Neutron Population Research Articles

A multi-region multi-energy formalism for the Feynman-alpha formulas

The stochastic transport equation, describing the dynamics in time of the neutron population in a nuclear system, is used to gain expressions for the higher moments of the neutron population in a sub-critical system. Such expressions are the bone structure of the so called Feynman-α method to analyze noise experiments, aimed to determine the reactivity of sub-critical systems. In the present study, a general formalism for the stochastic transport equation in an N regions system, under the M energy groups approximation will be introduced. In particular, expressions for the Feynman variance to mean (or the Feynman-Y function) under the above mentioned restriction will be sought by using the steady state mode of the solution.

Clustering of branching Brownian motions in confined geometries.

We study the evolution of a collection of individuals subject to Brownian diffusion, reproduction, and disappearance. In particular, we focus on the case where the individuals are initially prepared at equilibrium within a confined geometry. Such systems are widespread in physics and biology and apply for instance to the study of neutron populations in nuclear reactors and the dynamics of bacterial colonies, only to name a few. The fluctuations affecting the number of individuals in space and time may lead to a strong patchiness, with particles clustered together. We show that the analysis of this peculiar behavior can be rather easily carried out by resorting to a backward formalism based on the Green's function, which allows the key physical observables, namely, the particle concentration and the pair correlation function, to be explicitly derived.

Fissioncore: A desktop-computer simulation of a fission-bomb core

A computer program, fissioncore, has been developed to deterministically simulate the growth of the number of neutrons within an exploding fission-bomb core. The program allows users to explore the dependence of criticality conditions on parameters such as nuclear cross-sections, core radius, number of secondary neutrons liberated per fission, and the distance between nuclei. Simulations clearly illustrate the existence of a critical radius given a particular set of parameter values, as well as how the exponential growth of the neutron population (the condition that characterizes criticality) depends on these parameters. No understanding of neutron diffusion theory is necessary to appreciate the logic of the program or the results. The code is freely available in FORTRAN, C, and Java and is configured so that modifications to accommodate more refined physical conditions are possible.

G4-STORK: A Geant4-based Monte Carlo reactor kinetics simulation code

In this paper we introduce G4-STORK (Geant4 STOchastic Reactor Kinetics), a new, time-dependent, Monte Carlo particle tracking code for reactor physics applications. G4-STORK was built by adapting and expanding on the Geant4 Monte Carlo toolkit. The toolkit provides the fundamental physics models and particle tracking algorithms that track each particle in space and time. It is a framework for further development (e.g.for projects such as G4-STORK).G4-STORK derives reactor physics parameters (e.g.keff) from the continuous evolution of a population of neutrons in space and time in the given simulation geometry. In this paper we detail the major additions to the Geant4 toolkit that were necessary to create G4-STORK. These include a renormalization process that maintains a manageable number of neutrons in the simulation even in very sub- or supercritical systems, scoring processes (e.g.recording fission locations, total neutrons produced and lost, etc.) that allow G4-STORK to calculate the reactor physics parameters, and dynamic simulation geometries that can change over the course of simulation to illicit reactor kinetics responses (e.g.fuel temperature reactivity feedback).The additions are verified through simple simulations and code-to-code comparisons with established reactor physics codes such as DRAGON and MCNP. Additionally, G4-STORK was developed to run a single simulation in parallel over many processors using MPI (Message Passing Interface) pipes.

G4-STORK: A Monte Carlo Reactor Kinetics Simulation Code

G4-STORK (Geant4 Stochastic Reactor Kinetics) is a time-dependent Monte Carlo particle physics code for reactor physics applications. G4-STORK was built using the Geant4 Monte Carlo toolkit and is designed to model the continuous evolution of a population of neutrons in space and time. From this evolution, various important reactor physics quantities can be calculated, including the reactivity of the system and the entropy of the neutron spatial distribution. System properties, such as the temperature of a material, can be changed incrementally to approximate time dependence. Thus, G4-STORK can be used to model reactor kinetics and was used to simulate a system that underwent an instantaneous increase in temperature.

Constraints on the symmetry energy from observational probes of the neutron star crust

A number of observed phenomena associated with individual neutron star systems or neutron star populations find explanations in models in which the neutron star crust plays an important role. We review recent work examining the sensitivity to the slope of the symmetry energy L of such models, and constraints extracted on L from confronting them with observations. We focus on six sets of observations and proposed explanations: i) The cooling rate of the neutron star in Cassiopeia A, confronting cooling models which include enhanced cooling in the nuclear pasta regions of the inner crust; ii) the upper limit of the observed periods of young X-ray pulsars, confronting models of magnetic field decay in the crust caused by the high resistivity of the nuclear pasta layer; iii) glitches from the Vela pulsar, confronting the paradigm that they arise due to a sudden recoupling of the crustal neutron superfluid to the crustal lattice after a period during which they were decoupled due to vortex pinning; iv) the frequencies of quasi-periodic oscillations in the X-ray tail of light curves from giant flares from soft gamma-ray repeaters, confronting models of torsional crust oscillations; v) the upper limit on the frequency to which millisecond pulsars can be spun-up due to accretion from a binary companion, confronting models of the r-mode instability arising above a threshold frequency determined in part by the viscous dissipation timescale at the crust-core boundary; and vi) the observations of precursor electromagnetic flares a few seconds before short gamma-ray bursts, confronting a model of crust shattering caused by resonant excitation of a crustal oscillation mode by the tidal gravitational field of a companion neutron star just before merger.

An alternative stochastic formulation for the point reactor

The stochastic behavior of a point reactor is modeled with a system of Ito stochastic differential equations. This new approach does not require computing the square root of a matrix which is a great computational advantage. Moreover, the derivation procedure clearly demonstrates the mathematical approximations involved in the final formulation. Three numerical benchmarks show the accuracy of this model in predicting the mean and variance of the neutron and precursor population in a point reactor.

An adaptive p-refinement strategy applied to nodal expansion method in 3D Cartesian geometry

The aim of this work is to develop a coarse mesh treatment strategy using adaptive polynomial, p, refinement approach for average current nodal expansion method in order to solve the neutron diffusion equation. For performing the adaptive solution process, a posteriori error estimation scheme, i.e. flux gradient has been utilized for finding the probable numerical errors. The high net leakage in a node represents flux gradient existence between neighbor nodes and it may indicate the source of errors for the coarse mesh calculation. Therefore, the relative Cartesian directional net leakage of nodes is considered as an assessment criterion for mesh refinement in a sub-domain. In our proposed approach, the zeroth order nodal expansion solution is used along coarse meshes as large as fuel assemblies to treat neutron populations. Coarse nodes with high directional net leakage may be chosen for implementing higher order polynomial expansion in the corresponding direction, i.e. X and/or Y and/or Z Cartesian directions. Using this strategy, the computational cost and time are reduced relative to uniform high order polynomial solution. In order to demonstrate the efficiency of this approach, a computer program, APNEC, Adaptive P-refinement Nodal Expansion Code, has been developed for solving the neutron diffusion equation using various orders of average current nodal expansion method in 3D rectangular geometry. Some well-known benchmarks are investigated to compare the uniform and adaptive solutions. Results demonstrate the superiority of our proposed strategy in enhancing the accuracy of solution without using uniform high order solution throughout the domain and increasing the number of unknowns.

Undersampling diagnostics by fission matrix in Monte Carlo criticality calculations

Most Monte Carlo (MC) codes use the power iteration method for criticality calculations, where the fission bank of the previous cycle is taken as the source of the subsequent cycle. If the batch size (i.e. the neutron population per cycle) is insufficient, the power iteration may introduce biases in both keff and local tallies, which is known as the undersampling problem. Although the undersampling can be examined by independent runs with increasing batch sizes, it is important to diagnose undersampling directly upon a single calculation. In this paper, a method based on the fission matrix is proposed for undersampling diagnostics, and it is implemented in the RMC Monte Carlo code. Numerical results are presented for the “k-effective of the world” problem and the MC full-core performance benchmark. The results indicate that the fission matrix method is reliable for undersampling diagnostics. Besides, the effect of the mesh discretization on the fission matrix method is investigated.

A stochastic model for neutrons simulation considering the spectrum and nuclear properties with continuous dependence of energy

In this work we developed a stochastic model to simulate neutron transport in a heterogeneous medium, considering continuous neutron spectra and the nuclear properties with its continuous dependence on energy. This model was implemented using the Monte Carlo method for the propagation of neutrons in different media. Due to restrictions with respect to the number of neutrons that can be simulated in reasonable computational time we introduced a variable control volume together with (pseudo-) periodic boundary conditions in order to overcome this problem. This study allowed a detailed analysis of the influence of energy on the neutron population and its impact on the life cycle of neutrons. From the results, even for a simple geometrical arrangement, we can conclude that there is need to consider the energy dependence and hence defined a spectral effective multiplication factor per Monte Carlo step.

Analysis of Reactivity Induced Accident for Control Rods Ejection with Loss of Cooling

The understanding of the time-dependent behavior of the neutron population in a nuclear reactor in response to either a planned or unplanned change in the reactor conditions is of great importance to the safe and reliable operation of the reactor. In the present work the point kinetic equation is solved numerically using stiffness confinement method (SCM). The solution is applied to the kinetic equation in the presence of different types of reactivities and was compared with different analytical solutions. This method is also used to analyze reactivity induced accidents in two reactors. The first reactor is fueled with uranium and the second is fueled with plutonium. This analysis presents the effect of negative temperature feedback with the addition of positive reactivity of the control rods to overcome the occurrence of the control rod ejection accident and damaging of the reactor. Both the power and the temperature pulse following the reactivity initiated accidents are calculated. The results are compared with previous works and satisfactory agreement is found.

An accurate solution of point reactor neutron kinetics equations of multi-group of delayed neutrons

The understanding of the time-dependent behaviour of the neutron population in a nuclear reactor in response to either a planned or unplanned change in the reactor conditions is of great importance to the safe and reliable operation of the reactor. In this study, an accurate analytical solution of point reactor kinetics equations with multi-group of delayed neutrons for specified reactivity changes is proposed to calculate the change in neutron density. The method is based on formulating a coefficient matrix of the homogenous differential equations of the point reactor kinetics equations and calculating the eigenvalues and the corresponding eigenvectors of the coefficient matrix. A small time interval is chosen within which reactivity relatively stays constant. The analytical method was applied to solve the point reactor kinetics equations with six-groups delayed neutrons for a representative thermal reactor. The problems of step, ramp and temperature feedback reactivities are computed and the results compared with other traditional methods. The comparison shows that the method presented in this study is accurate and efficient for solving the point reactor kinetics equations of multi-group of delayed neutrons.

Statistical Theory of Fission Chains and Generalized Poisson Neutron Counting Distributions

The neutron counting probability distribution for a multiplying medium was shown by Hage and Cifarelli to be a generalized Poisson distribution that depends on the fission chain number distribution. An analytic formula is obtained for this number distribution, the probability to produce a number of neutrons in a fission chain. The formula depends on the probability that a fission spectrum neutron induces a subsequent fission and depends on the probability distribution for a specific number of neutrons to be produced in an individual induced fission. The formula is an exact solution to a functional equation due to Böhnel for the probability generating function. The Böhnel equation is derived as the t →∞ limit of a rate equation for a neutron population generating function, related to a rate equation studied by Feynman. The Böhnel equation is also shown to be a fixed point of an iteration problem, related to one studied by Hawkins and Ulam, where the iteration generates the chain a generation at a time. The discrete iteration problem is shown to be connected to the continuous time evolution of the chain. An explicit solution for the time evolution of the chain is given in the simplified approximation where at most two neutrons are created by an induced fission. The t →∞ limit of this equation gives a simple analytic expression for the solution to the Böhnel equation in this approximation. A generalized Poisson counting distribution constructed from the theoretical fission chain probability number distribution is compared to experimental data for a multiplying Pu sample.

Numerical simulation of stochastic point kinetic equation in the dynamical system of nuclear reactor

In the present paper, the numerical approximation methods, applied to efficiently calculate the solution for stochastic point kinetic equations (Hetrick, 1993; Kinard and Allen, 2004) in nuclear reactor dynamics, are investigated. A system of Itô stochastic differential equations has been analyzed to model the neutron density and the delayed neutron precursors in a point nuclear reactor. The resulting system of Itô stochastic differential equations are solved over each time-step size. The methods are verified by considering different initial conditions, experimental data and over constant reactivities. The computational results indicate that the methods are simple and suitable for solving stochastic point kinetic equations. In this article, a numerical investigation is made in order to observe the random oscillations in neutron and precursor population dynamics in subcritical and critical reactors.

The peculiar isolated neutron star in the Carina Nebula

While fewer in number than the dominant rotation-powered radio pulsar population, peculiar classes of isolated neutron stars (INSs) -- which include magnetars, the ROSAT-discovered "Magnificent Seven" (M7), rotating radio transients (RRATs), and central compact objects in supernova remnants (CCOs) -- represent a key element in understanding the neutron star phenomenology. We report the results of an observational campaign to study the properties of the source 2XMM J104608.7-594306. Its evolutionary state is investigated by means of deep dedicated observations obtained with XMM-Newton, the ESO Very Large Telescope, as well as publicly available gamma-ray data from the Fermi and AGILE missions. The observations confirm previous expectations and reveal a unique type of object. The source, which is likely within the Carina Nebula, has a soft spectrum with absorption features and no magnetospheric emission. The optical counterpart is fainter than V=27 and no gamma-ray emission is significantly detected. Very interestingly, while these characteristics are remarkably similar to those of the M7 or the only RRAT so far detected in X-rays, which all have spin periods of a few seconds, we found intriguing evidence of very rapid rotation, P=18.6 ms. We interpret these new results in the light of the observed properties of the currently known neutron star population, in particular those of standard rotation-powered pulsars, recycled objects, and CCOs. We find that none of these scenarios can satisfactorily explain the collective properties of 2XMM J104608.7-594306, although it may be related to the still poorly known class of Galactic anti-magnetars. Future XMM-Newton data, granted for the next cycle of observations (AO11), will help us to improve our current observational interpretation of the source, enabling us to significantly constrain the rate of pulsar spin down.

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.

Advanced Computational Models for Accelerator-Driven Systems

In the nuclear engineering scientific community, Accelerator Driven Systems (ADSs) have been proposed and investigated for the transmutation of nuclear waste, especially plutonium and minor actinides. These fuels have a quite low effective delayed neutron fraction relative to uranium fuel, therefore the subcriticality of the core offers a unique safety feature with respect to critical reactors. The intrinsic safety of ADS allows the elimination of the operational control rods, hence the reactivity excess during burnup can be managed by the intensity of the proton beam, fuel shuffling, and eventually by burnable poisons. However, the intrinsic safety of a subcritical system does not guarantee that ADSs are immune from severe accidents (core melting), since the decay heat of an ADS is very similar to the one of a critical system. Normally, ADSs operate with an effective multiplication factor between 0.98 and 0.92, whichmeans that the spallation neutron source contributes little to the neutron population. In addition, for 1GeV incident protons and lead-bismuth target, about 50% of the spallation neutrons has energy below 1MeV and only 15% of spallation neutrons has energies above 3MeV. In the light of these remarks, the transmutation performances of ADS are very close to those of critical reactors. This contributes to different research topics, including the following:

A General Approach to Nuclear Fission Sustainability and the Need for Specific Solutions. A Case Study on a New Coolant

Extensive exploitation of nuclear raw materials requires the use of “nuclear breeding”, which is a phenomenon that can be attained in fast reactors. However, those reactors have had a complex history with some drawbacks and some important nuclear-policy attacks, as the INFCE initiative launched inside IAEA in 1978. Two points were very relevant in that context: the extensive use of plutonium recycling and an inherent property of fast reactors that could induce positive feedback between reactivity and thermal-hydraulics. In fact, a partial or total loss of coolant could convey a tremendous injection of reactivity, which could produce a catastrophe. An alternative to breeding in critical fast reactors is presented by hybrids, which are subcritical reactors which need an external neutron source for keeping their neutron population alive. Besides that, design and natural responses of the reactor systems against accidental initiating events have to be considered for arriving to the concept of Residual Safety beyond Design Limits. Such a final safety level will depend quite a lot on the type of coolant and the way the fuel is conformed into elements of a given geometry.

FURTHER EVALUATION OF A STOCHASTIC MODEL APPLIED TO MONOENERGETIC SPACE-TIME NUCLEAR REACTOR KINETICS

In a previous study, the stochastic space-dependent kinetics model (SSKM) based on the forward stochastic model in stochastic kinetics theory and the Ito stochastic differential equations was proposed for treating monoenergetic space-time nuclear reactor kinetics in one dimension. The SSKM was tested against analog Monte Carlo calculations, however, for exemplary cases of homogeneous slab reactors with only one delayed-neutron precursor group. In this paper, the SSKM is improved and evaluated with more realistic and complicated cases regarding several delayed-neutron precursor groups and heterogeneous slab reactors in which the extraneous source or reactivity can be introduced locally. Furthermore, the source level and the initial conditions will also be adjusted to investigate the trends in the variances of the neutron population and fission product levels across the reactor. The results indicate that the improved SSKM is in good agreement with the Monte Carlo method and show how the variances in population dynamics can be controlled.

Neutron slowing-down time in matter

We formulate the neutron slowing-down time through elastic collisions in a homogeneous, non-absorbing, infinite medium. Our approach allows taking into account for the first time the energy dependence of the scattering cross-section as well as the energy and temporal distribution of the source neutron population in the results. Starting from this development, we investigate the specific case of the propagation in matter of a mono-energetic neutron pulse. We then quantify the perturbation on the neutron slowing-down time induced by resonances in the scattering cross-section. We show that a resonance can induce a permanent reduction of the slowing-down time, preceded by two discontinuities: a first one at the resonance peak position and an echo one, appearing later. From this study, we suggest that a temperature increase of the propagating medium in presence of large resonances could modestly accelerate the neutron moderation.