Page 316 - Intro to Tensor Calculus

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Various assumptions can be made concerning the particle flux. The resulting form of Boltzmann’s
equation must be modified to reflect these additional assumptions. As an example, we consider modifications
to Boltzmann’s equation in order to describe the motion of a massive ion moving into a region filled with a
homogeneous material. Here it is assumed that the mean-free path for nuclear collisions is large in comparison
with the mean-free path for ion interaction with electrons. In addition, the following assumptions are made
(i) All collision interactions are non-elastic.
(ii) The secondary particles produced have the same direction as the original particle. This is called the
straight-ahead approximation.
(iii) Secondary particles never have kinetic energies greater than the original projectile that produced them.
(iv) A charged particle will eventually transfer all of its kinetic energy and stop in the media. This stopping
distance is called the range of the projectile. The stopping power S j (E)= dE represents the energy
dx
loss per unit length traveled in the media and determines the range by the relation dR j = 1 or
dE S j (E)
R E dE 0
1
R j (E)= 0 . Using the above assumptions Wilson, et.al. show that the steady state linearized
0 S j (E )
Boltzmann equation for homogeneous materials takes on the form
1 ∂
~ ~ ~ ~
Ω ·∇φ j (~r, E, Ω) − (S j (E)φ j (~r, E, Ω)) + σ j (E)φ j (~r, E, Ω)
A j ∂E
Z (2.5.77)
X
= dE dΩ σ jk (Ω, Ω ,E,E )φ k (~r, E , Ω )
0 ~ 0
0
0 ~ 0
~ ~ 0
k6=j
~
where A j is the atomic mass of the ion of type j and φ j (~r, E, Ω) is the flux of ions of type j moving in
~
the direction Ω with energy E.
Observe that in most cases the left-hand side of the Boltzmann equation represents the time rate of
change of a distribution type function in a phase space while the right-hand side of the Boltzmann equation
represents the time rate of change of this distribution function within a volume element of phase space due
to scattering and absorption collision processes.

Boltzmann Equation for gases
~
Consider the Boltzmann equation in terms of a particle distribution function f(~r, V, t) which can be
written as
!
~
∂ F
~
~
~
+ V ·∇ ~r + ·∇ ~ V f(~r, V, t)= D C f(~r, V, t) (2.5.78)
∂t m
for a single species of gas particles where there is only scattering and no absorption of the particles. An
element of volume in phase space (x, y, z, V x ,V y ,V z ) can be thought of as a volume element dτ = dxdydz
for the spatial elements together with a volume element dτ v = dV x dV y dV z for the velocity elements. These
~
elements are centered at position ~ and velocity V at time t. In phase space a constant velocity V 1 can be
r
2
2
2
2
thought of as a sphere since V = V + V + V . The phase space volume element dτdτ v changes with time
1
y
x
z
~
r
since the position ~ and velocity V change with time. The position vector ~ changes because of velocity
r
1 John W. Wilson, Lawrence W. Townsend, Walter Schimmerling, Govind S. Khandelwal, Ferdous Kahn,
John E. Nealy, Francis A. Cucinotta, Lisa C. Simonsen, Judy L. Shinn, and John W. Norbury, Transport
Methods and Interactions for Space Radiations, NASA Reference Publication 1257, December 1991.