Fundamentals of Plasma Physics

13.1 Introduction 383

Adding Eqs.(13.1a) and (13.1b) gives

(mT+mF)R ̈=mT ̈rT+mF ̈rF=0 (13.3)

showing that the center of mass velocityR ̇ is a constant of the motion and so does not
change as a result of the collision. If∆ ̇rTand∆ ̇rFare defined as the change in respective
velocities of the two particles during a collision, then it is seen that these changes are not
independent, but are related by

(mT+mF)∆R ̇=0=mT∆ ̇rT+mF∆ ̇rF. (13.4)

It is useful to define the relative position vector

r=rT−rF (13.5)

and the reduced mass
1
μ

=

1

mT

+

1

mF

. (13.6)

Then, dividing Eqs.(13.1a) and (13.1b) by their respective masses and taking the difference
between the resulting equations gives an equation of motion for the relativevelocity

μ ̈r=

qTqF

4 πε 0 r^2

ˆr. (13.7)

Solving Eqs.(13.2) and (13.5) forrTandrFgives

rF = R−

μ

mF

r (13.8)

rT = R+

μ

mT

r

and so the respective test and field velocities are

̇rF = R ̇−

μ

mF

̇r (13.9)

̇rT = R ̇+

μ

mT

̇r. (13.10)

Since∆R ̇ = 0 the change of the test and field particle velocities as measured in the lab
frame can be related to the change in the relative velocity by

∆ ̇rF = −

μ

mF

∆ ̇r (13.11)

∆ ̇rT =+

μ

mT

∆ ̇r. (13.12)

The collision problem is first solved in the center of mass frame to find the change in
the relative velocity and then the center of mass result is transformed to the lab-frame to
determine the change in the lab-frame velocity of the particles.
Let us now consider a many-particle point of view. Suppose a mono-energetic beam of
particles impinges upon a background plasma. Several effects are expected to occur due
to collisions between the beam and the background plasma. First, there willbe a slowing
down of the beam as it loses momentum due to collisions. Second, there should be a
spreading out of the velocity distribution of the particles in the beam since collisions will
tend to diffuse the velocity of the particles in the beam. Meantime, the background plasma
should be heated and also should gain momentum due to the collisions. Eventually, the
beam should be so slowed down and so spread out that it becomes indistinguishablefrom
the background plasma which will be warmer because of the energy transferred from the
beam.