The Boltzmann Equation with Energy Dependence

The assumptions (7.106) are the only limitations to be imposed in the derivation of the Boltzmann equation, and it is important to note that no speciAc relation has been selected to describe the neutron-scattering function. Thus the results of this analysis will be generally applicable to many systems of practical importance. In order to demonstrate the application of these results a particular scattering law is later selected, namely, isotropic scattering in the center-of-mass system of coordinates. However, this specific calculation will not be introduced until the general relations have been developed. 

The purpose of the analysis which follows is to establish a suitable neutron-balance relation which takes into account the dependence of the relevant functions upon the neutron energy, direction of motion, time and spatial coordinates. The balance condition, which is an integro-differential equation, is then reduced to an infinite, but simpler, set of coupled integro-dilTerential equations by expanding the flux, the neutron sources, and the scattering function in spherical harmonics. The derivation of the Fermi age model from these results is then demonstrated. 

The first term gives the time rate of change of the neutrons which occupy the space dr dv dQy the second, the losses due to the net transport of neutrons through the boundaries of dr, the third, the losses from this space due to neutrons being absorbed or scattered out or slowed out (of the dv interval), and the fourth, the contributions from sources. 

Now for the scattering-integral term. The total number of neutrons scattered into dr dv dli per unit time from all possible speeds v and directions of motion O is obtained by taking the number of neutrons in dr dv dO which experience scattering collisions, namely, 

The product of the total scatterings and this probability integrated over all accessible v and O gives the total gain in the neutron population of the differential element dr dv dQ due to the neutrons scattered in from higher speeds and different directions of motion  

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