Multiplying Medium Research Articles

Neutron Transport Theory in a Ring Reactor

The one-speed transport equation is solved for a ring reactor. A complete solution is obtained for the space-time relaxation of a pulse of neutrons in a multiplying medium in which delayed neutrons are neglected. The solution consists of a fundamental mode, a finite number of harmonics, and an integral transient. A condition is deduced, which gives the maximum number of harmonics that can exist for a given ring circumferenced. The limitations of diffusion theory are pointed out with particular reference to the shortcomings of that theory in dealing with the early stages of evolution of the pulse. Delayed neutrons are included and a complete solution is obtained by means of the prompt jump approximation. The results are illustrated by numerical calculations designed to show the onset of instabilities in the harmonics when the reactor is sufficiently large.

Effect of 4PU on the dormant shoot tip culture of peach, nectarine, sweetcherry and plum.

Shoot tips of eight Prunus cultivars of peach, nectarine, sweetcherry and plum were cultured. Shoot tip of 0.5∼1.0mm taken from each dormant bud was placed on the paper-wick in the medium containing various kind of cytokinins. Survival rates were higher on the medium containing 2.0 mg/l N-(2-chloro-4-pyridyl)-N-phenyl-urea (4 PU, Kyowa Hakko Kogyo K.K.) than other medium containing 6-BA or kinetin. There were varietal differences among eight cultivars in survival rate and shoot growth. Among eight cultivars, three sweetcherry cultivars showed vigorous growth. When shoots were transferred to the multiplying medium with reduced cyto-kinin concentration of 0.5 mg/l 6-BA, 'Ooishiwasesumomo' and 'Taiyou', plum cultivars, multiplied 5.1 and 5.7 times respectively in one month. Roots were induced on the medium containing 0.5 mg/l 4PU. Acclimatized plants have been obtained in plum cultivar 'Taiyou'.

Propagation of almond by shoot tip culture.

Shoot tips 0.5mm long were taken from actively growing almond (Prunus amygdalus Batsch cv. hardy) shoots. These were cultured on the modified 1/2 MS medium (UEMATSU and AKIHAMA 1987) supplemented with 6-benzyladenin (6-BA) or N-(2-chloro-4-pyridyl)-N-phenylurea (4PU, Kyowahakko Kogyo K. K.). About 70 to 80% of shoots survived on the paper-wick liquid medium containing 2.0 or 4.0 mg/l 4PU. Growing shoots were transferred to the solid multiplying medium containing 2g/l gerlite and 0.5mg/l 6-BA. These shoots were subcultured on the same fresh medium at forty days intervals. After nine months of culture, the multiplication rate was 6.6-fold per L0rty days. The longest shoot reached to 3.8cm. Within a month after transferring to the rooting medium containing 0.5mg/l naphthalene aceticacid (NAA), root induction occurred.

On the factorial moments of the neutron multiplicity distribution of fission cascades

The effective number of combinations with which neutrons can be grouped together is an important quantity required for the interpretation of neutron time correlation or coincidence experiments with multiplying media. For this quantity analytical expressions are derived valid for a far subcritical assembly in a point and a one neutron energy group approximation. The derived expressions are applicable in models for the mass determination of spontaneous fission neutron emitters.

Kinetics of Driven Multiplying Media

The kinetics of multiplying media sustained by externally generated neutrons is examined. Point kinetics and diffusion theory considerations form the underlying basis. The results derived here constitute an extension of neutron kinetics analysis with ready applications to hybrid reactors and spallation accelerator blankets.

Analysis of some pseudo-discrete spatial decay modes of the two- and three-group neutron transport equations by analytic continuation

The nature and location of the poles of the dispersion function of the frequency-dependent two- and three-group neutron transport equations for an infinite 3-D homogeneous multiplying medium with constant cross sections and isotropic scattering is investigated. It is shown that on the right half of the complex plane, only one pole exists on the physical sheet of the Riemann surface, which is almost identical to the one obtained in two- and three-group frequency-dependent diffusion theory, and gives rise to the well-known, in neutron noise, long attenuation length global component of the detector response. In contrast to diffusion theory, in transport theory no other pole exists on the physical sheet; the poles associated with the short spatial relaxation length ‘local component’ in homogeneous diffusion theories approaches are bifurcated and are to be found on the adjacent ‘unphysical’ sheets of the Riemann surface of the generic multivalued dispersion function. Although not on the physical sheet, the bifurcated poles are located very close to it and give rise to pseudo-discrete spatial decay modes with very similar characteristics to the local component of the detector response in homogeneous diffusion theories.

A non-linear problem in neutronic fluctuations

The behaviour of the fluctuations in a multiplying medium has been studied using a simple birth and death process. The investigation differs from the classical approach to this problem in that it includes the effects of burnup, i.e. removal, of the absorbing particles. Thus we consider a mixture of two species one of which feeds upon the other. However, species ‘1’ is renewable by births, whereas species ‘2’ is not. We associate species ‘1’ with neutrons and species ‘2’ with the fissile isotope. A probability-balance equation is formed and the associated equation for the probability generating function is derived. From this equation we obtain values for the mean and variance by a cumulant expansion technique. Results show that the burnup effect leads to non-linear equations for the moments and that these in turn show a significant deviation from the conventional linear solutions for high supercriticality. The variance of the neutron density fluctuations is shown to tend to a constant value as time increases. The rate at which this occurs might be detectable experimentally and provides further evidence of the non-physical nature of the unboundedness of the classical linear variance with time.

Exact solution of transport equation in finite multiplying media

One speed neutron transport equation in a finite multiplying medium is exactly and uniquely solved following the first author's new technique (Das Gupta, 1978b) based on Laplace transformation and Wiener-Hopf technique, to obtain the angular neutron density at any depth. Criticality condition is derived.

Theory of stochastic space-dependent neutron kinetics with a Gaussian parametric excitation

Neutron kinetics and statics in a multiplying medium with a statistically fluctuating reactivity are unified and systematically studied by applying the Novikov-Furutsu formula. The parametric or multiplicative noise is spatially distributed and of Gaussian nature with an arbitrary spectral profile. It is found that the noise introduces a new definite production term into the conventional balance equation for the mean neutron number. The term is characterized by the magnitude and the correlation function of the random excitation. Its relaxation phenomena bring forth a non-Markoffian or a memory effect, which is conceptualized by introducing ‘pseudo-precursors’ or ‘pseudo-delayed neutrons’. By using the concept, some typical reactor physical problems are solved; they are (1) reactivity and flux perturbation originating from the random dispersal of core materials and (2) analysis of neutron decay mode and its relaxation constant, and derivation of the corresponding new inhour equation.

A discussion of the forward and backward Kolmogorov equations for multiplying systems

A parallel derivation is formulated of the forward Kolmogorov equation (FKE) and of the backward Kolmogorov equation (BKE) from detailed probability balance equations for a population of neutrons, precursors and detected neutrons living in a multiplying medium where processes of capture and leakage, fission, precursor decay, detection and source emission occur. The FKE and BKE are transformed into equations for the generating function of the related state probabilities. The work is characterized by the exposition of several mathematical details and implications. The material has been collected with the purpose of facilitating access and understanding.

An exact solution of the extinction problem in supercritical multiplying systems

Using the point model approximation and one-speed theory with no delayed neutrons, we have constructed a probability balance equation for neutrons by the backward method. This probability gives the distribution of neutrons in a multiplying medium at a given time and also the distribution that a chain will have generated a specified number of neutrons before extinction. We consider the limit of this probability for super and subcritical systems for long times after the initial triggering neutron. This leads to the extinction probability and to the individual probabilities of neutron population. To obtain specific results we have used a variety of models for the neutron multiplicity in the fission process, i.e. Poisson, birth and death, geometric and binomial. Exact solutions for the extinction probability have been obtained and its sensitivity to various parameters examined. Finally, we use the ‘quadratic approximation’ and assess its accuracy; it is found to overestimate the extinction probability and to be useful only for multiplication factors near unity.

Positive Solutions and Criticality of the Linear and Some Nonlinear Neutron Transport Problems

In this paper, we study the existence and nonexistence of a nonnegative solution for the linear and some nonlinear time-independent neutron transport problems in a multiplying transport medium. Our approach to the problem involves the construction of a “maximal” and a “minimal” sequence, each of which converges to a nonnegative solution of the problem. It is shown that these sequences converge monotonically to a maximal and a minimal solution, respectively, if and only if there exists an upper solution. Through the construction of an upper solution we obtain a simple criterion for insuring the criticality or subcriticality of the system when fission dominates absorption. This criterion leads to some explicit estimates on the critical size of the transport region in terms of the physical parameters of the transport medium. In particular, a simple relation for estimating the critical length of the slab problem is also given.

Measurement of Space Dependent Angular Thermal Neutron Spectra in Natural Uranium-Light Water Slab

Abstract Space dependent 0° angular thermal neutron spectra were measured by time-of-flight method at several positions in a spectrum source, which is a single natural uranium plate sandwiched between two layers of pure water. Theoretical analysis was performed by means of multigroup discrete S8 method, using Haywood's model for the scattering of thermal neutrons. Fairly good agreement was observed between experiment and theory. The experiment proved that; (1) The Haywoods model is valid for describing the thermal neutron spectra in heterogeneous multiplying media like the present spectrum source, under the condition that anisotropics present in the densities of either neutron scattering or the slowingdown of source neutrons to the thermal neutrons is taken into consideration. The discrepancy between the measured and calculated spectra is very small, and the error induced therefrom in the neutron absorption of 235U is below 3%. (2) The spatial distribution of the source neutrons requires to be taken into ...

A Comparison Between Neutron Waves Reflected from the Reflector and from the Core of a Water-Moderated and -Reflected Reactor

When a propagating neutron wave reaches an interface between two media, a part of it is transmitted and a part is reflected. In the present work, neutron waves reflected from the core and from the reflector are compared. The following is concluded: Reflection exists in both cases. When the first medium in which the wave is propagating is a multiplying medium, it is much easier to detect the reflected wave than it is in the nonmultiplying diffusive medium. The reflected wave amplitude and phase depend much more on the properties of the first medium than on the properties of the reflecting medium. Neutron waves reflected back into the core are in phase with the propagating waves and hence reinforce them. Neutron waves reflected by the core are out of phase with the propagating waves and hence weaken them. Other characteristics of the amplitudes and phases of waves in both cases are compared.

Reflector Effect on Neutron Wave Propagation in Multiplying Media

Neutron waves in a reflected reactor are analyzed using the twogroup diffusion equation. The solution is composed of a transient which disappears within a few milliseconds, a-steady-state part, and an oscillatory part. The last is comprised of four terms corresponding to four waves: two propagating waves and two reflected waves. When a sinusoidal source of thermal neutrons is applied to a water-moderated multiplying medium, the source wave amplitude decays with a relaxation length equal to the diffusion length of the medium. A wave of fast neutrons is then produced by fission which is induced by the source wave. This fastneutron wave carries the wave phenomenon across the core. The two propagating thermal-neutron waves are (a) the source wave, and (b) a wave of thermalized fission neutrons. The two reflected waves are the reflections of the two propagating waves. The amplitudes and phases of the four waves are calculated, together with those of the total wave which is the sum of the four partial waves. The agreement of the measured data of the total wave with the calculations is reasonable. Modulated poison and source oscillators were used for the experiments. The reactor response to both was essentially the same except inmore » the region near the oscillators; while the first decreased the number of neutrons, the second increased their number. (auth)« less

Estimated activity of a thick specimen in a multiplying medium (conjugate random walk method)

Estimated activity of a thick specimen in a multiplying medium (conjugate random walk method)

Neutron transport initial value problem in non-multiplying medium

The transport equation is discussed for neutron density in a non- multiplying medium assuming the initial neutron density distribution is known. The characteristics of both the medium and the neutron sources are considered to be a function of time. The existence and the uniqueness of the initial problem were proved and some of the earlier results obtained by the author were generalized. The behavior is briefly discussed of Knudsen's gas in a thermal bath. (auth)

The theory of space dependent reactor noise analysis using gamma radiation

The space dependent theory of the stochastic fluctuations of the prompt gamma flux in a homogeneous infinite multiplying medium at zero power is developed in the frequency domain. The Langevin technique has been used to obtain the cross power spectral density as well as the spatial coherence function of the prompt gamma flux fluctuations between two space points in the medium. For this purpose, a new quantity, the so-called Green's function of the gamma flux, has been introduced as the response of the prompt gamma flux to the injection of a neutron into the system. The treatment is based on the one and two group energy diffusion model. Special emphasis is devoted to the spatial correlation range of the gammas compared with the neutrons by introducing micro- and macroscales as measures of spatial coherence. The theory is applied numerically to an infinite light water-uranium-235 system to illustrate the effects of gamma rays upon the spatial correlation range.

Reactor Neutron Fluctuation Simulator Using Pulsed Neutron Source

The fission phenomena in a multiplying medium are simulated by neutron bursts in a non-multiplying medium produced by a pulsed neutron source. To simulate the chain reaction, the pulsed neutron source is triggered by neutronic signals from a detector installed in that medium. The kinetic parameters can be changed at will by regulating the detector sensitivity, the system medium and the burst yield of the pulsed neutron source. This simulator has been tested on experiments with Rossi-a, Feynman-a and pulse die away methods and is also applied to the experiments with waiting time method and observation of neutron families to verify the recently developed theory in this domain. The results prove that, in principle, the number of neutrons in this simulator can represent the probabilistic nature of the neutrons in a multiplying medium described by one point model. Furthermore, the newly developed theory for reactor parameter measurement has been substantiated with use of the simulator. The simulator in its pre...

Reactor Neutron Fluctuation Simulator Using Pulsed Neutron Source

The fission phenomena in a multiplying medium are simulated by neutron bursts in a non-multiplying medium produced by a pulsed neutron source. To simulate the chain reaction, the pulsed neutron source is triggered by neutronic signals from a detector installed in that medium. The kinetic parameters can be changed at will by regulating the detector sensitivity, the system medium and the burst yield of the pulsed neutron source. This simulator has been tested on experiments with Rossi-a, Feynman-a and pulse die away methods and is also applied to the experiments with waiting time method and observation of neutron families to verify the recently developed theory in this domain. The results prove that, in principle, the number of neutrons in this simulator can represent the probabilistic nature of the neutrons in a multiplying medium described by one point model. Furthermore, the newly developed theory for reactor parameter measurement has been substantiated with use of the simulator. The simulator in its present form is restricted in applicable range to keff=0–0.66 by limitations in the performance of the pulsed neutron source used.