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In a previous paper by Pazsit and Pal [“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 Bohnel [“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

In two earlier papers Pazsit et al. (2016), Nagy et al. (2018) we investigated the possibility of extracting the traditional multiplicity count rates from the cumulants of fission chambers signals in the current mode. The first three cumulants for up to three fission chambers were derived in an extended stochastic model of the detector signals, introduced earlier for simpler problems Pal et al. (2014), Pal and Pazsit (2015). It was shown that if all neutrons emitted from the sample simultaneously are also detected simultaneously, the multiplicity rates can be retrieved from the cumulants of the detector current, but the method breaks down if the detections of neutrons of common origin take place with a time delay spread wider than the pulse shape. It was seen that even if the pulses overlap partially, the retrieval of the multiplicity rates depends on the time delay distribution, which is usually not known. To remedy these shortcomings, in this work we extended the theory from the previous one-point (in time) treatment, where all moments (cumulants) refer to the same time instant, to two- and three-point distributions (correlations). It was found that the integrals of suitably chosen two- and three-point moments with respect to the time differences become independent of the probability density of the time delays of detections. With this procedure, within practical limits, the multiplicity rates can be retrieved from the detector current(s) for arbitrary time delay distributions, and hence also with thermalised neutrons. The underlying theory, the outline of the derivations and the full results are given in the paper.

In two earlier papers [1], [2] we investigated the possibility of extracting traditional multiplicity count rates from the cumulants of fission chamber signals in current mode. It was shown that if all neutrons emitted from the sample simultaneously are also detected simultaneously, the multiplicity rates can be retrieved from the first three cumulants of the currents of up to three detectors, but the method breaks down if the detections of neutrons of common origin take place with a time delay spread wider than the pulse shape. To remedy these shortcomings, in this work we extended the theory to two- and three-point distributions (correlations). It was found thatthe integrals of suitably chosen two- and three-point moments with respect to the time differences become independent of the probability density of the time delays of detections. With this procedure, the multiplicity rates can be retrieved from the detector currents for arbitrary time delay distributions. To demonstrate the practical applicability of the proposed method, a measurement setup was designed and built. The statistics (shape and amplitude distribution) of the detector pulse were investigated as important parameters of the theoretical model. Simulations were performed to estimate the expected value of the multiplicity rates in the built setup. Measurements were performed and two types of moments (the mean and the covariance function) of the recorded detector signals were calculated. Values of singles rates were successfully recovered.

In recent work, we have developed the theory of using the first three auto- and joint central moments of the currents of up to three fission chambers to extract the singles, doubles and triples count rates of traditional multiplicity counting (Pazsit and Pal, 2016; Pazsit et al., 2016). The objective is to elaborate a method for determining the fissile mass, neutron multiplication, and ( α , n) neutron emission rate of an unknown assembly of fissile material from the statistics of the fission chamber signals, analogous to the traditional multiplicity counting methods with detectors in the pulse mode. Such a method would be an alternative to He-3 detector systems, which would be free from the dead time problems that would be encountered in high counting rate applications, for example the assay of spent nuclear fuel. A significant restriction of our previous work was that all neutrons born in a source event (spontaneous fission) were assumed to be detected simultaneously, which is not fulfilled in reality. In the present work, this restriction is eliminated, by assuming an independent, identically distributed random time delay for all neutrons arising from one source event. Expressions are derived for the same auto- and joint central moments of the detector current(s) as in the previous case, expressed with the singles, doubles, and triples ( S , D and T ) count rates. It is shown that if the time-dispersion of neutron detections is of the same order of magnitude as the detector pulse width, as they typically are in measurements of fast neutrons , the multiplicity rates can still be extracted from the moments of the detector current, although with more involved calibration factors. The presented formulae, and hence also the performance of the proposed method, are tested by both analytical models of the time delay as well as with numerical simulations. Methods are suggested also for the modification of the method for large time delay effects (for thermalised neutrons).

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 Bohnel, 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 “Bohnel 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 Bohnel 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.

The extinction probability of a branching process (a neutron chain in a multiplying medium) is calculated for a system randomly varying in time. The evolution of the first two moments of such a process was calculated previously by the authors in a system randomly shifting between two states of different multiplication properties. The same model is used here for the investigation of the extinction probability. It is seen that the determination of the extinction probability is significantly more complicated than that of the moments, and it can only be achieved by pure numerical methods. The numerical results indicate that for systems fluctuating between two subcritical or two supercritical states, the extinction probability behaves as expected, but for systems fluctuating between a supercritical and a subcritical state, there is a crucial and unexpected deviation from the predicted behaviour. The results bear some significance not only for neutron chains in a multiplying medium, but also for the evolution of biological populations in a time-varying environment.

Neutron multiplicity counting (NMC) techniques are widely used for nuclear materials accountability and international safeguards applications to quantitatively evaluate characteristic properties pertaining to fissile material. Mathematical models for NMC moments have been previously derived for systems that use capture-based detectors; however, these models are not applicable when scatter-based detectors are used because of "neutron cross talk." Neutron cross talk occurs when a single neutron scatters and deposits energy above threshold into multiple detectors causing spurious increase in multiplicity counts; this, in turn, has caused fissile mass to be overestimated when not treated. In this paper, we propose new mathematical models derived from point kinetics to correct for neutron cross-talk effects up to any arbitrary order N, where N denotes the maximum number of counts a single neutron can cause. The new models were used to estimate the fissile mass of plutonium metal and oxide samples with effective 240Pu mass ranging from 2.5 to 250 g. The adequacy of the models was confirmed using simulations of a conceptual scatter-based neutron multiplicity counter (e.g., organic scintillators) using MCNPX v2.7e with the PoliMi fission event generating extension. The fissile mass estimates with no correction for neutron cross-talk events yielded an average relative deviation from the true 240Pueff mass of 55.94% and 84.56% for metal and oxide samples, respectively. When neutron cross-talk events of order N = 2 are included in the model, the fissile mass estimates yielded an average relative deviation of 11.89% for metal and 13.21% for oxide samples. Accounting for neutron cross-talk events of order N = 3 resulted in fissile mass estimates with an average relative deviation of 9.58% and 10.51% for metal and oxide samples, respectively. These mass estimates were compared to a reference case (i.e., no neutron crosstalk effects) that yielded an average relative deviation of 6.81% and 4.77% for metal and oxide samples, respectively. The discrepancy between the estimates from the proposed model and the reference case is attributed to the assumed value of N, which sets a finite upper bound on the order of cross-talk events the model treats (i.e., the model for N = 3 assumes that a neutron will never cause more than three counts).

This paper sets up a formalism that is sufficiently general to describe the effects of photofission, photonuclear, (n, xn), (n, n?x?), and (n, x?) reactions on the neutron-gamma Feynman-alpha variance-to-mean ratios. Such a formalism is obtained using the Chapman-Kolmogorov (master) forward equation for the above-mentioned set of nuclear reactions. Thereafter, the issue of estimating reaction intensities for gammas in the master equation is highlighted by the paper. As an example, a quantitative evaluation of reaction intensities is given for a case when (n, ?), photonuclear, and (n, 2n) reactions are relevant for the system. However, an evaluation of the influence of these types of reactions to the values of the Feynman variance-to-mean ratios is not within the scope of this paper. Overall, the results obtained in this paper are intended to give an extended systematic framework for the study of the neutron- and gamma-based nondestructive assay problems in nuclear reactor applications and materials control.

In nuclear safeguards, estimation of sample parameters using neutron-based non-destructive assay methods is traditionally based on multiplicity counting with thermal neutron detectors in the pulse mode. These methods in general require multi-channel analysers and various dead time correction methods. This paper proposes and elaborates on an alternative method, which is based on fast neutron measurements with fission chambers in the current mode. A theory of "multiplicity counting" with fission chambers is developed by incorporating Bohnel's concept of superfission [1] into a master equation formalism, developed recently by the present authors for the statistical theory of fission chamber signals [2,3]. Explicit expressions are derived for the first three central auto- and cross moments (cumulants) of the signals of up to three detectors. These constitute the generalisation of the traditional Campbell relationships for the case when the incoming events represent a compound Poisson distribution. Because now the expressions contain the factorial moments of the compound source, they contain the same information as the singles, doubles and triples rates of traditional multiplicity counting. The results show that in addition to the detector efficiency, the detector pulse shape also enters the formulas; hence, the method requires a more involved calibration than the traditional method of multiplicity counting. However, the method has some advantages by not needing dead time corrections, as well as having a simpler and more efficient data processing procedure, in particular for cross-correlations between different detectors, than the traditional multiplicity counting methods.

The Campbell theorem, relating the variance of the current of a fission chamber (a "filtered Poisson process") to the intensity of the detection events and to the detector pulse shape, becomes invalid when the neutrons generating the fission chamber current are not independent. Recently, a formalism was developed by the present authors, by which the variance of the detector current can be calculated for detecting neutrons in a subcritical multiplying system, where the detection events are obviously not independent. In the present paper, the previous formalism, which only accounted for prompt neutrons, is generalized to account also for delayed neutrons. A rigorous probabilistic analysis of the detector current was performed by using the same simple, but realistic detector model as in the previous work. The results of the present analysis made it possible to determine the bias of the traditional Campbelling techniques both qualitatively and quantitatively. The results show that the variance still remains proportional to the detection intensity, and is thus suitable for the monitoring of the mean flux, but the calibration factor between the variance and the detection intensity is an involved function of the detector pulse shape and the subcritical reactivity of the system, which diverges for critical systems.

Applying Higher Order Campbelling methods in neutron flux monitoring with fission chambers is advantageous due to their capabilities to suppress the impact of unwanted noises and signal contributions (such as gamma radiation). This work aims to verify through experimental results that the basic assumptions behind the Higher Order Campelling methods are valid in critical reactors. The experiments, reported in this work, were performed at the MINERVE reactor in Cadarache. It is shown that the calibration of a fission chamber and the associated electronic system is possible in higher order mode. With the use of unbiased cumulant estimators and with digital processing, it is shown that over a wide count rate range, accurate count rate estimation can be achieved based on signal samples of a few ms, which is a significant progress compared to similar experimental results in the literature. The difference between the count rate estimated by pulse counting and by the Higher Order Campelling is less than 4%. The work also investigates the possibility of monitoring transient events. For this purpose, a control rod drop event was followed in Higher Order Campbelling mode.

The signals of fission chambers are usually evaluated with the help of the co-called Campbelling techniques. These are based on the Campbell theorem, which states that if the primary incoming events, generating the detector pulses, are independent, then relationships exist between the moments of various orders of the signal in the current mode. This gives the possibility to determine the mean value of the intensity of the detection events, which is proportional to the static flux, from the higher moments of the detector current, which has certain advantages. However, the main application area of fission chambers is measurements in power reactors where, as is well known, the individual detection events are not independent, due to the branching character of the neutron chains (neutron multiplication). Therefore it is of interest to extend the Campbelling-type theory for the case of correlated neutron events. Such a theory could address two questions: partly, to investigate the bias when the traditional Campbell techniques are used for correlated incoming events; and partly, to see whether the correlation properties of the detection events, which carry information on the multiplying medium, could be extracted from the measurements. This paper is devoted to the investigation of these questions. The results show that there is a potential possibility to extract the same information from fission chamber signals in the current mode as with the Rossi- or Feynman-alpha methods, or from coincidence and multiplicity measurements, which so far have required detectors working in the pulse mode. It is also shown that application of the standard Campbelling techniques to neutron detection in multiplying systems does not lead to an error for estimating the stationary flux as long as the detector is calibrated in in situ measurements.

This chapter discusses the statics, kinetics, and dynamics of molten salt reactors in a simple model that allows analytical solutions. Due to this, and the introduction of some further limiting cases, the chapter offers substantial insight into and understanding of the physics and the neutronic behavior of molten salt systems with circulating fuel. After a discussion of the properties of the model and the limiting cases for the static equations, a treatment of space–time transients is introduced, and some limiting cases are solved explicitly. Thereafter the kinetic and dynamic response of the reactor is derived by calculating the space–frequency-dependent neutron fluctuations (neutron noise) induced by small stationary perturbations of the system parameters. The validity of the point kinetic approximation and the calculation of the point kinetic component of the noise are discussed. Finally, the neutron noise induced by various perturbations propagating with the circulating fuel is calculated and discussed.

The space-dependent neutron noise, induced by propagating perturbations (propagation noise for short) is investigated in a one-dimensional homogeneous model of various reactor systems. By using two-group theory, the noise in both the fast and the thermal group is calculated. The purpose is to investigate the dependence of the properties of the space-dependent fast and thermal propagation noise on the static neutron spectrum as well as on the presence of the fluctuations of several cross sections. The motivation for this study arose in connection with recent work on neutron noise in molten salt reactors (MSR) with propagating fuel of various compositions. Some new features of the induced noise were observed, but it was not clear whether these were due to the propagating perturbation alone, or to the propagation of the fuel and hence that of the delayed neutron precursors. The present study serves to clarify the significance of the spectral properties of the different cores through calculating the propagation noise in four different reactor systems, as well as considering the influence of the perturbation of the various cross sections. By comparing the results with those obtained in MSR, the effect of the moving fuel on the propagation noise is clarified. It is shown that in fast systems the noise in the fast group is much larger than in the thermal group and hence can gain diagnostic importance. It is also shown that the co-existence of several cross section fluctuations leads to qualitatively and quantitatively new characteristics of the noise, hence it is important to model the effect of e.g. temperature fluctuations of the coolant in a proper way.

In all cases where neutron fluctuations in a branching process (such as in multiplicity measurements) are treated in an energy dependent description, the energy correlations of the branching itself (energy correlations of the fission neutrons) need to be known. To date, these are not known from experiments. Such correlations can be theoretically and numerically derived by modelling the details of the fission process. It was suggested earlier that the fact that the prompt neutrons are emitted from the moving fission targets, will influence their energy and angular distributions in the lab system, which possibly induces correlations. In this paper the influence of the neutron emission process from the moving targets on the energy correlations is investigated analytically and via numerical simulations. It is shown that the correlations are generated by the random energy and direction distributions of the fission fragments. Analytical formulas are derived for the two-point energy distributions, and quantitative results are obtained by Monte-Carlo simulations. The results lend insight into the character of the two-point distributions, and give quantitative estimates of the energy correlations, which are generally small.

Multiplicity counters for neutron assay have been extensively used in materials control and accountability for nonproliferation, and nuclear safe- guards. Typically, neutron coincidence counters are utilized in these fields. In this paper we present a measurement system that makes use not only of neutron (n) multiplicity counting but also of gamma ray (γ) multiplicity counting and the combined higher-order multiples containing both neutrons and gamma rays. The benefit of this approach is in using both particle types available from the sample, leading to a reduction in measurement time when using more measurables. We present measurement results of n, γ, nn, nγ, γγ, nnn, nnγ, nγγ and γγγ multiples emitted by a 252Cf source and a 239Pu-Be 19 source. A dual radiation measuring system proposed here could use extra measurables either to improve the statistics when compared to a neutron-only system, or alternatively be used for extended analysis and interpretation of sample parameters. The study presented here provides an answer to the questions regarding the ability to measure both neutron and gamma ray coincidences at once. The results show that the measurements system proposed in this study has a potential to be valuable in the area of nuclear nonproliferation and homeland security.

We develop a fully analytic and self-contained description of the amplitude distribution of light pulses in an organic scintillation detector due to a monoenergetic source of fast neutrons. To this end, two classes of problems have to be handled. One is a formula for the light pulse amplitude distribution for the complete life history of neutrons slowing down in a mixture of hydrogen and carbon as a statistical average over all collision sequences that can occur, accounting also for neutron leakage. A complete solution is given in terms of a non-recursive convolution integral expansion with respect to the various possible collision histories. These latter are dependent on the collision probabilities of neutrons of a given energy. The second is the calculation of this collision probability from analytical expressions for the geometry of the detector, in the present case a right cylinder. This quantity was taken from Monte Carlo simulations in previous work. Recursive formulae are derived for the probabilities of arbitrary collision sequences, and quantitative results are given for up to five consecutive collisions of all combinations. These probabilities can be used to determine how to truncate the non-recursive expansion of the full light amplitude distribution in quantitative work. The calculational method serves to lend insight and understanding into the structure of the pulse height spectra, as well as it provides a computationally cheap method of generating a large number of such spectra for various detector compositions, sizes and neutron energies, for the development and test of new spectrum unfolding techniques. (C) 2010 Elsevier B.V. All rights reserved.

Measurement and monitoring of reactivity in a subcritical state, e.g. during the loading of a power reactor, has a clear safety relevance. The methods currently available for the measurement of k(eff) in stationary subcritical conditions should be improved as they refer to the critical state. This is also very important in the framework of ADS (accelerator driven systems) where the measurement of a subcritical level without knowledge of the critical state is looked for. An alternative way to achieve this is by mean of the Cf-252 source-detector method. The method makes use of three detectors inserted in the reactor: two "ordinary" neutron detectors and one Cf-252 source-detector which contains a small amount of Cf-252 that introduces neutrons in the system through spontaneous fission. By observing fissions through the detection system and correlating the signals of the three detectors, the reactivity rho (and hence the multiplication factor k) can be determined. Before the actual measurements took place, a suitable data acquisition system was realized in order to process the signals and compute the auto and cross power spectral densities. The measurements were then performed in the VENUS reactor, using the Cf-252 source-detector and two BF3 neutron detectors. The multiplication factor was determined using the Cf source method and compared with measurements using other methods and with computational results (Monte Carlo simulations). The Cf method was benchmarked at a UOX core to other experimental methods that used the critical state as reference and to calculations. Afterwards, the Cf source technique was analyzed in a MOX core to study the possible impact of a significant intrinsic source on the results. This benchmarking gives the possibility to validate the Cf method as a reliable technique for the measurement of subcritical levels in steady state and for cores with an intrinsic source like MOX or burnt fuel cores. (C) 2010 Elsevier Ltd. All rights reserved.

Formulas for multiplicity counting rates (singles, doubles, etc.), used for the unfolding of parameters of an unknown sample, can be derived from those for the corresponding factorial moments. So far such rates were derived only for neutrons. The novelty of this paper is related to the derivation of the individual gamma and mixed neutron-gamma detection rates as well as to investigation of the possibilities and actual algorithms for the sample parameter unfolding. Taking the individual gamma and mixed neutron-gamma moments up to third order, together with the neutron moments, there will be nine auto- and cross-factorial moments and corresponding multiplicity rates, as well as a larger number of unknowns than for pure neutron detection. The total number of measurable multiplicities exceeds the number of unknowns, but on the other hand the structure of the additional equations is substantially more complicated than that of the neutron moments. Since an analytical inversion of the highly non-linear system of over-determined equations is not possible, the use of artificial neural networks (ANNs) is suggested, which can handle both the non-linearity and the redundance in the measured quantities in an effective and accurate way. The use of ANNs is demonstrated with good results on the unfolding of various combination of multiplicities for certain combinations of the unknown parameters, including the sample fission rate, the leakage multiplication and the alpha and gamma ratios. (C) 2010 Elsevier B.V. All rights reserved.

The purpose of this paper is to present new analytical derivations to describe the emission and detection statistics of neutrons and photons generated in and emitted from fissile samples, with absorption included. The results of the analytical approach are compared with and validated by corresponding Monte Carlo simulations. The joint statistics of the generated and detected neutrons and photons are also described. The analytical model described in this paper accounts for absorption and detection, thus extending the model presented in previous studies. By using this new, improved model, one can investigate the relative feasibilities of measuring neutrons, gamma photons or combinations thereof, for the analysis of a specific fissile sample. It is seen that for larger mass samples photon absorption in the sample strongly decreases the multiplicity of emitted photons, whereas this is not the case for neutrons. The model correctly accounts for this strong self-shielding effect and confirm previous observations that although photons have a larger initial (source) multiplication, neutrons might be more favourable to measure in the case of large samples because of the increasing self-shielding effect for gamma photons.

Invariant imbedding theory is an alternative formulation of particle transport theory. Although stochastic foundations of invariant imbedding have been known from the beginnings, the method itself has so far exclusively been used for calculating first moments, i.e. expectations. The present paper attempts to set up a probability balance equation in the invariant imbedding approach from which equations for the first and second order densities are derived. It is shown that only the equations for the first order densities are non-linear, while subsequent order densities obey linear equations. This is expected to considerably simplify solution to those problems which involve second order density calculations where invariant imbedding techniques may be profitably used. Examples of such quantities are the variance or correlations between particles detected at two different energies or angles or the higher moments of the emitted multiplicity distribution such as the variance from a target bombarded by incident particles. One possible area of application of our equations is non-destructive estimation of fissile material by the active neutron assay technique. Another area is the study of particle cascade development in sputtering and positron backscattering from surfaces. The approach is illustrated by a simple forward–backward scattering model for these two problems.

The purpose of this note is to point out notational inconsistencies and actual errors in some of the basic and often cited expressions of Ensslin et al. [Application Guide to Neutron Multiplicity Counting, Los Alamos Report LA-13422-M, 1998], even if the final formulae are correct. These expressions describe the measured multiplicity counting rates, i.e. singles, doubles, and triples rates S, D, and T, as parameters of a multiplying sample, through analytical expressions of the factorial moments of the neutrons generated in the sample by one initial source event. They serve as the basis of unfolding these parameters by the inversion of the formulae. The motivation for this brief communication is twofold. First, Ref. [N. Ensslin,W.C. Harker, M.S. Krick, D.G. Langner, M.M. Pickrell, J.E. Stewart, Application guide to neutron multiplicity counting, Los Alamos Report LA-13422-M, 1998] is the most widely available and by far the most cited source of the derivation, which is supposedly also used as a tutorial for newcomers. Hence it is important that the correct derivation is available. Indeed it can be seen that the erroneous expressions have already propagated widely in citations. Second, similar derivations have become recently interesting for the multiplicities of gamma photons. Hence we want to create both a citable reference as well as guidance for such on-going work in further applications of the theory.

The concept and calculation techniques of multiplicities in nuclear safeguards are applied to the calculation of the traditional fast fission factor of reactor physics. The concept is the assumption that the original source neutrons from spontaneous or induced fission, and the further neutrons given rise through fast fission in the sample before leakage, are considered as being generated simultaneously with the source neutrons. The number distribution of the neutrons arising from such a "superfission" process will be different from that of the nuclear fission process. Concerning the mathematical treatment, in safeguards works the master equation approach is used to calculate the moments of such a distribution. Hence, to follow suite, a derivation of the fast fission factor is given here by a backward master equation. This method has the advantages that the derivation of the fast fission factor becomes more transparent than the traditional one, and that it allows also to determine higher order moments, notably the variance, of the total number of neutrons generated, i.e. when account is taken also of the contribution of fast fission to these moments. The results show that the relative standard deviation increases fast with the increase of the non-leakage probability of neutrons, and hence with the increase of the fast fission factor itself. Also the Diven factor of the superfission process (neutrons from fast fissions included) is significantly larger than that of thermal fission. We argue that the traditional model, in which the Feynman- and Rossi-alpha models are derived, does not account correctly for the extra branching represented by the fast fission process. Hence the Diven factor traditionally used in those formulae should be used in a modified form. We show how the effect of fast fission needs to be included into the model to obtain the correct formula, and give explicit expressions. Some quantitative examples are given for illustration.

The temporal evolution of the distribution of the number of neutrons in a time-varying multiplying system, producing only prompt neutrons, was treated recently with the master equation technique by some of the present authors. Such a treatment gives account of both the so-called zero power reactor noise and the power reactor noise simultaneously. In particular, the first two moments of the neutron number, as well as the concept of criticality for time-varying systems, were investigated and discussed. The present paper extends these investigations to the case when delayed neutrons are also taken into account. Due to the complexity of the description, only the expectation of the neutron number is calculated. The concept of criticality of a time-varying system is also generalized to systems with delayed neutrons. The temporal behaviour of the expectation of the number of neutrons and its asymptotic properties are displayed and discussed.

This paper presents a new technique for the analysis of neutron pulse height distributions generated in an organic scintillation detector. The methodology presented can be applied to techniques such as neutron spectrum unfolding, which have a variety of applications, including nuclear nonproliferation and homeland security. The technique is based on two independent approaches: (i) the use of the MCNP-PoliMi code to simulate neutron detection on an event-by-event basis with the Monte Carlo method and (ii) an analytical approach for neutron slowing down and detection processes. We show that the total neutron pulse height response measured by the organic scintillators is given by the sum of a large number of different neutron histories, each composed of a certain number of neutron scatterings on hydrogen and/or carbon. The relative contributions of each of these histories are described for a cylindrical liquid scintillator BC-501A. Simulations and measurements of neutron pulse height distributions are essential for neutron spectrum unfolding procedures. (c) 2007 Elsevier B.V. All rights reserved.

The zero-power reactor noise theory in a steady neutron multiplying medium was extended recently to a medium randomly varying in time to bridge the fields of the zero-power and the power reactor noise. For a time-varying medium in which the transition probability randomly fluctuates, only the use of the probability generating function technique based on the forward master equation approach is practical. However, with the forward master equation approach, the treatment of the joint moments of the neutron number and the medium state leads to a closure problem. Recently, the capability of the moment calculation technique for such cases was extended such that the closure problem could be solved exactly. The present paper describes and demonstrates this closure-free moment calculation technique in a time-varying binary multiplying medium, in which the medium state has two possible realizations. In addition to the first two moments of the neutron number N alone (irrespective of the medium state η), the joint moments of Nn and ηm, i.e., 〈Nnηm〉, were also obtained in a compact form for n = 1, 2 and arbitrary values of m, without a closure assumption. It was found that, for even m values, the asymptotic values of Nn and ηm are uncorrelated, whereas, for odd m values, they are negatively correlated, namely, their covariance is less than zero. The first two moments of the neutron number theoretically obtained were verified by the Monte Carlo technique. A perfect agreement was found between the Monte Carlo and the theoretical solutions. The closure-free moment calculation technique demonstrated in the present paper is expected to be applicable to various other problems related to the birth-and-death process with fluctuations of the transition probability, in which a closure problem occurs.

In a recent paper, we presented a simple analytical model to describe the statistics of the number of scattering collisions undergone by fast neutrons as they slow down until they are absorbed. In that study, we assumed that the moderator was infinite and homogeneous, and accounted for scattering and absorption by a single nuclear species. In the present paper, we extend that methodology to the more realistic case of neutron slowing down in a homogeneous mixture. The formulae are derived and evaluated numerically, and the results are found to be in very good agreement with corresponding Monte Carlo simulations. The average value of the number of collisions that a neutron undergoes before being captured is computed. The results for a capture-gated detector composed of hydrogen, carbon, and boron are discussed.

The subject of this paper is an analytical derivation of the full probability distribution of the number of neutrons and photons generated in a sample with internal multiplication by one source emission event (spontaneous fission), and its comparison with Monte Carlo calculations. We derive recursive analytic expressions for the probability distributions P(n) up to values of n=N for which the cumulative probability Click to view the MathML source is equal to or larger than 0.99. The derivation is achieved using the symbolic computation language Mathematica. With the introduction of a modified factorial moment of the number of neutrons and gamma photons generated in fission, the resulting expressions could be brought to a formally equivalent form with those for the factorial moments of the total number of neutrons and photons generated in the sample. The results were compared to Monte Carlo calculations, and excellent agreement was found between the analytical results and the simulations. The results show that the probability distributions change with increasing sample mass in such a way that the ?bulk? of the distribution changes only slightly, but a tail develops for higher n values. This tail is the main reason for the increase of the factorial moments with increasing sample mass, an effect that was observed in earlier studies.

The master equation approach, which has traditionally been used for the calculation of neutron fluctuations in multiplying systems with constant parameters, is extended to a case when the parameters of the system change randomly in time. A forward type master equation is considered for the case of a multiplying system whose properties jump randomly between two discrete states, both with and without a stationary external source. The first two factorial moments are calculated, including the covariance. This model can be considered as the unification of stochastic methods that were used either in a constant multiplying medium via the master equation technique, or in a fluctuating medium via the Langevin technique. The results obtained show a much richer characteristic of the zero power noise than that in constant systems. The results are relevant in medium power subcritical nuclear systems where the zero power noise is still significant, but they also have a bearing on all types of branching processes, such as evolution of biological systems, spreading of epidemics etc, which are set in a time-varying environment.

We propose a simple analytical model to describe the statistics of the number of collisions undergone by fast neutrons during slowing down until they are absorbed. We assume that the moderator is homogeneous and account for scattering and absorption, but do not consider thermalization. Although the problem cannot be solved in a compact form, a simple recurrent formula provides the solution in a very transparent way. The model can be readily evaluated numerically, and the results are in excellent agreement with the corresponding Monte Carlo simulations. Both the mean number and the variance of the number of collisions are calculated. The results are discussed and compared with the classical case of neutron slowing down to or past a given energy in a moderating medium without absorption.

The multiplicities, or factorial moments, of the distribution of the number of neutrons emerging from a fissile sample can be used to identify and quantify fissile isotopes, in particular even-N isotopes of transuranic elements. In fact, the spontaneously emitted source neutrons can induce further fissions in the sample, thereby changing the number distributions of the neutrons leaving the sample, and therefore their multiplicities. The multiplicities increase monotonically with sample mass, hence the measurement of the multiplicities can be used to quantify the sample mass. Analytical expressions for multiplicities that include induced fission effects have been derived for neutrons in the past. These expressions are given as functions of the probability of induced fission per neutron, and have been investigated both by Monte Carlo methods and in experiments using thermal neutron detectors. The object of this paper is to derive analytical formulae for the multiplicities of the gamma photons emitted by both spontaneous and induced fissions, and to perform a quantitative analysis. In addition, neutron and gamma multiplicities are calculated by Monte Carlo simulation using a modified version of the MCNP-PoliMi code. Good agreement is found between the analytical formulae and the Monte Carlo results. The results show the potential advantage of using gamma multiplicities when compared to neutron multiplicities: their higher quantitative values may, in principle, have the effect of leading to a larger sensitivity on the sample mass when compared to the analysis based on neutrons alone.

A class of semi-linear evolution equations arising in neutron fluctuation theory was studied. The local and global existence and uniqueness of the solution were proved.

The purpose of this paper is to develop a generalized formalism with which both the inherent neutron fluctuations in a steady system (the so–called zero power noise) and the neutron fluctuations induced by the fluctuation of the host material (power reactor noise or structural noise) can be treated simultaneously. To achieve this goal, a generalized form of the Rossi–alpha formula, i.e. the covariance function of neutron numbers, or detector counts, with a time delay τ is derived analytically for a subcritical system while the medium is randomly fluctuating between two states. The formalism used is backward–type master equations for the neutron probability distribution. The Rossi–alpha formula is suitable for calculating the auto–power spectral density of the neutron noise, and thus to make comparisons with power reactor noise calculations. The results obtained in this paper contain both the inherent fluctuations and the power reactor noise simultaneously, and reproduce either of these two separately in the corresponding limits.

Two versions of the neutron-gamma variance to mean (Feynman-alpha method or Feynman-Y function) formula for either gamma detection only or total neutron-gamma detection, respectively, are derived and compared in this paper. The new formulas have a particular importance for detectors of either gamma photons or detectors sensitive to both neutron and gamma radiation. If applied to a plastic or liquid scintillation detector, the total neutron-gamma detection Feynman-Y expression corresponds to a situation where no discrimination is made between neutrons and gamma particles. The gamma variance to mean formulas are useful when a detector of only gamma radiation is used or when working with a combined neutron-gamma detector at high count rates. The theoretical derivation is based on the Chapman-Kolmogorov equation with inclusion of general reactions and passage intensities for neutrons and gammas, but with the inclusion of prompt reactions only. A one energy group approximation is considered. The comparison of the two different theories is made by using reaction intensities obtained in MCNPX simulations with a simplified geometry for two scintillation detectors and a 252Cf-source enclosed in a steel container. In addition, the variance to mean ratios, neutron, gamma and total neutron-gamma, are evaluated experimentally for a weak 252Cf neutron-gamma source in a steel container, a 137Cs random gamma source and a 22Na correlated gamma source. Due to the focus being on the possibility of using neutron-gamma variance to mean theories for both reactor and safeguards applications, we limited the present study to the general analytical expressions for Feynman-Y formulas.

Classical particle transport, such as the diffusion and slowing down of neutrons in a multiplying system, is a random process. As a result, the number of neutrons in the system, or the number of detector counts, is a random number. Because of the branching property of the cascade, i.e. producing two or more neutrons in the fission process, correlations will exist between the neutrons in the system. This will lead to non-trivial statistics in the cascade, which, among others, carries information on the dynamical properties of the system. Neutron fluctuations have thus been used for a long time to measure either nuclear physics parameters such as the effective delayed neutron fraction, or the so-called multiplication constant, which describes the criticality properties of the system. In addition to the above described inherent random property of the particle transport in steady-state systems, another possibility of neutron fluctuations arises in systems whose properties fluctuate in time. This type of noise is observed in systems with high power. The reason for neutron fluctuations, that is neutron noise, is the fluctuations of the reactor material. There is boiling of the coolant, vibration of various components such as control rods, the core barrel etc., which all will lead to parametric excitation of the neutron noise. In this paper we shall however mostly deal with the neutron noise in low power systems, i.e. with inherent fluctuations of the neutron transport process, and the case of power reactor noise will only be touched upon. The neutron noise in low power systems has received a renewed strong interest recently, due to the appearance of a new concept, the so-called accelerator driven systems (ADS). An ADS is a subcritical reactor, driven by a strong external source, which utilizes spallation. The advantages of such a system is that it has excellent operational safety properties (a criticality accident is practically impossible), can utilize fertile nuclides as fuel such as thorium or U-238 easier than traditional reactors (hereby having access to more abundant fuel reserves as the current reactors), and finally, on the long run and with proper design, it would produce less high-level nuclear waste. As a matter of fact, an ADS can incinerate more waste than it produces, so it can also be used primarily for transmutation of nuclear waste with energy production as a by-product. In an ADS, however, the statistical properties of the neutron source are different from those of the classical radioactive sources that were used in the development of the theory in the past. A radioactive source emits one neutron at a time, and thus all source neutrons are independent (they obey Poisson statistics). In a spallation source, on the other hand, a large number of neutrons are produced in each spallation event, and the number of neutrons generated is a random number. Thus the source neutrons are correlated. In addition to time correlations, energy correlations may exist between spallation neutrons even if the neutron energies in a single spallation even are independent. This fact will be illustrated i the paper. All of the above mentioned (i.e. time and energy) correlations will influence the statistics of the neutrons in an ADS and thus they need to be accounted for.

Petrophysics, when applied to the fields of petroleum and groundwater resources, has the special connotation of fluid-rock interaction. Neutron radiography imaging techniques have been applied in the present study to evaluate fluid-rock behaviour for three specific experiments. The basic experimental set-up entails a beam of thermal neutrons intersecting a thin slice (5 mm thick) of fluid-filled porous rock, which is encased in aluminium to retain the fluid within the rock. Neutrons are attenuated by the fluid in the rock, and comparison of the neutron intensity in a dry and saturated state permits one to estimate the rock's porosity (ratio of pore volume to total volume) and relative fluid saturation, The three experiments are (I) Water inflitration into a porous rock, Displacement of heavy water by oil and 3 Water flooding of a clay-rich rock. Each experiment is analogous to a practical geological resource problem.

It was suggested recently [Pazsit, Ann. Nucl. Energy 24 (1997) 1257] that in addition to the traditionally measured scalar neutron flux, the information contained in the higher angular moments of the neutron flux, such as the neutron current or the flux gradient, could be used for monitoring and diagnostics of reactor cores. To explore this possibility, a detector was constructed with which vector components of the partial currents, and thus also components of the net current vector, can be measured. The two-dimensional current vector was measured in an experimental system containing a neutron source in a water tank, and was compared with calculated values. As one simple application, it was shown how the position of the source can be determined from measuring the scalar flux and the neutron current in one single spatial point. The experiments demonstrate both the suitability of the detector construction for the measurement of the neutron current vector and the use of the current in diagnostics and monitoring.

Studies of neutron fluctuations in spallation-driven subcritical systems require the use of energy-dependent master equations. In particular, calculation of the second moment of the neutron distribution requires knowledge on the energy correlations (two-point distributions) of the source particles. It is shown here that such correlations will exist even if the energies of all neutrons, generated in any single spallation event, are independent, provided that the energy distribution of the neutrons for separate spallation events is dependent on the number of neutrons generated. A simple model of number dependence of the energy spectrum is constructed, and the arising energy correlations are calculated. The error in calculating the second moment of the neutron distribution, arising when assuming zero correlations (i.e. using only one-particle energy spectra), is estimated in a simple model of neutron slowing down.

It has been proposed that the fluctuations of the neutron current, called the current noise, can be used in addition to the scalar noise in reactor diagnostic problems. The possibility of the localization of a vibrating control rod pin in a pressurized water reactor control assembly is investigated by using the scalar neutron noise and the two-dimensional radial current noise as measured at one central point in the assembly. An explicit localization technique is elaborated in which the searched position is determined as the absolute minimum of a minimization function. The technique is investigated in numerical simulations. The results of the simulation tests show the potential applicability of the method.

The Feynman- and Rossi-alpha formulas are calculated for subcritical systems driven by a multiple emission source, i.e., one that emits several neutrons on each source emission event. The prime exa...

The Feynman-alpha and Rossi-alpha formulae are derived analytically for subcritical systems driven by a multiple emission source,i.e. one that emits several neutrons on each source emission event. The prime example of such sources is a spallation source, which will be used in future accelerator driven subcritical systems (ADS) such as the energy amplifier. Such formulae for ADS have been derived recently but in simpler neutronic models. In this paper six groups of delayed neutron precursors are taken into account, and the full joint statistics of the prompt and all delayed groups is included. Thus the present results are generalisations of earlier ones. The involved problems that are encountered are solved with a combination of effective analytical techniques and symbolic algebra codes.

The spatial dependence of thermal and fast neutron flux was measured axially in the core of a 1 MW research reactor. The measurements were made by a thin optical fibre detector with a neutron sensitive ZnS(Ag) scintillation tip. For thermal neutrons 6 Li was used, whereas for fast neutrons 232 Th was used as neutron converter. The spatial dependence was measured by moving the fibre axially with a uniform speed. The measurement takes a few minutes, compared to up to 10 h with the conventional wire activation method. Comparison with traditional measurements shows a good agreement.

The flux of 14.1 MeV neutrons was measured with high spatial resolution in the vicinity of the target of a D–T neutron generator. The measurements were made by a thin optical fibre detector with a ZnS(Ag) scintillation tip mixed with a 232 Th neutron converter. The detector concept was originally developed at Nagoya University. The flux of 14.1 MeV neutrons could be measured with a spatial resolution of about 1 mm. The position of the impact area of the deuterium beam on the target surface was determined by the method as an application. A simple model for the calculation of the space dependence of the flux was developed, and it was shown to agree very well with the measurements. With the help of the model, both the vertical and horizontal position of the beam impact area centre can be determined from one single measurement through parameter fitting.

Dynamic neutron radiography is a method to image fluid flow in porous media, based on the tendency of neutrons to be preferentially attenuated by hydrogen. Fast neutrons with energies of several MeV can be produced by particle accelerators, and are suitable for detecting fluids in rock samples of 0.15 to 0.30 m (15 to 30 cm) thick. Slow (thermal) neutrons with energies of about 0.03 eV are produced by nuclear reactors, and are ideal for imaging fluid flow in rock samples in the order of 0.01 m (1 cm) thick. Thermal neutrons are attenuated by an exponential law, which relates the incident intensity of a neutron beam (Io) to the transmitted neutron beam (I): I = Io exp(-rmh), where r is the bulk density of the rock in the neutron path, m is the bulk neutron attenuation coefficient in the neutron path, and h is the thickness of the rock in the neutron path. The exponent term (rmh) can be linearly related to the components making up the rock: rmh = (r1m1V1 + r2m2V2 + rmmmVm)h, where V1 is the volume fraction of fluid 1, V2 is the volume fraction of fluid 2, Vm is the volume fraction of the rock matrix, the subscript "m" refers to the rock matrix, and "1" and "2"refer to fluid 1 and fluid 2, respectively. Knowing that V1 + V2 = f, which is the rock's porosity, an estimate of the relative fluid saturations can be made. These are the principal economic quantities required for either a hydrogeological or petroleum reservoir. Examples of water infiltration and petroleum imbibition into the Visingso Sandstone, Sweden, are provided to illustrate the application of the technique.

Neutron fluctuations are investigated in a subcritical system driven by a non-Poisson (correlated) source. Such sources arise in e.g. accelerator-driven systems (ADS) where several neutrons are born simultaneously with a certain fluctuation in the spallation process. The second moment of the neutron number and of the detector counts is calculated. In particular, the Feynman-alpha formula is derived both by a master equation technique and with a heuristic approach. It is shown that the dependence of the variance-to-mean of the counts on the time gate is the same as in the traditional case (Poisson source), but its magnitude is enhanced by the source fluctuations and multiplicity. Thus, the presence of a correlated source is beneficial for the application of the Feynman technique. Consequently, it may be possible to perform measurements with deeper subcriticalities than in the traditional case.

It is suggested that the information contained in the angular dependence of the flux, at least in the lower order moments such as the current, partial current or just the flux gradient, be used in core monitoring and diagnostics. The motivation for this proposal arises partly from the fact that with today's technology, angularly sensitive detectors, i.e. current or gradient detectors, suitable for industrial applications, can be constructed. The potentials of the use of the current (gradient), and the enhancement it may give to monitoring and diagnostic applications, is demonstrated through some simple model cases. It is shown that in spatial (noise) source reconstruction, i.e. locating both static and dynamic sources such as anomalies, discontinuities etc., the enhancement resulting from the use of the current is substantial.

A short letter to the Editor on the solutions of a static one-group diffusion equation in treating the criticality problem for a homogeneous reactor

A Feynman-alpha formula has been derived in a two region domain pertaining the stochastic differential self-interrogation (DDSI) method and the differential die-away method (DDAA). Monte Carlo simulations have been used to assess the applicability of the variance to mean through determination of the physical reaction intensities of the physical processes in the two domains. More specifically, the branching processes of the neutrons in the two regions are described by the Chapman - Kolmogorov equation, including all reaction intensities for the various processes, that is used to derive a variance to mean relation for the process. The applicability of the Feynman-alpha or variance to mean formulae are assessed in DDSI and DDAA of spent fuel configurations., 15 pages, 5 figures. Submitted to EPJ Plus

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.

Traditional theories of neutron fluctuations in subcritical stationary multiplying systems with analytical solutions disregard the energy dependence of the neutrons, i.e. they treat only one type of neutrons in the branching process. In several recent applications, such as certain coincidence measurements in nuclear safeguards and Accelerator-Driven reflected fast-reactor Systems (ADS), energy dependence of the neutrons must be included into the description. The simplest of such models, still allowing analytical solutions, is to distinguish between two types of neutrons, fast and thermal, corresponding to the two-group theory of traditional reactor physics. The purpose of the present paper is to extend the traditional theory of neutron fluctuations to two neutron types, fast and thermal. We will consider the special case when branching is only induced by one of the two types, giving rise to the progeny of the other type only (thermal fission generating fast neutrons). Stationary injection of neutrons by an extraneous source, as well as the detection process, are taken into account. The results obtained have applications in current problems of nuclear safeguards and reactivity measurements in ADS.