Page 315 - Intro to Tensor Calculus

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which can be written as
~
dN ∂N F
~
= + V ·∇ ~r N + ·∇ ~ V N (2.5.73)
dt ∂t m
where d ~ V = ~ F represents any forces acting upon the particles. The Boltzmann equation can then be
dt m
expressed as
~
∂N F
~
+ V ·∇ ~r N + ·∇ ~ V N = Gains − Losses. (2.5.74)
∂t m
If the right-hand side of the equation (2.5.74) is zero, the equation is known as the Liouville equation. In
the special case where the velocities are constant and do not change with time the above equation (2.5.74)
can be written in terms of the flux φ and has the form

1 ∂
~
~
+ Ω ·∇ ~r +Σ s (E,~r)+Σ a (E,~r) φ(~r, E, Ω,t)= D C φ (2.5.75)
v ∂t
where
Z Z
~
~
0
~ 0
0
0 ~ 0
~ 0
D C φ = dΩ dE Σ(E → E, Ω → Ω)φ(~r, E , Ω ,t)+ S(~r, E, Ω,t).
The above equation represents the Boltzmann transport equation in the case where all the particles are
the same. In the case of atomic collisions of particles one must take into consideration the generation of
secondary particles resulting from the collisions.
Let there be a number of particles of type j in a volume element of phase space. For example j = p
(protons) and j = n (neutrons). We consider steady state conditions and define the quantities
~
(i) φ j (~r, E, Ω) as the flux of the particles of type j.
(ii) σ jk (Ω, Ω ,E,E ) the collision cross-section representing processes where particles of type k moving in
0
~ ~ 0
~
~ 0
direction Ω with energy E produce a type j particle moving in the direction Ω with energy E.
0
(iii) σ j (E)=Σ s (E,~r)+Σ a (E,~r) the cross-section for type j particles.
The steady state form of the equation (2.5.64) can then be written as
~
~
~
Ω ·∇φ j (~r, E, Ω)+σ j (E)φ j (~r, E, Ω)
Z
X (2.5.76)
~ ~ 0
0 ~ 0 ~ 0
0
= σ jk (Ω, Ω ,E,E )φ k (~r, E , Ω )dΩ dE 0
k
where the summation is over all particles k 6= j.
The Boltzmann transport equation can be represented in many different forms. These various forms
are dependent upon the assumptions made during the derivation, the type of particles, and collision cross-
sections. In general the collision cross-sections are dependent upon three components.
(1) Elastic collisions. Here the nucleus is not excited by the collision but energy is transferred by projectile
recoil.
(2) Inelastic collisions. Here some particles are raised to a higher energy state but the excitation energy is
not sufficient to produce any particle emissions due to the collision.
(3) Non-elastic collisions. Here the nucleus is left in an excited state due to the collision processes and
some of its nucleons (protons or neutrons) are ejected. The remaining nucleons interact to form a stable
structure and usually produce a distribution of low energy particles which is isotropic in character.