Neutron Spectrum Unfolding

Introduction

The neutron flux is a quantity that describes the neutron population in a nuclear reactor and it is a function of space, neutron energy and time. The neutron energy distribution in a nuclear reactor spans several decades, typically from meV to MeV range. Physics tells us that the probability for an interaction of a neutron with a nucleus of some material present in the reactor strongly depends on the energy of that neutron. These probabilities are called “neutron cross sections” and are tabulated as a function of neutron energy and more and more variance and co-variance data is becoming available.

Measuring the energy dependence of the neutron flux is not something that can be done “online”. Typically, this involves inserting some very specific isotopes in the reactor and measure their activation after some time of reactor operation by means of gamma ray spectrum measurements. These activation levels can be transformed in reaction rates.

Objectives

The goal of “Neutron Spectrum Unfolding” is to obtain an expression for the energy spectrum of the neutron flux in the reactor, assuming knowledge of the cross sections Σ(E) and measured reaction rates RR_i for a number (preferably many but typically between 10 and 20) isotopes. Often, the neutron flux is approximated as either a piecewise constant (histogram) or piecewise linear function. The cross sections are considered to be piecewise linear functions. This problem is known as a coupled set of Fredholm integral equations of the first kind and is notoriously ill-conditioned. Therefore, regularization techniques like Tikhonov regularization can (should?) be employed. The goal of the master thesis is to analyze different numerical schemes and regularization techniques to solve this problem and to estimate the impact of the ill-conditioning (i.e. how “wrong” can the result be). The final outcome should be a Python script and demo notebook implementing and demonstrating the best approach. This code will be used in the future by the reactor dosimetry group in their daily work.

Objectives

1. Brief introduction on cross section data, reactor physics by SCK CEN staff
2. Literature review on existing methods (many date from the 1970s)
3. Develop a numerical scheme without regularization but taking into account variance and covariance information in the cross sections (least squares problem)
4. Analyze different regularization schemes
5. Analyze impact of ill-conditioning on the credibility of the obtained results
6. Deliver a master thesis text + implementation in Python

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