Analytical solution for the Doppler broadening function using the Kaniadakis distribution

doi:10.1016/j.anucene.2018.11.023

Annals of Nuclear Energy

Volume 126, April 2019, Pages 262-268

https://doi.org/10.1016/j.anucene.2018.11.023 Get rights and content

Highlights

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Analytical solution for general Doppler broadening function through Kaniadakis distribution.
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The analytical solution is based on Frobenius and parameter variation methods.
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The analytical solution obtained is consistent with the numerical solution from its integral form.
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The maximum error obtained is smaller than 1% when compared to the numerical solution.

Abstract

Several works have been done for the development of models that generalize the Maxwell-Boltzmann distribution, aimed at encompassing physical phenomena that lie outside the thermal equilibrium. Amongst these, there are distributions that result from the non-extensive statistics of Tsallis and Kaniadakis. Starting from these generalized distributions, a Doppler broadening function was proposed in recent papers, using the deformed Kaniadakis distribution, which was numerical, evaluated with the use of a Gauss-Legendre quadrature. From this perspective, this paper presents an analytical solution for the generalized Doppler broadening function through obtaining a partial differential equation, considering the Kaniadakis distribution. This equation is solved analytically using the methods of Frobenius and variation of parameters, in order to obtain a generalized solution for the Doppler broadening function, containing a deformation parameter $κ,$ that measures the deviation in relation to the Maxwell-Boltzmann distribution. Finally, the results were produced considering several values for $κ,$ with the intent of making a comparison with the reference values. For the validation of the deformed Doppler broadening function’s analytical solution, a numerical solution of the partial differential equation was generated. It was possible to use this numerical solution as a benchmark for the analytical solution that was derived. It was demonstrated that the analytical solution obtained is consistent, because when $κ$ tends to zero, the solution falls in the conventional form, when the Maxwell-Boltzmann distribution is considered. Apart from this, the results were shown to be good, especially when we consider the temperature and power ranges for practical applications, as the maximum error obtained was smaller than 1%.

Introduction

In a thermal neutron reactor, high-energy neutrons are produced by nuclear fission. In most operating reactors, these neutrons go through a process of moderation, in which they lose energy until reaching a thermal range, where new fissions are produced.

However, in the process of moderation, the neutrons can be lost through resonance absorption, which is one of the main causes of neutron loss in thermal reactors. The effect of this resonance absorption varies according to the temperature of the nuclear fuel inside the reactor core, as the so-called Doppler broadening effect takes place (Stacey, 2007).

The adequate treatment of nuclear resonance absorption is fundamental to the study of neutron moderation in a reactor, and to determining the group constants.

The most-used formalisms to describe the cross-section resonance behavior in a nuclear reactor are the Maxwell-Boltzmann (MB) statistics. In this model, the nuclei of a reactor core are considered to be one single gas in which molecules are in a thermal motion with different velocities (Pathria and Beale, 2011). This thermal motion is described by the Maxwell-Boltzmann distribution: $f (V, T) = (\frac{M}{2 {π K}_{B} T}) e x p (- \frac{{MV}^{2}}{2 K_{B} T})$

In Eq. (1), $K_{B}$ represents the Boltzmann constant, M is the nuclei mass, T is the temperature of the medium, and V is the velocity of the target nuclei. The Maxwell-Boltzmann distribution describes with precision the physical neutron-nuclei phenomena in a condition of thermal equilibrium.

However, the physical phenomena that lie outside the condition of thermal equilibrium cannot be described through the Maxwell-Boltzmann statistics, making it necessary to use a generalized statistic. This generalization has been the object of research and continuous development, as seen, for example, in the distributions proposed by Tsallis, 1988, Kaniadakis, 2001.

In this context, considering a distribution of nuclei velocities that can describe phenomena that lie outside the thermal equilibrium (Guedes et al., 2017), it is possible to carry out simulations for systems composed of particles that are always in a state of motion. Apart from their application in nuclear reactor physics, these generalized statistics can also be applied to the description of effects in other fields, such as, in the study of the dynamics of open stellar clusters (Carvalho et al., 2010) and in the study of the estimation of gas temperatures in optical and electrical diagnostics of a spark-plug discharge in air (Oliveira et al., 2012).

Section snippets

Doppler broadening with the Kaniadakis distribution

In this paper we propose an analytical solution for the Doppler broadening function according to the generalized Kaniadakis statistics, so that the effect of the Doppler broadening can be better represented, as a result of the neutron-nuclei interactions that lie slightly outside the thermal equilibrium. The Kaniadakis (2001) generalized distribution is given by: $f_{\{κ\}} (V, T) = A (κ) \exp_{\{κ\}} (- \frac{{MV}^{2}}{2 K_{B} T})$ where, $A (κ) = {(\frac{|κ| M}{π K_{B} T})}^{n / 2} (1 + \frac{n |κ|}{2}) \frac{Γ (\frac{1}{2 |κ|} + \frac{n}{4})}{Γ (\frac{1}{2 |κ|} - \frac{n}{4})}$

and the κ-exponential function is defined by: $\exp_{\{κ\}} (x) = {(\sqrt{1 + κ^{2} x^{2}} + κ x)}^{\frac{1}{κ}}$

When κ → 0,

Obtaining the differential equation for the broadening function

The integral Eq. (5) may be converted into an ordinary differential equation by differentiating it in relation to the variable x, which represents the neutron energy. In doing so, one will find: $\frac{\partial ψ_{κ} (ξ, x)}{\partial x} = \frac{ξ}{2 \sqrt{π}} B (κ) \underset{- \infty}{\int^{+ \infty}} \frac{1}{1 + y^{2}} \frac{\partial}{\partial x} i \exp_{\{κ\}} [z] d y$ where: $z = \frac{- ξ^{2} {(x - y)}^{2}}{4}$

The differential term in the integral of Eq. (9) is given by: $\frac{\partial i \exp_{\{κ\}} (z)}{\partial x} = \frac{- ξ^{2} (x - y)}{2} \frac{i \exp_{\{κ\}} (z)}{(\frac{κ^{2} z - \sqrt{1 + κ^{2} z^{2}}}{κ^{2} - 1})}$

Thus, one can write that: $\frac{\partial ψ_{κ} (ξ, x)}{\partial x} = \frac{- ξ^{3}}{4 \sqrt{π}} B (κ) \underset{- \infty}{\int^{+ \infty}} \frac{(x - y)}{(1 + y^{2})} \frac{i \exp_{\{κ\}} (\frac{- ξ^{2} {(x - y)}^{2}}{4})}{C (z)} d y$

where: $C (z) = (\frac{κ^{2} z - \sqrt{1 + κ^{2} z^{2}}}{κ^{2} - 1})$ .

The function $i \exp_{\{κ\}} (- z)$ decreases rapidly when

Analytical solution of the differential equation for the Doppler broadening function

In order to obtain the solution for the generalized differential Eq. (32), we will use the Frobenius method and that of the variation of parameters.

The Frobenius method is used to obtain the solution for the homogeneous differential equation, using the following series expansion: $ψ_{h} (ξ, x) = \sum_{n = 0}^{\infty} c_{n} x^{n + s}, c_{0} \neq 0$ where s is the indeterminate variable. Calculating the first and second derivative of Eq. (34) and replacing in the homogeneous equation results in: $\sum_{n = 0}^{\infty} c_{n} (n + s) (n + s - 1) x^{n + s - 2} \sum_{n = 0}^{\infty} c_{n} ξ^{2} [(n + s) + \frac{(ξ^{2} + 2 (1 - κ^{2}))}{4}] x$

Results

With the goal of validating the analytical solution for the Doppler broadening function obtained in this paper as represented by Eq. (50), a numerical solution for the integral Eq. (5) was implemented.

Table 1, Table 2, Table 3, Table 4, Table 5, Table 6 show the results of the analytical calculations for function $ψ_{{κ}} (ξ, x)$ and the percentage deviations in relation to the numerical values from Eq. (5) for κ = 0,01, κ = 0,1 κ = 0,2, respectively.

Based on the analytical and numerical solutions for

Conclusions

This paper describes the analytical solution for the Doppler broadening function $ψ_{{κ}} (ξ, x),$ obtained using Kaniadakis’ velocity distribution. This distribution is a deformation of the Maxwell-Boltzmann distribution, as ruled by term $κ$ . The higher the value of parameter k, the greater the deformation will be in relation to the Maxwell-Boltzmann distribution.

To produce a reliable analytical approximation for $ψ_{{κ}} (ξ, x)$ it was converted from the integral form to a partial differential equation. Based

Acknowledgments

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior Brasil (CAPES) - Finance Code 001. The authors thank Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for support.

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