(3.89) |

where

(3.90) |

Here, is the relative velocity prior to a collision, and , where is the angle of deflection, and is an azimuthal angle that determines the orientation of the plane in which a given two-body collision occurs. Recall that , , , and are short-hand for , , , and , respectively. Finally, .

The type and type particle velocities prior to the collision are and , respectively, so that . Let us write the corresponding velocities after the collision as (see Section 3.3)

(3.91) | ||

(3.92) |

Here, is assumed to be small, which implies that the angle of deflection is also small. Expanding to second order in , we obtain

(3.93) |

Likewise, expanding , we get

(3.94) |

Note that, in writing the previous two equations, we have neglected the and dependence of , et cetera, for ease of notation. Hence,

(3.95) |

It follows that

where

(3.97) |

Let , , and be a right-handed set of mutually orthogonal unit vectors. Suppose that . Recall that . Now, in an elastic collision for which the angle of deviation is , we require , , and when . In other words, we need , , and when . We deduce that

(3.98) |

Thus,

(3.99) |

and

(3.100) |

where use has again been made of the fact that is small.

Now,

(3.101) |

where and are the maximum and minimum angles of deflection, respectively. However, according to Equation (3.82), small-angle two-body Coulomb collisions are characterized by

where is the impact parameter. Thus, we can write

where the quantity

is known as the

It follows from the previous analysis that

(3.105) |

If we define the tensor

then it is readily seen that

(3.107) |

Here, , , et cetera, run from to , and correspond to Cartesian components. Moreover, we have made use of the

(3.108) |

Integration by parts yields

(3.109) |

However,

because is a function of . Thus, we obtain the so-called

where

It is sometimes convenient to write the Landau collision operator in the form

where

and