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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

The possibility of measuring the gas-phase velocity in a two-phase mixture through the use of neutron noise techniques is demonstrated in the zero-power reactor CROCUS of the Ecole Polytechnique Fe...

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.

The conventional time-domain reactor noise techniques analyse the number of neutron detector pulse signals, so that they sometimes encounter serious difficulties owing to the count-loss effect due to the dead time of detector systems. To avoid all the difficulties coming from the count-loss effect, a novel time-domain technique was recently proposed (Pal and Pazsit, 2015). This technique analyses the auto-covariance function of continuous current signals arising from ionization chambers such as the fission chamber, so that it is inherently insensitive to the count-loss effect. In the present study, a different time-domain technique that analyses the integral values of current signals is proposed. With regard to these two techniques, the experimental conditions under which they successfully measure the subcriticality through determination of the prompt neutron decay constant are clarified.

Recently a joint project has been carried out between the Paul Scherrer Institut, the Ecole Polytechnique Federale de Lausanne and swissnuclear, an industrial partner, in order to determine the axial void distribution in a channel installed in the reflector of the zero power research reactor CROCUS, using neutron noise techniques. The main objective of the present paper is to report on the validation of the results against an alternative measurement technique using gamma-ray attenuation and simulations with the TRACE code. For the gamma-ray attenuation experiments, the channel used in CROCUS is installed out of the core in a Plexiglass water tank. The source and detector are fixed and the channel is moved axially to keep the geometry of the source/detector arrangement untouched. This is key to measure the void effect by gamma attenuation due to the low contrast of this technique. The paper compares the experimental results obtained with both techniques, with the outcomes of simulations carried out with the TRACE code. Even though the quantitative void fraction estimations are not consistent, the trends obtained with the simulation and experimental techniques are the same. The discrepancies between the various experimental techniques and the simulation outcomes are related to the heterogeneous distribution of the water-air mixture in the radial sections of the channel.

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.

The method of Heun's differential equation is demonstrated in studying a fractional linear birth-death process (FLBDP) with long memory described by a master equation. The exact analytic solution of the generating function for the probability density is obtained on the basis of Heun's differential equation. The multi-fractal nature of FLBDP associated with long memory is demonstrated in conjunction with the present simple birth death process. Finally, the subtle multi-fractal nature of critical fluctuations under long memory is also displayed in the present FLBDP. Further, discussions are also given on the features of transient fluctuation in systems with long memory.

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).

The concept of eigenvalue separation (ES) was introduced in the past for the characterisation of the space-time kinetics of reactor transients, and the stability properties of large loosely coupled cores. However, most of the investigations reported so far concern the determination of the ES itself either from static calculations, or from measurements of the flux tilt or neutron noise cross-correlations. Conclusions on system behaviour were only drawn from the properties of the static eigenfunctions, comparing non-perturbed and perturbed systems, without explicitly solving the time- or frequency-dependent problem. In this paper, we explore the role of the ES on the neutronic response of a critical core to small stochastic perturbations (neutron noise); in particular, the spatial and frequency characteristics of the arising neutron noise as a function of the ES, as well as the spatial structure of the perturbation. It is shown that for systems with small ES and non-uniform perturbations, point kine...

Determination of the local void fraction in BWRs from in-core neutron noise measurements requires the knowledge of the axial velocity of the void. The purpose of this paper is to revisit the problem of determining the axial void velocity profile from the transit times of the void between axially placed detectors, determined from in-core neutron noise measurements. In order to determine a realistic velocity profile which shows an inflection point and hence has to be at least a third order polynomial, one needs four transit times and hence five in-core detectors at various axial elevations, whereas the standard instrumentation usually consists only of four in-core detectors. Attempts to determine a fourth transit time by adding a TIP detector to the existing four LPRMs and cross-correlate it with any of the LPRMs have been unsuccessful so far. In this paper we thus propose another approach, where the TIP detector is only used for the determination of the axial position of the onset of boiling. By this approach it is sufficient to use only three transit times. Moreover, with another parametrisation of the velocity profile, it is possible to reconstruct the velocity profile even without knowing the onset point of boiling, in which case the TIP is not needed, although at the expense of a less flexible modelling of the velocity profile. In the paper the principles are presented, and the strategy is demonstrated by concrete examples, with a comparison of the performance of the two different ways of modelling the velocity profile. The method is tested also on velocity profiles supplied by system codes, as well as on transit times from neutron noise measurements.

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.

The purpose of this work is to elaborate a master equation formalism for the evolution of the probability distribution of the cumulative energy generated by fissions in a multiplying system with delayed neutrons. The formalism accounts for the fact that the fission energy chi is also a random variable, thus the fluctuations of the total energy generated are due to both the fluctuations of the number of fissions, as well as to the fluctuations of the energy per fission. By comparing to the case where the fission energy is taken as constant, the significance of the fluctuations of the fission energy can be assessed. The first two moments of the cumulative fission energy are determined explicitly, and the time dependence of the expectation and the variance is calculated for different reactivities. As expected, the variance of the energy per fission does not play a significant role in the variance of the cumulative fission energy.

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 problem of determining the axial velocity profile from the in-core neutron noise measurements is revisited, with the purpose of developing an objective method for the determination of the void fraction. Until now it was assumed that in order to determine a realistic velocity profile which shows an inflection point and hence has to be at least a third order polynomial, one needs four transit times and hence five in-core detectors at various axial elevations. However, attempts to determine a fourth transit time by adding a TIP detector to the existing four LPRMs and cross-correlate it with any of the LPRMs were unsuccessful so far. In this paper we thus propose another approach, where the TIP detector is only used for the determination of the axial position of the onset of boiling. By this approach it is sufficient to use only three transit times. Moreover, with another parametrisation of the velocity profile, it is possible to reconstruct the velocity profile even without knowing the onset point of boiling, in which case the TIP is not needed. In the paper the principles are explained and the strategy is demonstrated by concrete examples.

International audience; This paper investigates, through numerical simulations, the performance of a signal analysis method by which a high temperature fission chamber can be used over a wide range of count rates. Results reported in a previous paper (Elter et al., 2015 [I]) indicated that the traditional Campbell method and the pulse mode cannot provide a sufficient overlap at medium count rates. Hence the use of the so-called Higher Order Campbell (HOC) methods is proposed and their performance is investigated. It is shown that the HOC methods can guarantee the linearity (i.e. correctness) of the neutron flux estimation over a wide count rate, even during transient conditions. The capabilities of these methods for suppressing parasitic noise (originating from various sources) are verified.

The neutron noise induced by propagating perturbations in a simple model of a Molten Salt Reactor (MSR) is calculated and analyzed using one/two-group diffusion theory. The novelty, as compared to previous works, is that the noise source includes also the fluctuations of the fission cross sections and the fluid velocity, in addition to the previous case when only the fluctuations of the absorption cross section were accounted for. Another novelty is that the solution is obtained through the matrix Green's function of the flux and precursor equations, these two being kept separate. Inclusion of each of these two new noise sources leads to a structure of the noise source, and hence also that of the neutron noise, which is conceptually different from the case when only the fluctuations of the absorption cross sections are treated, with some surprising features. The use of the matrix Green's function is advantageous to understand the new features, and it helps to point out some new aspects of the neutron noise even in traditional systems, which have not been noticed before. The results contribute to the understanding and interpretation of the neutron noise in MSRs. (C) 2015 Elsevier Ltd. All rights reserved.

The derivation of the point-kinetic component of the neutron noise in two-group diffusion theory in molten salt reactors (MSRs), based on different techniques, is discussed. First, the point-kinetic component is calculated by projecting the corresponding full space-frequency-dependent solution onto the static adjoint. Then, following the standard procedure in reactor physics, the point-kinetic solution is determined by solving the linearized point-kinetic equations. Both results are thereafter analyzed and compared quantitatively. Such a comparison clearly indicates that the solution obtained by the conventional derivation, i.e., from the point-kinetic equations, significantly differs from the exact one and is not able to reproduce certain features of the latter. Similar discrepancies between the two methods were also pointed out and confirmed earlier in one-group MSR calculations.

Core-barrel motion (CBM) surveillance and diagnostics, based on the amplitude of the peaks of the normalized auto power spectral densities (APSDs) of the ex-core neutron detectors, have been performed and continuously developed in Sweden and were applied for monitoring of the three PWR units, Ringhals 2 to 4. From 2005, multiple measurements were taken during each fuel cycle, and these revealed a periodic behavior of the 8-Hz peak of the beam-mode motion: the amplitude increases within the cycle and returns to a lower value at the beginning of the next cycle. The work reported in this paper aims to clarify the physical reason for this behavior. A combination of a mode separation method in the time domain and a nonlinear curve fitting procedure of the frequency spectra revealed that two types of vibration phenomena contribute to the beam-mode peak. The lower frequency peak around 7 Hz in the ex-core detector APSDs corresponds to the CBM, whose amplitude does not change during the cycle. The higher frequency peak around 8 Hz arises from the individual vibrations of the fuel assemblies, and its amplitude increases monotonically during the cycle. This paper gives an account of the work that has been made to veri,b, the above hypothesis.

The concept of the dynamic adjoint was introduced by Hugo van Dam for calculating the in-core neutron noise in boiling water reactors in the mid-70's. This successful approach found numerous applications for calculating the neutron noise in both PWRs and BWRs since then. Although the advantages and disadvantages of using the direct (forward) or the adjoint (backward) approach for the calculation of the neutron noise were analysed in a number of publications, the direct relationship between the forward Green's function and the dynamic adjoint has not been discussed. On the other hand, in particle transport theory the relationship between the direct and adjoint Green's function has been discussed in detail, in which Mike Williams has had many seminal contributions. In this note we analyse the relationship between the direct Green's function and the dynamic adjoint in the spirit of Mike's work in neutron transport and radiation damage theory. The paper is closed with some personal remarks and reminiscences.

It is known that the stability of boiling water reactors (BWRs), when quantified with the so-called decay ratio as the stability parameter, may show seemingly abrupt changes, despite the smooth variations of the control parameters (reactor power and core flow). There is also evidence of the fact that the stability properties can exhibit a hysteresis effect when moving back and forth on the same path on the power-flow map. The most common explanation of the abrupt change is based on the co-existence of two different types of instabilities (global and regional), and their interplay (van der Hagen et al., 1994; Pazsit, 1995). In this paper we suggest an alternative phenomenological explanation, which only assumes the existence of one single mode of instability. We propose the hypothesis that the decay ratio of one single mode of a complex, many-variable non-linear system might obey a cusp catastrophe as a function of the control parameters. Such a phenomenological model can explain both the discontinuous variation of the decay ratio, as well as the hysteresis effect. Since a cusp-type behaviour implies that the decay ratio is many-valued in a certain region of the power-flow map, a mechanism is suggested how a Hopf bifurcation with multiplicative noise can lead to such a behaviour.

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.

The effect of cycle burnup on the ex-core detector noise at the frequency of the pendular core barrel vibrations in the Ringhals-3 PWR core was investigated using a neutron noise simulator. The purpose of the investigations was to confirm or disprove a hypothesis raised by Sweeney et al. (1985) that fuel assembly vibrations could affect the ex-core detector noise and cause the corresponding peak in the auto power spectral density (APSD) to increase during the cycle due to the effects of fuel burnup, the change of boron concentration, flux redistribution etc. Numerical calculations were performed by modelling the vibrations of fuel assemblies at different locations in the core and calculating the induced neutron noise at three burnup steps. The APSD of the ex-core detector noise was evaluated with the assumption of vibrations either along a straight-line or along a random two-dimensional trajectory, with two different representations of the cross section perturbations caused by the vibrations. The results show the obvious effect of in-core fuel vibrations on the ex-core detector noise, but the monotonic increase of the APSD does not occur for all fuel elements, vibration types and cross section perturbation models. Such an increase of the of APSD occurs predominantly for peripheral assemblies with one of the perturbation models. However, assuming simultaneous vibrations of a number of fuel assemblies uniformly distributed over the core with random vibrations and the more realistic perturbation model, the effect of the peripheral assemblies will dominate and hence the surmised monotonic increase of the amplitude of the ex-core neutron noise during the cycle can be confirmed.

The detectors of the neutron flux monitoring system of the foreseen French GEN-IV sodium-cooled fast reactor (SFR) will be high temperature fission chambers placed in the reactor vessel in the vicinity of the core. The operation of a fission chamber over a wide-range neutron flux will be feasible provided that the overlap of the applicability of its pulse and Campbelling operational modes is ensured. This paper addresses the question of the linearity of these two modes and it also presents our recent efforts to develop a specific code for the simulation of fission chamber pulse trains. Our developed simulation code is described and its overall verification is shown. An extensive quantitative investigation was performed to explore the applicability limits of these two standard modes. It was found that for short pulses the overlap between the pulse and Campbelling modes can be guaranteed if the standard deviation of the background noise is not higher than 5% of the pulse amplitude. It was also shown that the Campbelling mode is sensitive to parasitic noise, while the performance of the pulse mode is affected by the stochastic amplitude distributions.

The calculation of the time-dependent extinction probability (EP) in an infinite medium of fissile material is calculated including delayed neutrons. This extension of the model reveals some interesting consequences, one of which is the appearance of an initial plateau in the EP in the range 10(-6) to 10(-4) s which is significantly less than the final asymptotic plateau value reached as time tends to infinity; i.e. the steady state EP. This steady state is delayed by the presence of the delayed neutrons by a time of the order of 3/lambda (similar to 36 s) where lambda is the mean decay constant of the precursors. The presence of the initial plateau is due specifically to delayed neutrons and has not been commented on before. Several other situations are considered, with differing initial conditions, and are illustrated by numerical results.

This paper reports on theoretical investigations of the stochastic properties of the signal series of ionisation chambers, in particular fission chambers. First, a simple and transparent derivation is given of the higher order moments of the random detector signal for incoming pulses with a non homogeneous Poisson distribution and random pulse heights and arbitrary shape. Exact relationships are derived for the higher order moments of the detector signal, which constitute a generalisation of the so-called higher order Campbelling techniques. The probability distribution of the number of time points when the signal exceeds a certain level is also derived. Then, a few simple pulse shapes and amplitude distributions are selected as idealised models of the detector signals. Assuming that the incoming particles form a homogeneous Poisson process, explicit expressions are given for the higher order moments of the signal and the number of level crossings in a given time interval for the selected pulse shapes.

The signals of fission chambers are traditionally evaluated with Campbelling methods at medium count rates. Lately there has been a growing interest in the extension of Campbelling methods to cover a wider range of count rates. These methods are applied to measure the neutron flux in the stationary state of the reactor. However, there has not been any attempt to generalize these techniques for transient reactor states. This short note is devoted to a discussion of this question. It is shown through analytic and numerical calculations that for practical reasons the traditional, stationary Campbelling methods can be applied for transient scenarios as well. (C) 2016 Elsevier B.V. All rights reserved.

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 subject of this paper is the calculation of the neutron noise induced by small-amplitude stationary radial variations of the core size (core expansion/compaction, also called core flowering) of a large sodium-cooled fast reactor. The calculations were performed on a realistic model of the European Sodium Fast Reactor (ESFR) core with a thermal output of 3600 MW(thermal), using a multigroup neutron noise simulator. The multigroup cross sections and their fluctuations that represent the core geometry changes for the neutron noise calculations were generated by the code ERANOS. The space and energy dependences of the noise source represented by the core expansion/compaction and the induced neutron noise are calculated and discussed.