A Many-Resonance Approximation for the Neutron Energy Spectrum

The neutron energy spectrum of fast reactors in the energy range from several keV to several tens of keV is influenced by a multitude of resonances of the fertile and fissile elements. A single elastic scattering in this range distributes the neutrons across many resonances. Since the resonance parameters are randomly distributed about average values, the collision rate below any energy point is the sum of many, uncorrelated, resonant scattering rates above the point. Hence the collision density, as a function of energy, is a smooth curve dominating over small local fluctuations. It is demonstrated, both analytically for simplified cases and numerically for realistic cases, that the deviations from a smooth curve are negligible.In lethargy units, the smooth collision density is [a (u)/v(u)] exp[-v(u)]. The definitions of the parameters a(u) and v(u) involve only average properties of the resonance population, namely the averages over many resonances of the scattering probabilities s_i ≡ ∑ (scattering, element)/∑ (total, mixture). The average absorption probability is a(u); ν(u) is given implicitly by the transcendental equation 1 - v = ∑_i [⟨s_i⟩ /α_i] [1-(1-α_i )^1-v], where α_i is the maximum relative energy loss per scattering in the i’th element. An accurate solution of the transcendental equation is found most essential for an accurate prediction of integral reaction rates. For this purpose a series solution for v in terms of ⟨s_i⟩ is developed.