The use of symmetry and other features of neutron paths for speeding up computations by the Monte Carlo method

Abstract

SEVERAL means for improving the convergence for the Monte Carlo method are recommended in [1–6]. The most important of them involves rational organization of the computer program. This includes: 1. (1) achieving a solution of the problem as a whole, i.e. seeking some dependence or formula, and not computing separate points of the dependence in sequence and independently; 2. (2) considering all the points of the dependence simultaneously and in a correlated way, i.e. so that they have matched fluctuations; 3. (3) sharply reducing the amount of computation by exchanging computational information at adjacent points; ideally, the random path can be drawn at a single point of the dependence. Recommendations (2) and (3) are fairly easily realized in practice when solving Laplace's equation and problems on gamma quanta propagation, but are extremely difficult in the case of slow neutrons. The reasons for this lie, firstly, in the variety of types of neutron elastic scattering by different nuclei, which makes it difficult to obtain a correlation of the neutron paths in media of different compositions; secondly, in the large number of collisions when a neutron is slowed from millions to a few parts of an electron-volt, which makes it impossible in practice to use a single path of given “value” for different media; and thirdly, in the low probability of neutron migration at great distances from point sources. ▪ In the present paper we propose a special algorithm for the draw of neutron directions, ensuring frequent coincidence of the directions of neutron motion after scattering in different media, even in the case of collisions with different nuclei and differences between the directions prior to scattering. To achieve high accuracy in the probability of neutron migration at great distances, we propose making use of the high degree of symmetry in individual parts of the path, even when there is no symmetry in the geometry of the problem. All these methods, discussed below, have enabled us to evaluate by computer for the first time the dependence of the readings of a neutron gamma counter (NGC) on the hydrogen content of strata and the pore diameter for a standard NGC (as used in all the industrial-geophysical offices of the Union). Since this problem will often be used to illustrate the proposed methods, we shall state it more fully. The NGC is mounted eccentrically in a cylindrical pore (see Fig.) filled with salt water. It consists of an iron tube one centimetre thick (its height can be assumed infinite), in which a lead/iron filter is mounted, for isolating the polonium-beryllium neutron source from the three SI-4G gamma quantum counters, located so that the distance between the source and mid-point of the counters is 60 cm. We want to find the dependence of the gamma quantum counter readings on the nuclear composition of the mineral stratum and the liquids filling the pore, the pore diameter and certain other parameters. The physical processes occurring in the NGC are: slowing of fast neutrons to energies of 0.1 eV, thermal ization and diffusion of hot neutrons, radiative capture of neutrons with radiation of the corresponding spectrum of gamma quanta, and propagation of the latter in conditions of Compton, photo- and other effects with the possibility of their being registered in the indicator (the energy efficiency of this is known). Not all these processes are of equal importance, and some can be neglected, depending on the physical features of the specific problem of nuclear geophysics.