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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).

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.

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.

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 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 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.

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 one-group diffusion theory of molten salt reactors in a homogeneous reactor model is revisited. First, the integral terms in the equation for the flux, obtained after the elimination of the delayed neutron precursors, are given a physical interpretation. This gives an understanding of the physical meaning of the concept of infinite fuel recirculation velocity, which eliminates one of the two integral terms, introduced in earlier work in order to find analytic solutions. In the light of the physical interpretation, another approximation, representing a different limiting case can be defined, corresponding to long recirculation times, i.e. no recirculation of the delayed neutron precursors to the core. This approximation incurs the neglecting of the other integral term, and it can also be solved analytically. Finally it is shown that the full equation, without neglecting any of the integral terms, has also a compact analytical solution and it is demonstrated how the case of the infinite velocity can be obtained as a limit case of the full solution. The analytical solutions open up the possibility to study a number of questions in analytical form, such as the calculation of the point kinetic response of the reactor, or the effect of the different boundary conditions. As an illustration, the solutions corresponding to the vanishing of the flux at the extrapolated boundary are compared to those obtained from the logarithmic boundary conditions.

The probabilistic aspects of the detection of nuclear particles are discussed with the application of the methods of renewal theory. It is assumed that the inter-event times in the investigated event series are independent, identically distributed random variables. The detector efficiency and various types of the dead time are accounted for. Exact analytical results are derived for the probability distribution functions, the expectations and the variances of the number of detected particles. Also, a precise analysis of the coincidence problem is given. The results should serve for the evaluation of the measurements in view of the dead time correction effects for the higher moments of the detector counts.

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.

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 regard the safety analysis models as non-linear input–output maps. We assume random input and perform a statistical analysis of the output variables, define statistical test functions to decide whether two outputs are correlated, whether they are linearly dependent or there is a non-linear relationship between them. The methodology is applied to ATHLET calculations and we show chaotic behavior to occur in a domain of the investigated time interval. The time dependence of the upper tolerance region for two strongly correlated output variables is determined.

In former years, thermal hydraulics phenomena have been analyzed mostly by means of conservative models. Recently there is a tendency to revert to best estimate safety analysis of thermal hydraulics phenomena in a nuclear power plant. In a best estimate analysis we have to know the uncertainty of the estimated values. We investigate two aspects of the error analysis. Firstly, we define a black box model of the thermal hydraulic calculation and derive the precise number of sample calculations needed for a given tolerance level even for several statistically dependent output variables. Then we point out possible chaotic behavior of calculation along with suggesting methods for recognizing the appearance of chaos.

This paper presents a full derivation of the variance-to-mean or Feynman-alpha formula in a two-energy-group and two-spatial-region treatment. The derivation is based on the Chapman-Kolmogorov equation with the inclusion of all possible neutron reactions and passage intensities between the two regions. In addition, the two-group one-region and the two-region one-group Feynman-alpha formulas, treated earlier in the literature for special cases, are extended for further types and positions of detectors. We focus on the possibility of using these theories for accelerator-driven systems and applications in the safeguards domain, such as the differential self-interrogation method and the differential die-away method. This is due to the fact that the predictions from the models which are currently used do not fully describe all the effects in the heavily reflected fast or thermal systems. Therefore, in conclusion, a comparative study of the two-group two-region, the two-group one-region, the one-group two-region and the one-group one-region Feynman-alpha models is discussed.

The stochastic behavior of one of the simplest autocatalytic reactions, \( A+X\rightarrow X+X\) has been analyzed. Assuming the perfect mixing of substrate and autocatalytic particles in a closed volume to be uniform, an exactly solvable system of differential equations for the probability of finding n autocatalytic particles at a given time moment was derived. It was found that the time dependence of the mean number of the autocatalytic particles is clearly different from that obtained from the corresponding rate equation. The hierarchial system of the moment equations derived from the generating function of the number of the autocatalytic particles has been also analyzed. It was shown that neglecting the semi-invariants of order larger than two brings about the most suitable approximation for the first moment. In the case of closed reaction constituents, the probability distribution of the system lifetime has been determined, and the dependence of the expectation and variance of the lifetime on the number of substrate particles has been calculated.

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.

In this paper the theory of the joint neutron-gamma photon distributions emitted from fissile samples with an intrinsic neutron source, i.e. spontaneous fission, is described. In a sample of finite size short fission chains will develop for each initial source event, thereby changing the number distributions (multiplicities) of the emitted neutrons and gamma photons as compared to the elementary source events. Although in the fission process the neutrons and the gamma photons are generated independently of each other, since new gamma photons are also generated in the fission chain, the number of total emitted neutrons and gamma photons will develop correlations which increase with increasing sample mass. In the paper the general theory of the joint distributions is derived through master equations for the generating functions. The first few joint factorial moments are calculated explicitly, including the covariance between the neutron and gamma photon numbers.

The RAD Laboratory measured annual means of radon activity concentrations in 15 277 first-floor rooms of dwellings and in 325 rooms on upper floors in Hungary (1994–2004). The original purpose of the survey was to find radon-prone area in Hungary. The maximum measured value was 5800 Bq m −3 , while the minimum was 10 Bq m −3 . Due to geological diversity and different structures of buildings, the data set of first-floor rooms did not follow the lognormal distribution. Therefore, strata were chosen so that the measured data fitted the lognormal distribution. The numbers of dwellings above a given radon level were determined in each stratum. The national distribution was then taken as the sum of the individual distributions of all strata. This distribution was not lognormal. The parameters of the best fitting lognormal distribution were GM = 58 Bq m −3 , GSD = 2.2. The weighted averages of strata values GM = 62 Bq m −3 , GSD = 2.1 were obtained corresponding to 92% of Hungarian dwellings.

The theory of particle transport with branching in a medium randomly varying in time is developed in this paper. We consider an evolution equation for the probability distribution of the number of particles in a multiplying system whose properties jump randomly between two discrete states. A forward type master equation is derived for the probability distributions, and from these, the first two factorial moments are calculated, including the variance. This model can be considered the unification of stochastic methods that were used either for the particle fluctuations in a constant multiplying medium via the master equation technique, or in a fluctuating medium via the Langevin technique. The results obtain show a much richer variety of behaviour than any of the above two methods separately can reconstruct.

This paper presents a review of all the aspects of slowing down phenomena in which stochastic considerations of the process need to be taken into account. The cases both with and without absorption are treated. A consistent treatment is given of the energy distribution functions, the moments of collisions to slow down past a certain energy, the speed of neutrons and the slowing down time, with explicit expressions for the expectation and variance in all cases. Besides a compact and uniform derivation of known results, some new results are also presented.

A detailed mathematical analysis is given of the prompt neutron period measurement based on the fluctuation of the number of neutrons in a reactor. In the case of the Feynman method the effect of the delayed neutrons on the accuracy of the measurement is investigated and the correlation between the numbers of neutrons counted in consecutive time intervals is examined so as to make possible appropriate choice of the conditions of measuring. For the Rossi method a general theory is developed taking into account the role of the delayed neutrons. It was found that the Rossi method can be used in the case of thermal reactors provided the average neutron lifetime is not longer than about 10−4 sec. Mogilner's ‘zero probability method’ is discussed and the theoretical basis of this method is given.Last, a new method (differential method) is proposed which is less sensitive to the correlation than the Feynman method. The Rossi and Mogilner methods proved to be faster and more reliable than the other two methods.

The general properties of the first-order magnetic phase transitions between ordered structures have been investigated. The number of possible models has been limited by supposing the sensitivity of the exchange energy to the interatomic distance to be responsible for the phase transition. It is shown that the magnetic phase transition occurs as a result of a signchange in the proper combination of the exchange parameters. The transition may be influenced by the variation of the pressure and the magnetic field. The actual phase transition is, however, hindered in both directions by the elastic energy wall, separating the free energy minima of the different phases. TheT-P phase diagram has been calculated and the conditions for the existence of a triple point in theT-P plane have been investigated in the case of antiferromagnetic ⇆ ferromagnetic transitions. The temperature hysteresis, i.e. the difference between the upper (T sup) and lower (T inf) transition temperatures has been determined at different pressures and for both temperatures a linear dependence has been obtained. The character of the disappearance of the magnetic order has been also studied and it has been found that under certain conditions the order of the ferromagnetic-paramagnetic transition may be different from that of the antiferromagnetic-paramagnetic one. TheT-H coexistence curve between the ferromagnetic and the spin-flop antiferromagnetic phases has been determined. The upper and the lower phase transition temperature versus magnetic field curves are calculated and it is found that with increasing magnetic field and decreasing temperature the difference between these two curves (i. e. the hysteresis width) becomes increasingly small until at a critical field (H t) and temperature (T t) they end in a common point with the coexistence curve. BelowT t the variation of the magnetic field does not lead to a first-order antiferromagnetic ⇆ ferromagnetic transformation. The ferromagnetic phase is nothing, but the antiferromagnetic phase in a magnetic field strong enough to turn the moments parallel to the field direction. AboveH t but belowH c the variation of the temperature brings about also a second-order ferromagnetic ⇆ antiferromagnetic transition. The discontinuous change of the lattice parameter and that of the sublattice magnetization have been calculated at the coexistence temperature and at both the upper and lower transition temperatures. It has been shown that the entropy change of the transition has two contributions: one of them is associated with the volume change while the other with the magnetization change. Under certain conditions the order of magnitude of both contributions may be the same.

Neutron fluctuations in a constant multiplying medium (zero power noise) and those in a fluctuating medium (power reactor noise) have been traditionally considered as two separate disciplines that exist in two opposing limiting areas of operation (low and high power, respectively). They have also been treated by different mathematical methods, i.e., master equations and Langevin equation, respectively. In this paper we develop a theory of neutron fluctuations in a medium randomly varying in time, based on a forward-type master equation approach. This method accounts for both the zero power and the power reactor noise simultaneously. Factorial moments and related quantities (variance, power spectrum, etc.) of the number of the neutrons are calculated in subcritical systems with a stationary external source. It is shown that the pure zero power and power reactor noise results can be reconstructed in the cases of vanishing system fluctuations and high power, respectively, the latter being a nontrivial result. Further, it is shown that the effect of system fluctuations on the zero power noise is retained even in the limit of vanishing neutron number (reactor power). The results have thus even practical significance for low-power systems with fluctuating properties. The results also have a bearing on other types of branching processes such as evolution of biological systems, germ colonies, epidemics, etc., which take place in a time-varying environment.