Page 313 - Intro to Tensor Calculus
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will be interested in examining how the particles in a volume element of phase space change with time. We
introduce the following notation:
r
(i) ~ the position vector of a typical particle of phase space and dτ = dxdydz the corresponding spatial
volume element at this position.
~
(ii) V the velocity vector associated with a typical particle of phase space and dτ v = dV x dV y dV z the
corresponding velocity volume element.
~
~
~
(iii) Ω a unit vector in the direction of the velocity V = vΩ.
1
2
(iv) E = mv kinetic energy of particle.
2
~
~
~
(v) dΩ is a solid angle about the direction Ωand dτ dE dΩ is a volume element of phase space involving the
solid angle about the direction Ω.
~
r
(vi) n = n(~r, E, Ω,t) the number of particles in phase space per unit volume at position ~ per unit velocity
~
~
~
~
at position V per unit energy in the solid angle dΩattime t and N = N(~r, E, Ω,t)= vn(~r, E, Ω,t)
~
the number of particles per unit volume per unit energy in the solid angle dΩattime t. The quantity
~
~
r
N(~r, E, Ω,t)dτ dE dΩ represents the number of particles in a volume element around the position ~ with
~
~
energy between E and E + dE having direction Ω in the solid angle dΩat time t.
~
~
2
(vii) φ(~r, E, Ω,t)= vN(~r, E, Ω,t) is the particle flux (number of particles/cm − Mev − sec).
~
~ 0
0
(viii) Σ(E → E, Ω → Ω) a scattering cross-section which represents the fraction of particles with energy E 0
~
~ 0
and direction Ω that scatter into the energy range between E and E + dE having direction Ωin the
~
solid angle dΩ per particle flux.
(ix) Σ s (E,~r) fractional number of particles scattered out of volume element of phase space per unit volume
per flux.
(x) Σ a (E,~r) fractional number of particles absorbed in a unit volume of phase space per unit volume per
flux.
r
Consider a particle at time t having a position ~ in phase space as illustrated in the figure 2.5-4. This
~
~
~
particle has a velocity V in a direction Ω and has an energy E.In terms of dτ = dx dy dz, Ωand E an
~
~
~
element of volume of phase space can be denoted dτdEdΩ, where dΩ= dΩ(θ, ψ)= sin θdθdψ is a solid angle
~
about the direction Ω.
The Boltzmann transport equation represents the rate of change of particle density in a volume element
~
dτ dE dΩ of phase space and is written
d
~
~
~
N(~r, E, Ω,t) dτ dE dΩ= D C N(~r, E, Ω,t) (2.5.71)
dt
where D C is a collision operator representing gains and losses of particles to the volume element of phase
space due to scattering and absorption processes. The gains to the volume element are due to any sources
~
S(~r, E, Ω,t) per unit volume of phase space, with units of number of particles/sec per volume of phase space,
together with any scattering of particles into the volume element of phase space. That is particles entering
the volume element of phase space with energy E, which experience a collision, leave with some energy
E − ∆E and thus will be lost from our volume element. Particles entering with energies E >E may,
0