TwoBody Coulomb Collisions
Consider a twobody Coulomb collision between a species particle, with mass and charge , and a species particle,
with mass and charge . The equations of motion of the two particles take the form
where

(3.40) 
and
denotes a time derivative.
Here, and
are the respective position vectors, and
is
the relative position vector.
It is easily demonstrated that
where

(3.43) 
is the vector position of the center of mass (which does not accelerate). Equations (3.38) and (3.39) can be combined to give a single
equation of relative motion,

(3.44) 
Two relations that immediately follow from the previous equation are
where

(3.47) 
is the conserved angular momentum per unit mass, and

(3.48) 
the conserved energy.
Equation (3.47) implies that
. This is the equation of a plane that passes through the origin, and whose normal is parallel
to the constant vector
. We, therefore, conclude that the relative position vector is constrained to lie in this plane,
which implies that the trajectories of both colliding particles are coplanar.
Let the plane
coincide with the  plane, so that we can write
. It is convenient to
define the standard plane polar coordinates
and
.
When expressed in terms of these coordinates, the conserved angular momentum per unit mass becomes

(3.49) 
where

(3.50) 
Furthermore, the conserved energy takes the form

(3.51) 
Suppose that , where
and
. It follows that

(3.52) 
Hence, Equation (3.51) transforms to give

(3.53) 
It is convenient to define the relative velocity at large distances,

(3.54) 
as well as the impact parameter,

(3.55) 
The latter parameter is simply the distance of closest approach of the two particles in the situation in which there is no Coulomb force acting between them,
and they, consequently, move in straightlines. (See Figure 3.1.) The previous three equations can be combined to give

(3.56) 
Figure 3.1:
A twobody Coulomb collision.

Figure 3.1 shows the collision in a frame of reference in which the species particle remains stationary at the origin, ,
whereas the species particle traces out the path . Point corresponds to the closest approach of the
two particles. It follows, by symmetry (because Coulomb collisions are reversible), that the angles and
shown in the figure are equal to one another. Hence, we deduce that

(3.57) 
Here, is the angle through which the path of the species (or the species particle) is deviated as a consequence of the collision,
whereas
is the angle through which the relative position vector,
, rotates as the species particle
moves from point (which is assumed to be infinity far from point ) to point . Suppose that point
corresponds to . It follows that

(3.58) 
Here,
, where
is the distance of closest approach.
Now, by symmetry,
, so Equation (3.56) implies that

(3.59) 
Combining Equations (3.56) and (3.58), we obtain

(3.60) 
where
, and

(3.61) 
Integration (Speigel, Liu, and Lipschutz 1999) yields

(3.62) 
Hence, from Equation (3.57), we get

(3.63) 
which can be rearranged to give

(3.64) 
Note that the analysis in this section tacitly assumes that Coulomb collisions are not significantly modified by
any magnetic field that might pervade the plasma. This assumption is justified provided that the particle gyroradii are all much
larger than the Debye length.