Collision Times

(3.167) |

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

(3.168) |

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

where and . The net force per unit volume acting on type particles due to collisions with type particles is thus

Suppose that the drift velocity, , is much smaller than the thermal velocity, , of type particles. In this case, we can write

(3.171) |

Hence, Equations (3.169) and (3.170) yield

However, it follows from symmetry that

(3.173) | ||

(3.174) | ||

(3.175) |

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

(3.176) |

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

(3.177) |

Integration by parts gives

(3.178) |

which reduces to

The *collision time*,
, associated with collisions of particles of type
with particles of type
, is conventionally
defined via the following equation,

(3.180) |

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

Consider a quasi-neutral plasma consisting of electrons of mass
, charge
, and number density
, and ions
of mass
, charge
, and number density
. Let the two species both have Maxwellian distributions characterized by a
common temperature
, 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*,

(3.182) |

which is the mean time required for the direction of motion of an individual electron to deviate through approximately as a consequence of collisions with other electrons. Second, the

(3.183) |

which is the mean time required for the direction of motion of an individual electron to deviate through approximately as a consequence of collisions with ions. Third, the

(3.184) |

which is the mean time required for the direction of motion of an individual ion to deviate through approximately as a consequence of collisions with other ions. Finally, the

(3.185) |

which is the mean time required for the direction of motion of an individual ion to deviate through approximately 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,

(3.186) |

which implies that electrons scatter electrons (through ) 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

(3.187) |

Given that (see Section 3.10), where is the plasma parameter (see Section 1.6), we obtain the estimate (see Section 1.7)

(3.188) |

where is the electron plasma frequency (see Section 1.4). Likewise, the ion-ion collision frequency is such that

(3.189) |

where is the ion plasma frequency.