Monte Carlo Calculation of the Distribution of Thermal Neutrons in a Semi-Infinite Medium from a Point Isotropic Source with a Reflector

Yu. P. Pakhomov, G. Sh. Pekarskii

(Presented by the scientific seminar of the Research Institute of Electronic Introscopy)

At present, neutron methods of logging, moisture measurement, determination of the elemental composition of substances, etc., have become widely used. Since the main characteristics of these methods (sensitivity, productivity, depth, etc.) are determined by the magnitude and spatial distribution of the neutron flux in the volume of the substance under study, a number of works have investigated the distribution of the neutron flux as a function of the parameters of the medium and of the neutron spectrum of the source [1–4].

The difficulty of taking boundary conditions and the energy dependence of interaction cross sections into account has limited the existing theoretical studies to consideration of an infinite medium [1–3], or of a semi-infinite medium and a unidirectional source [4].

In view of the fact that in a number of works [5, 6] a significant influence of the neutron reflector, its material and dimensions, on the magnitude and distribution of the neutron flux has been obtained, we have carried out a study of the influence of the parameters of a point isotropic source (neutron spectrum, presence of a reflector and its dimensions) and of the parameters of the medium (moisture content) on the magnitude and spatial distribution of the thermal-neutron flux in a semi-infinite volume of soil.

The studies were carried out for neutrons of the spectrum of Pu—Be, Pu—LiF sources and for neutrons with an energy of 14 MeV. Sand (silicon dioxide) with a mineralogical density of 2.65 g/cm³ and a porosity of ~40% is used as the medium under study. The influence of moisture was evaluated for low (4 vol.%) and full water saturation of the medium.

The maximum dimensions of the reflector (radius and height) were taken equal to two mean free paths of the source neutrons, which provides sufficiently complete utilization of back-scattered neutrons. Water was used as the reflector. The source is located on the lower surface along the axis of the cylindrical reflector at a height of 10 cm above the surface of the medium.

The determination of the thermal-neutron flux in the specified geometry was carried out by calculating neutron transport by the Monte Carlo method. The calculation program consists of two parts: calculation of the slowing down of fast neutrons and calculation of the diffusion of thermal neutrons. In the calculations, multigroup operating constants [7] are used. They cover practically the entire range of neutron-energy variation, from thermal energies to 10.5 MeV. The interaction cross section for neutrons with energy greater than 10.5 MeV was approximated by the group cross section in the range

6.5–14 MeV, using data from [8, 9]. To reduce the statistical error in calculating the neutron flux, the calculation uses one of the modifications of the Monte Carlo method, consisting in analytical averaging of the absorption process [10]. In calculating slowing down, energy loss was taken into account only in elastic collisions [1–2]. The anisotropy of scattering was calculated in the transport approximation, except for hydrogen, for which the scattering anisotropy was taken into account exactly.

The thermal-neutron flux was estimated from the number of collisions of thermal neutrons. All calculated results are normalized to a neutron source emitting isotropically 1 neutron/sec into the full solid angle and, for clarity, are presented in the form of smooth curves obtained from histograms. The results shown in the graphs were averaged over the following volumes: for distributions of the thermal-neutron flux with depth—within 0–10 cm in radius from the axis coinciding with the source axis, and within the depth \(h \div h+5\) cm; for distributions with radius—within the depth 0–5 cm (at the surface) and over radius within \(r \div r+10\) cm.

Figure 1 gives the results of calculations of the thermal-neutron flux with depth for a medium moisture content \(W=4\) wt.% and a source location height \(H=10\) cm above the surface of the medium. Dashed lines denote the data for sources without a reflector; solid curves, for sources with a reflector. Similar results for \(W=40\) wt.% are given in Fig. 2.

Fig. 1. Distribution of the thermal-neutron flux with depth for sand with a moisture content of 4 wt.%. Dashed lines—source without a reflector; solid lines—with a reflector. Numbers by the curves: 1—neutron generator with energy 14 MeV, 2—Pu–Be source, 3—Pu–LiF source.

Fig. 2. Distribution of the thermal-neutron flux with depth for sand with a moisture content of 40 wt.%. The notation is the same as in Fig. 1.

Data on the radial distribution of the thermal-neutron flux for sources with a reflector, located at a height of 10 cm, are presented in Fig. 3. Dashed curves denote curves for a medium moisture content \(W=4\) wt.%, and solid curves—for \(W=40\) wt.%. Similar re-

Fig. 3. Distribution of the flux of thermal neutrons in sand with radius from sources with a reflector. Dashed lines — moisture content 4 wt.%, solid lines — 40 wt.%. Numbers by the curves: 1 — neutron generator with energy 14 MeV, 2 — Pu—Be source, 3 — Pu—LiF source.

Fig. 4. Distribution of the flux of thermal neutrons in sand with radius from sources without a reflector. The designations are the same as in Fig. 3.

The results for neutron sources without a reflector are shown in Fig. 4.

From the results obtained it follows:

1. The use of a reflector makes it possible to increase significantly the flux of thermal neutrons in a semi-infinite medium. In this case:

a) for low moisture saturation the influence of the reflector is more substantial. Thus, for a depth of 20 cm the neutron flux increases for a neutron generator by a factor of 20, for a Pu—Be source by a factor of 5, and for a Pu—LiF source by a factor of 4; while at a moisture content of 40 wt.% the flux increases, respectively, by factors of 1.2, 1.4, and 1.5;

b) of the sources investigated, the greatest flux of thermal neutrons at small depths is provided by the Pu—LiF source (in the calculation per one source neutron);

c) the use of a reflector substantially changes the character of the distribution of the thermal-neutron flux with depth.

1. The neutron spectrum of the source substantially affects the distribution of the thermal-neutron flux in the medium. In this case:

a) for low moisture saturation the increase of the flux in the medium is especially substantial and, for a medium with \(W=4\) wt.% at a depth of 20 cm, amounts, relative to the flux created by a neutron generator, to a factor of 10 for a Pu—Be source and a factor of 26 for a Pu—LiF source;

b) even for complete moisture saturation (\(W=40\) wt.%) the flux at a depth of 20 cm increases, respectively, by factors of 1.4 and 1.9.

1. The radial distribution of the thermal-neutron flux with the use of a reflector makes it possible to increase significantly the flux of thermal neutrons within a local volume located under the source.

To study the effect of thermal and fast neutrons from the reflector on the magnitude of the flux and on the form of its distribution in the medium, calculations were performed for various reflector sizes, as well as for a reflector shielded with cadmium of a thickness ensuring complete absorption of thermal neutrons. Figure 5 presents the results of calculations of the distribution of the thermal-neutron flux with depth at medium moisture content \(W=4\) wt.% and source placement height \(H=10\) cm.

Fig. 5. Effect of reflector size on the distribution of the thermal-neutron flux with depth. The numbers by the curves indicate the diameter and thickness of the cylindrical reflector: 1 — \(20\times 7.5\), 2 — \(15\times 5\), 3 — \(20\times 3\), 4 — \(20\times 7.5+\mathrm{Cd}\), 5 — without reflector.

Analysis of the results shows:

1. The main contribution to the increase in the thermal-neutron flux in the medium is due to the thermal neutrons of the reflector. The magnitude of the contribution is determined by the reflector dimensions.

2. The use of a reflector in a cadmium shield makes it possible to obtain a flux distribution with depth whose character is similar to the distribution from a source without a reflector. The increase in the flux is due to fast and intermediate neutrons of the reflector, which have a softer spectrum than the source spectrum. As a result, the maximum of the distribution with depth is shifted toward the surface. For a reflector \(20\times 7.5\) cm in diameter in a cadmium shield, the increase in the flux at the maximum of the distribution with depth is \(\sim 30\%\).

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