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