Corresponding author: Kirill S. Kupriyanov (bdf1@mail.ru)
Academic editor: Georgy Tikhomirov
The task of determining the radiation situation, including neutron and gammaquantum flux density, radiation spectrum, specific volumetric activity of radioactive gases in the air, etc. behind the protective composition having inhomogeneities, has always been important in matters of radiation safety. One of the ways to solve the problem of determining gamma radiation fluxes was to divide the total ionizing radiation flux into four components: lineofsight (
Finally, the authors present calculations for the four components of the total ionizing radiation flux for various parameters of the cylindrical inhomogeneity in the reactor protection. Based on the obtained values, conclusions are made about the importance of taking into account the leakage albedo component in the formation of the radiation situation behind the core vessel.
Inhomogeneities in protective compositions are subdivided into elementary (simple) and complex ones. Elementary inhomogeneities include those in which the radiation field does not depend on the field of a neighboring inhomogeneity. The study of complex inhomogeneities is a more general problem and, as a rule, does not have an analytical solution (
The authors consider an elementary straight cylindrical channel with a diameter of 2
Cylindrical inhomogeneity in the protection
In the figure, the ray shows the formation of the leakage albedo component — the ray leaves the zone outside the channel, goes into the channel region, being physically attenuated, is reflected from the channel wall and hits the detection point
• m_{0} º m – for gamma quanta ;
• m_{0} º [l(
• m_{0} º S
For simplicity, we shall first consider the plane problem, and then move on to the spatial solution. Let us consider a top view (from the end of the channel) and mark the elements responsible for the formation of the leakage component (Fig.
Top view of the inhomogeneity
Let us consider the right section: at radius
where
Assuming that the source is isotropic (
Similarly, we can obtain the total incident flux from the part of the ring on the left side. The difference between the right and left sides along the length of the attenuation section (see Fig.
where
Now, we shall turn to the spatial problem (Fig.
Channel in space
To illustrate how the new geometric attenuation lengths are calculated, Fig.
Length of physical attenuation for
The length of geometric attenuation
Then formulas (2) and (3) can be represented as
In most practical cases, we can consider the source as an infinite plane and perform integration over the radius with an upper limit equal to infinity. Thus, the total flux incident at the point
To obtain the values of the reflected flux, we shall use the value of the numerical differential albedo: this value depends on the angle of incidence q, the angle of reflection y and the energy of the ionizing radiation flux. The angle of incidence depends on the angle a, therefore, this value must be taken into account even before the first integration over the angle a.
The task is axisymmetric; therefore, integrating the differential of the flux incident onto the lateral surface along a narrow ring, we finally obtain a solution for the leakage albedo component as the sum of two components:
F_{leak.alb} = F_{leak.alb1} + F_{leak.alb2};
Using the Heaviside function H(a – a_{bd}), we can write the solution in a single integral:
To check the obtained formula, Monte Carlo calculations were performed using SERPENT (a multipurpose threedimensional Monte Carlo particle transport code) (
Geometric model in SERPENT:
In the model calculation, aluminum was used as a protective material, and the source was set to be monoenergetic with energy of gamma quanta equal to 1.25 MeV. To obtain a flux of gamma quanta on the channel axis at the outlet from the protection, a small finite volume was set in the model, sufficient to register gamma quanta emitted from the source. The results of model calculations, as well as calculations by analytical formulas, are presented in Tab.
Results of calculations using the Serpent code and analytical formulas
SF_{Serpent}, cm^{–2}s^{–1}  SF_{analyt.}, cm^{–2}s^{–1}  e = [(SF_{analyt.} – SF_{Serpent})/ SF_{Serpent}] ×100, %  

15  135  0.011  0.01  –9.1 
15  270  3.501×10^{–3}  2.291×10^{–3}  –34.6 
30  135  0.037  0.036  –2.7 
30  405  5.252×10^{–3}  3.875×10^{–3}  –26.2 
10  200  2.241×10^{–3}  1.968×10^{–3}  –12.2 
Let us now analyze the results obtained.
• Comparing the fluxes obtained analytically and by the Monte Carlo method, one can see that the analytical result always gives lower values.
• Minimum errors are obtained in the case when the inhomogeneity is large and the main contribution to the flux formation is made by the lineofsight component. If the influence of the leakage components grows, the error increases, which is caused by the violation of the correctness of the assumptions made in the derivation of the analytical formulas.
Let us estimate the contribution of the leakage albedo component to the total ionizing radiation flux density at the detection point. Calculations for a particular case will be presented below.
The monoenergetic ionizing radiation source is ^{60}Co; the reflective surface is aluminum. The values of
Calculations of individual components of the ionizing radiation flux density








15  350  0.3  9.18×10^{–4}  3.93×10^{–6}  1.786×10^{–4}  7.451×10^{–5}  0.063 
5  350  0.3  1.02×10^{–4}  1.087×10^{–9}  6.795×10^{–6}  1.462×10^{–5}  0.118 
5  115  0.3  9.443×10^{–4}  5.759×10^{–5}  1.856×10^{–4}  2.44×10^{–4}  0.17 
30  350  0.3  3.66×10^{–3}  5.341×10^{–5}  1.17×10^{–3}  2.043×10^{–4}  0.04 
15  115  0.3  8.435×10^{–3}  5.265×10^{–4}  3.104×10^{–3}  9.932×10^{–4}  0.076 
15  115  3  8.435×10^{–3}  8.383×10^{–6}  3.104×10^{–3}  9.452×10^{–5}  0.0081 
5  350  3  1.02×10^{–4}  0  6.795×10^{–6}  1.377×10^{–6}  0.012 
Let us analyze the results presented in this table:
– For substances with high m_{0} the contribution of the leakage components is orders of magnitude less than that of the lineofsight components.
– As height
– With
Therefore, it is important to take into account the leakage albedo in the case of small channels in protections with ‘low’ values of m_{0}. For a particular case from the given example, the leakage albedo can be 17% of the total flux.
Despite the rapid development of numerical methods in the calculation of the radiation environment, analytical solutions still find their application in the initial estimates of radiation fields, in the study of the dependences of the obtained fluxes of ionizing radiation, as well as for verification of software systems (
* Russian text published: Izvestiya vuzov. Yadernaya Energetika (ISSN 02043327), 2021, n. 1, pp. 133–142.