A random walk in reactor physics and neutron transport

The title of this paper alludes to two different meanings of “random”. First, the phrase “Random walk” refers to the fact that I selected, at random, a few topics which I myself found fascinating, surprising, and hence hopefully entertaining, in the hope that the reader will also find them entertaining. The phenomena that will be described and discussed here will reveal some unexpected features, which in some cases are puzzling or even counter-intuitive, and their explanation sometimes discloses commonly accepted misbeliefs or misunderstandings. I always found such cases very intriguing. Inevitably, such subjects do not constitute a continuous story, rather they are picked randomly, hence the first meaning of the phrase “random walk” in the title. Curiosities similar to the types that will be discussed in this note are usually published as a “Letter to the Editor” or a “Technical Note”, since they do not contain new research results. A few examples are given in Ref [1] (meaning of the flux) and Refs [2] - [4] (number of collisions until slowing down). The readers are encouraged to check up these letters or technical notes. Many are, in contrast to the present article, quite short, often only one page, hence the “output/input ratio” in intellectual entertainment is quite high. I can also recommend the readers to watch out for such short notes by themselves (although, sadly, the number of such notes seems to be decreasing). The second reason why the word “random” appears in the title is because the curious facts and phenomena which will be discussed here concern the randomness of neutron transport, manifesting itself in the fact that the number of neutrons in the system, or the number of detector counts during a time period, is a random number or random process (hence often referred to as neutron fluctuations or neutron noise). Random processes in general, whether about neutrons or other processes, have themselves fascinating and surprising properties. The subjects discussed in this small essay will hopefully also expedite a wider understanding of the properties and use of neutron fluctuations in nuclear systems. With this introduction, I invite the reader to follow me on the random walk in the fascinating world of random particle transport

Ringhals Diagnostics and Monitoring: An overview of 30 years of collaboration 1993 - 2023

A joint research project between Chalmers University of Technology and the Ringhals power plant was conducted regarding development of noise analysis methods and their application to reactor diagnostics between 1995 - 2023. The project was financially supported by Ringhals. The actual contacts and collaboration started actually in 1993, although at the beginning with support from SKI. This report gives a historic overview of the project; its origins, start-up, the problems tackled, and the results obtained. The emphasis is more on providing a full descriptive inventory of the methods and results with explaining their significance, without going in deeply into technical details. For these latter, references will be made to the project reports and other publications. In addition to the research items, the report also includes lists of papers published and conference talks presented from the results of the collaboration, as well as the list of persons contributing to the results, the list of PhD and Licentiate exams, and finally a list of various prizes obtained by the Chalmers participants of the project. This report constitutes the closing part of the above mentioned long-term collaboration, and is supported financially by with Ringhals, Vattenfall AB, contract No. 4501756928-062. The work in the contract was performed between 1 July 2022 and 30 June 2023. The work was performed by Imre P\ue1zsit, who was the Principal Investigator for the whole long-term project. Contact person from Ringhals was Henrik Nyl\ue9

Multiplicity theory beyond the point model

Passive methods of nuclear safeguards determine the important parameters of an unknown sample from the statistics of the detection of the neutrons emitted from the item. These latter are due to spontaneous fissions and (α,n) reactions, enhanced by internal multiplication before leaking out. Based on the original work of B\uf6hnel, the methodology of traditional multiplicity counting is based on the first three factorial moments of the number of neutrons, emitted from the sample due to one source event. These “B\uf6hnel moments” were derived in the so-called “point model”, in which no space-dependence is accounted for, rather a uniform first collision probability is assumed for each neutron, irrespective of the position of its birth and its velocity direction, and, more important, it is assumed to be the same for all generations in the fission chain as for the source neutrons. The purpose of the present work is to derive the same factorial moments in a one-speed space-dependent model, in which the position and direction of the neutrons is accounted for, but (similarly to the original B\uf6hnel model), no energy dependence is assumed. The integral equations for the moments are solved numerically with a collision number expansion. It is shown that compared to the space-dependent calculations, the unfolding method using the point model underestimates the fissile mass and the underestimation increases with increasing both of fissile mass and the value of α

Transport calculations of the multiplicity moments for cylinders

In a previous paper by P\ue1zsit and P\ue1l [“Multiplicity Theory Beyond the Point Model,” Ann. Nucl. Energy, Vol. 154 (2021)], a general transport theory calculation of the factorial moments of the number of neutrons emitted spontaneously from a sample was elaborated. In contrast to the original derivations by Hage and Cifarelli [“On the Factorial Moments of the Neutron Multiplicity Distribution of Fission Cascades,” Nucl. Instrum. Meth. Phys. Res. A, Vol. 236 (1985)] and B\uf6hnel [“The Effect of Multiplication on the Quantitative Determination of Spontaneously Fissioning Isotopes by Neutron Correlation Analysis,” Nucl. Sci. Eng., Vol. 90 (1985)], also referred to as the point model, in the transport model the spatial and angular dependence of the internal fission chain is taken into account with a one-speed transport theory treatment. Quantitative results were given for a spherical item, and the bias of the point model regarding the estimation of the fission rate as compared to the more exact space-dependent model was estimated as a function of the size of the sphere and the α factor.In the present paper the formalism and the quantitative work are extended to the treatment of items with cylindrical shapes, which are more relevant in many practical applications. Results are presented for both square cylinders (D=H) and for tall (H/D>1) and flat (H/D<1) cylinders. This way the differences between the cylinder and the sphere on one hand and those between the various cylinder shapes on the other hand can be estimated. The results show that the bias depends on the geometry of the cylinder quite moderately, but similarly to the case of the sphere, the bias of the point model is quite significant for larger item sizes and α values, and it is nonconservative (underestimates the fissile mass) as well

Derivation and quantitative analysis of the differential self-interrogation Feynman-alpha method

A stochastic theory for a branching process in a neutron population with two energy levels is used to assess the applicability of the differential self-interrogation Feynman-alpha method by numerically estimated reaction intensities from Monte Carlo simulations. More specifically, the variance to mean or Feynman-alpha formula is applied to investigate the appearing exponentials using the numerically obtained reaction intensities.Comment: Proceedings 52nd INMM conference, Palm Desert, 17-21 July 201

Space-Dependent Calculation of the Multiplicity Moments for Shells With the Inclusion of Scattering

In recent work, we extended the methodology of multiplicity counting in nuclear safeguards by elaborating the one-speed stochastic transport theory of the calculation of the so-called multiplicity moments, i.e., the factorial moments of the number of neutrons emitted from a fissile item, following a source event from an internal neutron source [spontaneous fission and ((Formula presented.)) reactions]. Calculations were made for solid spheres and cylinders, with the source being homogeneously distributed within the item. Recent measurements of the Rocky Flats Shells during the Measurement of Uranium Subcritical and Critical (MUSIC) campaign conducted by Los Alamos National Laboratory and assisted by the University of Michigan inspired us to extend the model to spherical shell geometry with a point source in the middle of the central cavity. Comparison of the calculated results with the experimental ones indicated that accounting for fission as the only neutron reaction (the standard procedure in the point model, adapted also in our work so far) was not sufficient for reaching good agreement with measurements. The model was therefore extended to include elastic scattering into the one-speed formalism, whereas the effect of inelastic scattering was accounted for in an empirical way. After these extensions, good agreement was found between the calculated and the measured values. The paper describes the extension of the theory and provides concrete quantitative results

Identification of diversions in spent PWR fuel assemblies by PDET signatures using Artificial Neural Networks (ANNs)

Spent nuclear fuel represents the majority of materials placed under nuclear safeguards today and it requires to be inspected and verified regularly to promptly detect any illegal diversion. Research is ongoing both on the development of non-destructive assay instruments and methods for data analysis in order to enhance the verification accuracy and reduce the inspection time. In this paper, two models based on Artificial Neural Networks (ANNs) are studied to process measurements from the Partial Defect Tester (PDET) in spent fuel assemblies of Pressurized Water Reactors (PWRs), and thus to identify at different levels of detail whether nuclear fuel has been replaced with dummy pins or not. The first model provides an estimation of the percentage of replaced fuel pins within the inspected fuel assembly, while the second model determines the exact configuration of the replaced fuel pins. The two models are trained and tested using a dataset of Monte-Carlo simulated PDET responses for intact spent PWR fuel assemblies and a variety of hypothetical diversion scenarios. The first model classifies fuel assemblies according to the percentage of diverted fuel with a high accuracy (96.5%). The second model reconstructs the correct configuration for 57.5% of the fuel assemblies available in the dataset and still retrieves meaningful information of the diversion pattern in many of the misclassified cases

Conceptual design and initial evaluation of a neutron flux gradient detector

Identification of the position of a localized neutron source, or that of local inhomogeneities in a multiplying or scattering medium (such as the presence of small, strong absorbers) is possible by measurement of the neutron flux in several spatial points, and applying an unfolding procedure. It was suggested earlier, and it was confirmed by both simulations and pilot measurements, that if, in addition to the usually measured scalar (angularly integrated) flux, the neutron current vector or its diffusion approximation (the flux gradient vector) is also considered, the efficiency and accuracy of the unfolding procedure is significantly enhanced. Therefore, in support of a recently started project, whose goal is to detect missing (replaced) fuel pins in a spent fuel assembly by non-intrusive methods, this idea is followed up. The development and use of a dedicated neutron detector for within-assembly measurements of the neutron scalar flux and its gradient are planned. The detector design is based on four small, fiber-mounted scintillation detector tips, arranged in a rectangular pattern. Such a detector is capable of measuring the two Cartesian components of the flux gradient vector in the horizontal plane. This paper presents an initial evaluation of the detector design, through Monte Carlo simulations in a hypothetical scenario

Ringhals Diagnostics and Monitoring, Final Research Report 2012-2014

This report gives an account of the work performed by the Department of Nuclear Engineering, Chalmers, in the frame of research collaboration with Ringhals, Vattenfall AB, contract No. 630217-031. The contract constitutes a 3-year co-operative research work concerning diagnostics and monitoring of the BWR and PWR units. The work in thecontract has been performed between January 1st 2012, and December 31st, 2014. During this period, we have worked with four main items as follows:1. Development and application of the analysis method of core barrel vibrations, developed in the previous Stages, to three ex-core measurements performed during several cycles in R2, R3 and R4. What regards R2, this was the first attempt to analyze ex-core measurements taken at BOC, MOC and EOC, with the new curve-fitting procedure;2. Investigation of the ultra-low frequency oscillations in reactor power in R4;3. Development of the theory and simulations in order to determine the void content in R1 from the analysis of in-core measurements;4. Evaluation of the measurements made in R1 with the use of 4 LPRMs and one TIP detector, for testing the velocity and void fraction profile reconstruction methods.This work was performed at the Department of Nuclear Engineering, Chalmers University of Technology by Victor Dykin, Cristina Montalvo (visitor from the TechnicalUniversity of Madrid), Imre P\ue1zsit (project co-ordinator) and Henrik Nyl\ue9n, who was also the contact person at Ringhals