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# Landau Collision Operator

The fact that two-particle Coulomb collisions are dominated by small-angle scattering events allows some simplification of the Boltzmann collision operator in a plasma. According to Equations (3.23) and (3.88), the Boltzmann collision operator for two-body Coulomb collisions between particles of type (with mass and charge ) and particles of type (with mass and charge ) can be written

 (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

 (3.96)

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

 (3.102)

where is the impact parameter. Thus, we can write

 (3.103)

where the quantity

 (3.104)

is known as the Coulomb logarithm.

It follows from the previous analysis that

 (3.105)

If we define the tensor

 (3.106)

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 Einstein summation convention (that repeated indices are implicitly summed from to ) (Riley 1974). Hence, we deduce that

 (3.108)

Integration by parts yields

 (3.109)

However,

 (3.110)

because is a function of . Thus, we obtain the so-called Landau collision operator (Laudau 1936),

 (3.111)

where

 (3.112)

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

 (3.113)

where

 (3.114)

and

 (3.115) (3.116)

Next: Coulomb Logarithm Up: Collisions Previous: Rutherford Scattering Cross-Section
Richard Fitzpatrick 2016-01-23