# A Monte Carlo calculation of the resonance escape probability of thorium in a homogeneous reactor

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## Abstract

We have undertaken to develop a method !or determining the resonance escape probability of homogeneous reactors containing aqueous solutions of a resonance absorber. TH(NO₃)₄ was selected as a salt since it is readily dissolvable in water. Even at the highest concentration, the atomic densities of thorium and nitrogen are small compared to hydrogen and oxygen. Only the latter two contribute significantly to the slowing down in the high energy range; therefore, a water Monte Carlo serves adequately as the neutron supply to the resonance region of thorium.

In order to find an escape probability we determine the ratio of the number of neutrons that attain energies below the resonance region to the number that appear in the resonance region for the first time. The difference between the two numbers in the ratio occurs due to absorption by the resonance absorber, in this case thorium, and due to leakage.

The initial source of neutrons comes from H²(d,n)He³ reaction where the deuteron beam strikes the deuterium target at the center of one face of the containing parallelepiped.

Once the water Monte Carlo is run one has a set of data cards which can be used as the input data to a second Monte Carlo designed to calculate the resonance escape probability. Our prime concern is this second Monte Carlo.

The resonance escape Monte Carlo is a direct analog type using known physical distributions and laws. The decisions about what events occur in the history of a neutron are based on these distributions and laws. The detailed effect of the thorium resonances on the number of neutrons absorbed is described by the cross-section as a function of energy. A Breit-Wigner Doppler broadened single level formulation gives the scattering and absorption parts of the cross-section as a function of energy. Given a neutron-thorium interaction, the neutron is either absorbed or scattered with probabilities σ_{ab}/(σ_{ab}+σ_{sc}) and σ_{sc}/(σ_{ab}+σ_{sc}). A new statistical weight is assigned the neutron by taking the latter fraction of the old weight. A new neutron is picked from the source supply only after the previous neutron has delivered a weight to low energy or to the exterior of the pile.

The heart of the Monte Carlo method lies in the determination of a random variable, distributed according to some known probability function. This requires a source of random numbers which in our case is provided by the IBM-650 computer.

The resonance escape probability as a function of thorium number density fits very closely to a straight line with small negative slope throughout the whole range of stable concentrations. The Monte Carlo statistics provides probable errors of less than 0.3% for all five points calculated. Each of the five points lies close enough to the straight line of least-squares fit to give an error less than 0.3%.

A theoretical determination of the resonance escape probability based on resonance integral theory for the eight peaks used in the Monte Carlo calculation gives good enough agreement to make both methods plausible.

Although other resonance escape calculations have been done by the Monte Carlo method, they have been for radically different systems. In all such cases the system was heterogeneous with "lumped" uranium for the absorber. Therefore comparisons between existing Monte Carlo calculations is impossible. We hope to be able to provide experimental verification for the applicability of the Monte Carlo model.