EXPERIMENTAL STUDY OF THE INFLUENCE OF A REFLECTOR ON THE SPATIAL DISTRIBUTION OF THERMAL NEUTRONS IN A SEMI-INFINITE MEDIUM FROM A FAST-NEUTRON SOURCE

Yu. P. Pakhomov, G. Sh. Pekarskii, V. P. Shabalin

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

At present, neutron methods are widely used in thickness gauging, flaw detection, moisture measurement, and in a whole range of geophysical problems. The source of information in these methods is both transmitted or reflected radiation and the products of nuclear reactions. Naturally, the spatial distribution of neutrons in the object under study has a determining influence on the principal parameters of the method (sensitivity, productivity, depth, etc.). In turn, the spatial distribution of neutrons in a medium is determined by the spectral composition of the neutrons emitted by the source, the presence of a reflector, the material and linear dimensions of the reflector, and also by the chemical composition of the medium and its density. In this case, the use of a reflector makes it possible, first, to increase the absolute value of the thermal-neutron flux in the object under study and, second, by changing the dimensions and material of the reflector, to actively influence the spatial distribution of neutrons in it with the aim of optimizing the control parameters.

Available results of theoretical studies of the spatial distribution of thermal neutrons from a fast-neutron source have been obtained for special cases of point and plane sources in an infinite medium [1, 2]. For a reflector of finite dimensions, spaces between the reflector and the medium under study, and a source with a continuous neutron spectrum, an analytical solution of the neutron-transport equation is impossible, while the statistical-trials method (the Monte Carlo method) is rather complicated and requires a large amount of computation (especially at large distances from the source). Experimental studies have definite advantages in this case; therefore, we carried out an experimental determination of the influence of the dimensions and material of a reflector on the spatial distribution of thermal neutrons in a semi-infinite medium.

The thermal-neutron flux was measured by the method of radioactive indicators. Indium foils with a thickness of 100 mg/cm² were used as the thermal-neutron detector. To eliminate the effect of activation of the indium foils by epithermal neutrons on the measurement of the thermal-neutron flux, the cadmium-difference method was used. The indium foils were irradiated twice at the same point, with a cadmium screen and without it. The thickness of the cadmium screen was 0.5 mm.

The thermal-neutron flux was calculated by the formula [3]

\[ \Phi = \frac{\tau e^{\lambda t_2}} {k \cdot \Sigma_{\mathrm{act}} \cdot V \left(1-e^{-\lambda t_1}\right)} \left(n - F_{cd}\cdot n_{cd}\right) \left[\frac{1}{\mathrm{cm}^2\,\mathrm{min}}\right], \]

where \(\Sigma_{\mathrm{act}}\) is the macroscopic activation cross section of \(In^{115}\), \(\mathrm{cm}^{-1}\);
\(V\) is the foil volume, \(\mathrm{cm}^3\);
\(t_1\) is the foil irradiation time, min;
\(t_2\) is the foil cooling time, min;
\(n\) and \(n_{id}\) are the count rates of the indium foil without a cadmium shield and with a shield, respectively (after background subtraction), \(\mathrm{min}^{-1}\);
\(F_{cd}\) is a correction for the absorption of resonance neutrons in cadmium;
\(\tau\) is a correction for the measurement time;
\(k\) is a coefficient characterizing the registration efficiency of \(\beta\)-radiation.

In the work a Pu—Be source with a yield of \(4\cdot 10^6\) n/sec was used, with paraffin and graphite reflectors. The regularities of the influence of the linear dimensions of the reflector on the spatial distribution of thermal neutrons were studied for a paraffin reflector. A semi-infinite volume of sand was used as the medium under investigation. During the measurements the moisture content was kept constant and equal to 2.7 percent by weight.

The measurement geometry is shown in Fig. 1. Figures 2 and 3 present the distributions of thermal neutrons over radius and depth, respectively.

It follows from the results obtained that the use of a reflector makes it possible:

1. To increase the thermal-neutron flux at depths down to 10 cm by more than a factor of 10, and at depths greater than 10 cm by a factor of 2–3, in comparison with the thermal-neutron flux from a source without a reflector.

2. To actively affect the distribution of thermal neutrons with depth and obtain the desired character of the distribution. Thus, a source without a reflector creates a thermal-neutron field with a maximum at a depth of 20÷30 cm. The use of a graphite reflector with linear dimensions comparable to those of a paraffin reflector makes it possible to shift the maximum into the region of smaller depths, of the order of 10 cm. The use of a hydrogen-containing reflector (with linear dimensions not less than one slowing-down length) leads to a maximum of the thermal-neutron flux at the surface of the medium. It is also possible to obtain a uniform distribution down to a depth of \(\sim 40\) cm without a pronounced maximum (geometry 3, Fig. 1).

3. To obtain a certain “focusing” of the thermal-neutron flux in comparison with a source without a reflector. This is illustrated by the data of Fig. 4, where the dependences are presented of the number of neutrons \(N\), crossing an area of radius \(r\), on the radius of this area.

In estimating the measurement error it was established that the main contributions to the error in measuring the absolute value of the flux are: statistical error, the error in determining the foil volume, and the error associated with allowing for absorption, backscattering, and self-absorption of \(\beta\)-radiation in the indium foils. The largest error of the absolute measurements in the experiment was 40%.

Comparison of the obtained experimental results with the results of calculation by the Monte Carlo method [4] indicates that, within the limits of the experimental accuracy and the statistical accuracy of the calculation, satisfactory agreement is observed. The results of the comparison are given in Table 1.

Fig. 1. Geometry of the experiment: \(S\) — source of fast neutrons; в–з — reflectors made of paraffin; и — reflector made of graphite.

Subfigure labels: а., б., в., г., д., е., ж., з., и.

Visible annotations and dimensions: \(S\); \(100\); \(\varnothing 200\); \(150\); \(120\); \(\varnothing 300\); \(\varnothing 400\); \(250\); \(90\); \(150\); \(H_3BO_3\); \(120\); \(400\); \(200\); \(120\).

Fig. 2. Spatial distribution of the heat flux of neutrons:
a — geometry b, Fig. 1;
b — geometry v, Fig. 1;
c — geometry g, Fig. 1;
d — geometry i, Fig. 1;
e — geometry zh, Fig. 1;
f — geometry z, Fig. 1.

Panel a: vertical axis \(\Phi \times 10^{4}\,[\mathrm{neutr.}/\mathrm{cm}^{2}\,\mathrm{sec}]\); horizontal axis \(r\,[\mathrm{cm}]\). Legend: \(o-h=0\ \mathrm{cm}\); \(x-h=10\ \mathrm{cm}\); \(\nabla-h=20\ \mathrm{cm}\); \(+-h=40\ \mathrm{cm}\).

Panel b: horizontal axis \(r\,[\mathrm{cm}]\). Legend: \(o-h=0\ \mathrm{cm}\); \(x-h=10\ \mathrm{cm}\); \(\nabla-h=20\ \mathrm{cm}\); \(+-h=40\ \mathrm{cm}\).

Panel c: vertical axis \(\Phi \times 10^{4}\,[\mathrm{neutr.}/\mathrm{cm}^{2}\,\mathrm{sec}]\); horizontal axis \(r\,[\mathrm{cm}]\). Legend: \(o-h=0\ \mathrm{cm}\); \(x-h=10\ \mathrm{cm}\); \(\nabla-h=20\ \mathrm{cm}\); \(+-h=40\ \mathrm{cm}\).

Panel d: horizontal axis \(r\,[\mathrm{cm}]\). Legend: \(o-h=0\ \mathrm{cm}\); \(x-h=10\ \mathrm{cm}\); \(\nabla-h=20\ \mathrm{cm}\); \(+-h=40\ \mathrm{cm}\).

Panel e: vertical axis \(\Phi \times 10^{4}\,[\mathrm{neutr.}/\mathrm{cm}^{2}\,\mathrm{sec}]\); horizontal axis \(r\,[\mathrm{cm}]\). Legend: \(o-h=0\ \mathrm{cm}\); \(x-h=10\ \mathrm{cm}\); \(\nabla-h=20\ \mathrm{cm}\); \(+-h=40\ \mathrm{cm}\).

Panel f: horizontal axis \(r\,[\mathrm{cm}]\). Legend: \(o-h=0\ \mathrm{cm}\); \(x-h=10\ \mathrm{cm}\); \(\nabla-h=20\ \mathrm{cm}\); \(+-h=40\ \mathrm{cm}\).

Fig. 3. Number of neutrons through a disk of radius \(r\): 1 — source without a reflector (a, Fig. 1); 2 — paraffin reflector (v, Fig. 1); 3 — paraffin reflector (z, Fig. 1); 4 — paraffin reflector (zh, Fig. 1); 5 — paraffin reflector (i, Fig. 1).

Fig. 4. Dependence of the distribution of thermal neutrons with depth on the material and dimensions of the reflector:

1 — geometry b, Fig. 1;
2 — geometry c, Fig. 1;
3 — geometry i, Fig. 1;
4 — geometry g, Fig. 1.

Axis labels in the figure: depth, cm; \(\Phi \times 10^{4}\,[\text{neutrons}/\text{cm}^{2}\cdot\text{min}]\).

Table 1
\(r = 0\) cm

Comparison of the results of the experimental determination of the flux (geometry c, Fig. 1) and the calculation by the Monte Carlo method: curve 2, Fig. 1 in [4].

REFERENCES

1. B. Davison. Neutron Transport Theory. Translated from English. Edited by G. I. Marchuk. Moscow, Atomizdat, 1960.

2. K. Bekurts, K. Wirtz. Neutron Physics. Translated from English. Edited by L. A. Mikaelyan and V. I. Lebedev. Moscow, Atomizdat, 1968.

3. G. Bowen, D. Gibbons. Radioactivation Analysis. Translated from English. Edited by I. P. Alimarin. Moscow, Atomizdat, 1968.

4. P. Yu. Pakhomov, G. Sh. Pekarskii. Calculation by the Monte Carlo method of the distribution of thermal neutrons in a semi-infinite medium from a point isotropic source with a reflector. In the present collection.