Estimating changes in the doppler effect in correcting for nuclear thermal motion and the resonant behavior of the scattering cross section in the scattering indicatrix

Abstract

A major aspect of reactor physics is research on the effects of nuclear thermal motion on resonant neutron absorption. At present, corrections are applied for the thermal nuclear motion in the resonant energy range only via changes in the total resonant cross sections. The scattering indicatrix describes the neutron energy change in elastic scattering and is considered to be as if the scattering occurred at immobile nuclei, i.e., as a step, even in reference calculations performed by Monte Carlo methods. It has been shown [1] that the scattering indicatrix incorporating the nuclear thermal motion for a nuclide having a resonant scattering cross section differs appreciably from a step even at higher energies. This could be explained briefly by considering (1) and (2), which are equations used for the scattering indicatrix in the gas approximation: ~.(~) we.'-. ~. r) : ~ j" v.g<,..-,) z~M(v, r)~,dv. (l) y'., 0 -I in which v' and v are the neutron velocities before and after the collision, V r the relative velocity of the neutron and nucleus, V the nuclear velocity, M(V, T) a Maxwell distribution, /z the cosine of the angle between the neutron and nuclear velocity vectors, and I t .~ f ~,(v)v,~,,-,,) z~Mcv, r~,dV. (2) ~(,.,) w(,,,'-v, 7"3 = .,,~ 0 -I The scattering cross section in (1) is considered as constant and extracted from the integral. Then the integral is readily derived and one gets a standard formula for the scattering indieatrix in the gas approximation for light nuclei [2]. This indicatrix quite rapidly becomes a step as the temperature of the moderating medium decreases or the moderation energy increases (although it must be noted that it becomes a pure step function only at absolute zero). In the case of a resonant nuclide, the scattering cross section is dependent on the relative velocity of the neutron and nucleus, and the corresponding resonant scattering indicatrix is calculated from (2), i.e., the Doppler effect is incorporated into the indicatrix. Figure la shows the 238U scattering indicatrix near the 6.67 eV resonance for a fixed initial neutron energy to left and right of the resonance at 1000 K. Here and subsequently, the dashed lines denote the steps, while the points denote the scattering indicatrix from (1), and the solid line is the resonant indicatrix from (2). Figure lb shows in the same sequence the indicatrix near the 661 eV resonance at 1000 K. The resonant indicatrix is very different from a step function.We have used the TVS-M program [3] to examine the effects of difference between the actual indicatrix and a step on the reactor functionals, as it solves the neutron moderation equation in the resonant energy range. In the case of a multizone cell, the corresponding equations for the collision density F(u) take the form vF(u) = ~, v.P..n(u)Q,(u).(3) n'in which ~'n is the volume of zone n, u the lethargy reckoned from the upper bound of the resonant region and taken as 4.65 keV, Pn,.n(u) is the probability that a neutron having lethargy u and generated in zone n' undergoes its firs...