Calculation of the Neutron Importance Function and the Delayed Neutron Effective Fraction by the Monte Carlo Method
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ulation of the Neutron Importance Function and the Delayed Neutron Effective Fraction by the Monte Carlo Method E. A. Gomin*, V. D. Davidenko, A. S. Zinchenko**, and I. K. Kharchenko National Research Center Kurchatov Institute, Moscow, 123182 Russia *e-mail: [email protected] ** e-mail: [email protected] Received December 14, 2015
Abstract—An algorithm for calculation of the neutron importance function and the delayed neutron effective fraction by the Monte Carlo method implemented in the KIR program is presented. The results of calculation of the delayed neutron effective fraction in some critical experiments are given in comparison with the experimental results. Keywords: neutron kinetics, Monte Carlo method, delayed neutron effective fraction, importance function DOI: 10.1134/S1063778817080051
INTRODUCTION In 2013, we started to develop the KIR-P program intended for calculation of nuclear reactor kinetics in adiabatic and quasi-static approximations based on the Monte Carlo method. The Monte Carlo method is used in this program to find the point kinetics parameters and calculate the shape function. The KIR-P program is being developed to solve the neutron transport equations, which are presented in [1] in the integral form. Obviously, the point kinetics parameters obtained with the help of KIR-P can be used in any engineering program intended for the same purposes as KIR-P. The procedure of calculating fission neutron lifetimes is presented in [2]. The present paper gives a description of the techniques for calculating the effective fraction of delayed neutrons. The calculation of the effective fraction of delayed neutrons, like the calculation of fission neutron lifetimes, is tied to evaluation by the Monte Carlo method of the following type of functionals:
∫
+
J = Q(r )ψ(r )Q (r )dr,
(1)
where Q(r) is the integrated function, ψ(r) is collision density, and Q+(r) is the importance function. A detailed analysis of the techniques for evaluation of functional (1) by the Monte Carlo method is presented in [2], while in the present paper, we note that the effective fraction βeff of delayed neutrons is the ratio of functionals of type (1), namely [3]:
+
∫
∫
+
β eff = Q del(r )Qdel (r )dr / Q(r )Q (r )dr,
(2)
where Q(r) and Qdel(r) are densities of the sources of fission neutrons and delayed neutrons, respectively:
∫∫ νΣ (r, E )Φ(r, Ω, E )d ΩdE, (r ) = ∫∫ ν Σ (r, E )Φ(r, Ω, E )d Ω dE ;
Q(r ) = Qdel
f
del
f
+ Q+(r) and Qdel (r) are the importance functions of fission neutrons and delayed neutrons, respectively:
χ(r, E ) + Φ (r, Ω, E )d Ω dE , 4π χ del (r, E ) + + Qdel (r ) = Φ (r, Ω, E )d Ω dE ; 4π χ(r, E) and χdel(r, E) are the spectra of fission neutrons and delayed neutrons, respectively; and Φ+(r, Ω, E) is the importance function. For implementation of some calculation algorithms using the Monte Carlo method to determine the nuclear reactor kinetics, it is necessary to know the average fraction of delayed neutrons upon fission of one nucleus of a fissile substance (βav). One of the applicable defi
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