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We present an overview of fission chamber’s functioning modes, theoretical aspects of the nonnegative matrix factorization methods, and the opportunities that offer neutron data processing in order to achieve neutron flux monitoring tasks. Indeed, it is a part of research project that aimed at applying Blind Source Separation methods for in-core and ex-core neutron flux monitoring while analyzing the outputs of fission chamber. The latter could be used as a key issue for control, fuel management, safety concerns, and material irradiation experiments. The Blind Source Separation methods had been used in many scientific fields such as biomedical engineering and telecommunications. Recently, they were used for gamma spectrometry data processing. The originality of this research work is to apply these powerful methods to process the fission chamber output signals. We illustrated the effectiveness of this tool using simulated fission chamber signals.

The major reason to control neutron flux inside any nuclear reactor is its proportionality to the power density. For this reason, we are concerned, in our research, with this variable which can be measured using one of the five following types of neutron detectors: Boron Trifluoride (BF3) proportional counters, Boron (^{10}B) lined detectors, fission chamber (FC), ^{3}He proportional counters, and self-powered neutron detectors (SPND) [

The TRIGA Mark II Reactor of the Nuclear Studies Centre of Maâmora (CNESTEN-Morocco) of the National Centre for Nuclear Energy, Sciences and Techniques (

It is important to know neutron flux distribution inside such reactor as accurately as possible. This can be possible using fission chambers [

To improve the neutron flux monitoring, we seek to use new methods based on Blind Source Separation (BSS) algorithms. In fact, BSS methods require no hypothesis on the way that the signal and the noise are mixed inside a given system. That is why they have been used in several scientific fields to solve numerous problems (audio signal processing, astronomy, biomedical engineering, telecommunications, and geophysics) [

This paper is organized as follows: first, we briefly discuss the FC’s principle and their different operating modes (Section

The FCs are simple generic ionization chambers with fissile deposit on at least one of the electrodes (Figure

(a) Overall plan of FC with fissile element ^{235}U [

The fission product emitted to the gas creates about 1^{5} to 1^{6} electron-ion pairs along its path [

The electric field that is applied to the electrodes separates electrons and ions. The anode collects the negative charges, while the cathode collects the positive ones. This causes an observable pulse of current. The resulting signal reflects the interaction of the neutron flux with the fissile deposit. It can also be gamma ray (

As mentioned above, there are three operating modes of an FC according to its design and external coupled electronic devices. Indeed, they are commonly used to process the FC output signal that corresponds to the induced current.

At a low fission rate, the sensitivity range of FCs in pulse mode is between 0.01 cps/nv and 4 cps/nv. Here, the pulses are separated from each other. The probability spectrum of the pulses charge is estimated to extract the signal from background noise. The monitoring of the pulse counting rate allows measuring a quantity that is proportional to the flux and the fission rate. The maximum count rate depends on various chamber parameters such as size, filling gas pressure and composition, and interelectrode distance. According to Böck and Villa [^{6} cps (for pulse widths of the order of 80 ns); so a high sensitive chamber (1 cps/nv) can be operated in pulse mode up to 1^{6} nv and a low sensitivity chamber (0.01 cps/nv) up to 1^{8} nv. The chamber can withstand gamma radiation up to 1^{4} Gy/h [

At a higher fission rate, individual pulses cannot be separated. The mean value of the current, generated by the pileup pulses is measured according to the Campbell theorem which states that the first two statistical moments of current average and variance are proportional to fission rate [^{−14} A/nv and 1^{−12} A/nv. A very sensitive FC with 6^{−13} A/nv can measure maximum fluence of about 2^{9} nv. The gamma sensitivity in this mode is around 1^{−9} A/Gy/h [

For high neutron fluence rate, the generation of the pulses is limited by the pulse pileup phenomenon. The signal fluctuation is a measurement of the neutron fluence rate. The greater the energy deposit caused by a neutron event, the more accurate the measurement [

In general, Campbell’s theorem gives the relationship between the intensity of a shot noise process and cumulates. The Campbelling mode [

Schematic of the electronic system used in the IRINA project [

In this section, we present a general view on the actual methods used in the control of the neutron flux. It consists of using the FC in pulse, Campbelling, and current modes. We also present the state of the art about Blind Sources Separation algorithms as alternative techniques which may improve the use of FC’s output signal in neutron flux control.

In TRIGA MARK II Reactor, the FC is a key to monitor and measures the on-line neutron flux. From processing the FC’s signals, we estimate the fission rate related directly to the neutron flux. Different methods are used for this processing with different electronics according to the statistic order of the count rates and depending on low, medium, and high detection rate, respectively. At a low count, the signal has the shape of individual pulses. For this reason, pulse counting technique, which is based on level crossing, is applied. At medium and higher count rate, the pulses are overlapped. Therefore, the counting becomes impossible and the detection rate is estimated with both Campbelling and current techniques. To solve this problem, several investigations have been carried in this sense. Among these scientific works is Elter et al. survey which suggests applying Higher Order Campbelling methods (HOC) in neutron flux monitoring with FCs to suppress the impact of gamma ray contribution [

The works have been carried out with these methods for monitoring and controlling power in nuclear reactors ensure their reliability to better reactor’s control and increase the reactor’s safety parameters during operation. However, they are implemented electronically in analog data acquisition chains. For this reason, we encourage applying other techniques to digitally process the FC’s output signal by using new signal processing methods.

Many recent publications have dealt with digital processing of nuclear signals in general. We cite here as an example Elbadri et al. works in which they have applied linear and nonlinear adaptive filters to improve the quality of HPGe preamplifier’s output signals which are used in environment monitoring gamma ray spectrometry [

The application of new digital signal processing methods, the so-called Blind Source Separation (BSS), on nuclear data was introduced by Mekaoui et al. [

Moreover, BSS tools now raise great interest and play an important role in many application areas. Their first application solved the “cocktail party” problem [

The BSS principle can be schematized as shown in Figure

Block diagram of the basic BSS problem [

In theory, the measured observations

Mathematically speaking, according to Cichocki et al. (

The power of BSS techniques resides in their ability to determine the matrix

The estimated sources are given by the vector

There are several BSS algorithms which are grouped into four approaches as shown in Figure

Basic approaches for Blind Sources Separation [

In addition to these classical methods, the linear instantaneous mixtures can be processed using Nonnegative Matrix and Tensor Factorizations (NMF and NTF), Sparse Component Analysis (SCA), and Morphological Component Analysis (MCA). In our research, we are interested in the use of NMF and NTF approaches to analyze FC’s output signals.

The NMF is a technique of the dimension reduction adapted to sparse matrix containing positive data. It was introduced by Lee and Seung [

The NMF of a matrix

The factorization is solved by finding a local optimum of the optimization problem:

Several algorithms and implementations are used to compute the NMF [

For some problems, the matrices can be seen as second-order tensors. In some cases, they can go up to the third or higher order. For this reason, the NMF can be generalized to Nonnegative Tensor Factorization (NTF).

The NTF problem can be expressed as nonnegative canonical decomposition/parallel factor decomposition denoted by CANDECOMP and PARAFAC, respectively [

Given an

The tensor

In order to compute the nonnegative component matrices

NTF is a technique for computing and decomposing a nonnegative value tensor into sparse and reasonably interpretable components. Four algorithms are implemented in NTF [

Simple and Fast Hierarchical Alternating Least Squares (Simple/Fast HALS).

Multiplicative alpha divergence algorithms.

Multiplicative beta-divergence algorithms.

Block principal pivoting.

In our project, we aim at applying NMF and NTF algorithms to analyze the FC’s output signals. Indeed, the recorded signals are considered as time-series mixtures of several components (sources) which we try to extract using these new signal processing methods. Figure

Project steps for the neutron flux monitoring using BSS methods.

As an example, we are going to represent in Figures

The observations used as input of the NMF algorithm.

Signal to interference ratio for the mixing matrix.

The estimated source.

Since we achieve the separation task, we can then characterize each isolated component through the computation of the auto- and cross-correlation functions and the power spectral densities and/or according to time-frequency analysis in order to extract meaningful features permitting achieving the neutron flux monitoring task.

The state of the art concerning classical and new methods used to control the neutron flux within the experimental and research nuclear reactors has been summarized. Indeed, the on-line flux neutron monitoring is most important task for control, fuel management and safety concerns, and material irradiation experiments. Consequently, it is a challenge shared by wide range of scientific areas, from the fundamental physics to the control reactors. The fission chamber is privileged instrument for recording directly the fast fluctuations of neutron flux. Therefore, it is the most appropriate neutron detector that fulfills that need.

The output signal of the fission chamber can be processed through pulse mode, Campbelling mode, and current mode. Each of these modes operates within a specific range of the neutron flux. The originality of our research work is to apply the nonnegative matrix factorization algorithms to achieve Blind Sources Separation task of fission chamber output signals operating at any neutron flux range. Indeed, we seek better characterization of neutron signals with very high precision through the analysis of extracted independent components. Forthcoming works are going to deal with the implementation of the Blind Sources Separation Algorithms and their application in neutron flux monitoring area.

The authors declare that they have no conflicts of interest.