Characteristic Ray Solutions of the Transport Equation

Abstract

With the onset of increasingly sophisticated reactor designs in the late 1950’s, the problem of explaining the behavior of thermal neutron physics in reactors became more acute. This behavior is governed by the Maxwell-Boltzmann equation, which in its time-independent form is:

$$\left( {\underline \Omega\;\nabla+ \sum\limits_s {\left( {E,\underline r } \right)}+ \sum\limits_a {\left( {E,r} \right)} } \right){\kern 1pt} \phi \left( {\underline {r,E,\underline \Omega} } \right) = \int {d\underline \Omega\int {dE'\sum\limits_s {\left( {E' \to E,\underline {\Omega '}\to \underline \Omega,\underline r } \right)\phi \left( {\underline {r,E',{{\underline \Omega}^\prime }} } \right) + S\left( {\underline r ,E,\underline \Omega} \right)} } } $$

((1.1))

Here the basic variable is the angular neutron flux at position r, energy E and in a direction Ω indicated by ϕ (r, E, Ω) ∑_S (E,r) is the neutron scattering cross-section at energy E and position r, ∑_a (E,r) is the corresponding neutron absorption cross-section.

$$\sum\limits_s {\left( {E' \to E;{{\underline \Omega}^\prime };\underline r } \right) = \sum\limits_s {\left( {E',\underline r } \right)P\left( {E' \to E,{{\underline \Omega}^\prime } \to \underline \Omega,\underline r } \right)} } $$

where ∑_S (E’, r) is the neutron scattering cross-section at energy E’ and position r, and PCE’→E,Ω’→Ω, r) is the probability that a scattering collision at position r by a neutron with energy E’ and direction Ω’ will cause the neutron to change its energy to E and its direction to Ω.