On the Zero-Neutron Density in Stochastic Nuclear Dynamics (original) (raw)
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Calculation of the Parameters of Stochastic Neutron Kinetics in Zero Power Nuclear Reactors
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. SERIES: NUCLEAR AND REACTOR CONSTANTS, 2019
The majority of neutron-physical problems of nuclear power plants design can be solved on the basis of various approximations to the Boltzmann transport equation in terms of the averaged characteristics of the reactor: the effective multiplication factor, neutron flux, average neutron lifetime, etc. However, the neutron chain reaction itself is always stochastic. There are situations in which the stochastic nature of the chain reaction cannot be ignored. This is the so-called “blind” start-up problem with a weak external neutron source, the work of physical assemblies of “zero” power, the analysis of the reactivity noise of such assemblies, etc. Despite the well-developed theoretical basis for the stochastic description of the behavior of neutrons in a nuclear reactor, there are still not enough calculation algorithms and programs for stochastic kinetics analysis. The paper presents two computational algorithms for point reactor model, which are developed on the basis of the theory ...
Stochastic methods for the neutron transport equation II: Almost sure growth
The Annals of Applied Probability, 2020
The neutron transport equation (NTE) describes the flux of neutrons across a planar cross-section in an inhomogeneous fissile medium when the process of nuclear fission is active. Classical work on the NTE emerges from the applied mathematics literature in the 1950s through the work of R. Dautray and collaborators, [8, 9, 24]. The NTE also has a probabilistic representation through the semigroup of the underlying physical process when envisaged as a stochastic process; cf. [8, 22, 23, 25]. More recently, [6] and [18] have continued the probabilistic analysis of the NTE, introducing more recent ideas from the theory of spatial branching processes and quasi-stationary distributions. In this paper, we continue in the same vein and look at a fundamental description of stochastic growth in the supercritical regime. Our main result provides a significant improvement on the last known contribution to growth properties of the physical process in [25], bringing neutron transport theory in line with modern branching process theory such as [15, 13]. An important aspect of the proofs focuses on the use of a skeletal path decomposition, which we derive for general branching particle systems in the new context of non-local branching mechanisms.
A Lumped Stochastic Model of Coupled Neutronic Assemblies
2020
Large amplitude random fluctuations in the neutron number are a characteristic of fissile assemblies containing intrinsic random sources and cannot be adequately represented by low order statistical moments such as the mean and variance, except when the population is nearly deterministic. The non-zero probability that an individual neutron chain (defined as a source neutron and all of its progeny) may become extinct in finite time or grow without bound manifests as strong stochastic variation of the population as a whole. A precise characterization of random process describing the time variation of the neutron number requires specification of the infinite order probability distribution function (pdf) lim k→∞ P (n 1 , t 1 ; n 2 , t 2 ; • • • n k , t k), t 1 < t 2 • • • < t k , defined as the joint probability of finding n 1 neutrons at time t 1 , n 2 neutrons at time t 2 ,. . ., n k neutrons at time t k , and so on. Needless to say, it is not feasible to construct this distribution under general conditions because of the need to compute time correlations of arbitrarily high order, even when the neutron phase dependence (position, energy, direction) is ignored, as is the case in this work. However, a major simplification is afforded by the observation that the random process associated with the neutron population is Markovian to an excellent approximation. The joint probability distribution function then factors into a product of lower order pdfs, in the form P (n 1 , t 1) P (n 2 , t 2 |n 1 , t 1) • • • P (n k , t k |n k−1 , t k−1), so that the entire distribution is characterized by just the conditional probability P (n j , t j |n i , t i), t i ≤ t j , and the marginal distribution P (n i , t i). Moreover, the theory of branching Markov processes shows that the conditional probability satisfies the Chapmann-Kolmogorov equation or, equivalently, the forward and backward Master equations, from which the stochastic dynamics can be computed by Monte Carlo simulation or by directly solving the Master equation itself. Joint distribution functions can then in principle be constructed by composing the conditional probabilities at successive times. Thus, a knowledge of the conditional probability suffices to completely characterize the neutron population for all time. 2 Definitions Consider M coupled neutronic assemblies and define P (n 1 , n 2 , • • • n M ; t) as the joint probability of finding n 1 neutrons in assembly-1, n 2 in assembly-2,. . ., and n M neutrons in assembly-M at time t, conditioned on an initial distribution of neutrons at t = 0. Using the multindex notation n = {n 1 , n 2 , • • • n M }, | n| = n 1 + n 2 , • • • + n M , this joint probability distribution function (jpdf) is compactly expressed as P (n; t). Let λ c,i (t)dt, λ f,i (t)dt, and λ a,i (t)dt be the probabilities of capture, fission, and absorption (capture plus fission), respectively, in a short time interval dt in assembly-i, i = 1, 2, • • • M. Also let
An application of reactor noise techniques to neutron transport problems in a random medium
Annals of Nuclear Energy, 1989
~Neutron transport problems in a random medium are considered by defining a joint Markov process describing the fluctuations of neutron population and the random changes in the medium. Backward Chapman-Kolmogorov equations are derived which yield an adjoinI transport equation for the average neutron density. It is shown that this average density also satisfies the direct transport equation as given by the phenomenological model.
Chinese Physics C, 2022
Dissipative dynamics of nuclear fission is a well confirmed phenomenon described either by a Kramers-modified statistical model or by a dynamical model employing the Langevin equation. Though dynamical models as well as statistical models incorporating fission delay are found to explain the measured fission observables in many studies, it nonetheless shows conflicting results for shell closed nuclei in the mass region 200. Analysis of recent data for neutron shell closed nuclei in excitation energy range 40−80 MeV failed to arrive at a satisfactory description of the data and attributed the mismatch to shell effects and/or entrance channel effects, without reaching a definite conclusion. In the present work we show that a well established stochastic dynamical code simultaneously reproduces the available data of pre-scission neutron multiplicities, fission and evaporation residue excitation functions for neutron shell closed nuclei 210 Po and 212 Rn and their isotopes 206 Po and 214,216 Rn without the need for including any extra shell or entrance channel effects. The calculations are performed by using a phenomenological universal friction form factor with no ad-hoc adjustment of model parameters. However, we note significant deviation, beyond experimental errors, in some cases of Fr isotopes.
Variances and Covariances of Neutron and Precursor Populations in Time-Varying Reactors
Nuclear Science and Engineering, 1968
The fluctuating populations of particles in a reactor are described in usual kinetics studies by the ''numbers'' of neutrons and precursors which, in the absence of feedback mechanisms, can be identified with first moments of the population distributions. At a higher level of description, variances and covariances of the neutron and precursor populations are determined from equations similar to the first-moment equations. The behavior of these first and second moments for time-varying reactors is explored here analytically and numerically, and inferences are made as to the effect of initial reactor conditions and modes of reactivity change on this dynamic behavior.