Feynman-alpha Formula Research Articles

On the relation between Rossi alpha and Feynman alpha methods

The relation between the Rossi alpha and Feynman alpha formulae is well known. The second factorial moment of counts in an interval, which is related to the variance, can be obtained by a double integration of the Rossi alpha expression. However in a recent paper, Munoz Cobo et al. have cast doubts on the generality of this relation stating that this relation is valid only for point models and is not true for space dependent models.The present paper revisits the issue. We examine the multi mode formulae obtained by Munoz Cobo et al. in order to assess the validity of the above relationship. We show that indeed it is possible to derive the Feynman alpha formula by a double integration of the Rossi alpha formula followed by some elementary algebraic manipulation. To demonstrate the generality of this relationship we use two distinct approaches. In the first of these, we show, using the (backward) stochastic transport equation and the space dependent Bartlett formula, that the Rossi alpha and Feynman alpha functions obey the above relationship. In the second, we start from the theory of stochastic point processes and show that the relationship is quite general.

A general analytical solution for the variance-to-mean Feynman-alpha formulas for a two-group two-point, a two-group one-point and a one-group two-point cases

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.

Two-point theory for the differential self-interrogation Feynman-alpha method

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.

Derivation and quantitative analysis of the differential self-interrogation Feynman-alpha method

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.

Feynman-Alpha and Rossi-Alpha Formulas with Delayed Neutrons for Subcritical Reactors Driven by Pulsed Non-Poisson Sources with Correlation between Different Pulses

Through our earlier papers, we have shown that reactor noise in accelerator-driven systems (ADS) is different from that in critical or radioactive source-driven subcritical systems due to periodically pulsed source and its non-Poisson character. We have developed a theory of reactor noise for ADS, taking into account the non-Poisson character of the source. Various noise descriptors, such as Rossi-alpha, Feynman-alpha (or variance to mean), power spectral density, and cross power spectral density, have been derived for a periodically pulsed source, including correlation between different pulses and finite pulses of different shapes. For mathematical simplicity, the theory was restricted to the case of prompt neutrons only. Recently, we extended the theory to the delayed neutron case and derived Feynman-alpha and Rossi-alpha formulae by considering the source to be a periodically pulsed non-Poisson source, without correlations between different pulses. The present paper extends the treatment to account for the possibility of correlations between pulses. Feynman-alpha and Rossi-alpha formulas are derived by considering the source to be a periodic sequence of delta function non-Poisson pulses, with exponential correlations.

Reactor noise in accelerator driven systems – II

Reactor Noise in accelerator driven systems (ADS) is expected to show differences from critical or radioactive source driven sub-critical systems due to the periodically pulsed source and its non-Poisson character. A theory of ADS Reactor Noise, incorporating these features was worked out by us and has been published earlier. The present paper is a continuation of the earlier work and constitutes an extension and generalization of the theory. We present theoretical arguments as well as empirical evidence to elucidate the non-Poisson character of the ADS neutron source. We solve the delta-function pulse train problem by treating the source as an exponentially correlated process as against the uncorrelated non-Poisson process considered earlier. In yet another direction, the theory is extended to treat pulses having finite width by describing the source as a doubly stochastic Poisson point process. It is shown that the results of the extended theory reduce to those of our earlier formulation (or to other published results) under appropriate conditions. Evaluation of various descriptors gets simplified by going to the Laplace domain but results in infinite series. Calculations in the time domain are more difficult but yield closed form expressions. We describe both approaches in the course of this paper. We use the Rossi alpha and Feynman alpha formulae, which are evaluated for all the cases considered, for illustrating our approach. In addition to these, we also obtain the auto correlation function, cross-correlation function and the corresponding power spectral in one of the cases. A brief discussion on the various noise techniques vis-a-vis other methods for sub-critical monitoring is included.

Feynman-alpha experiment with stationary multiple emission sources

In order to verify the Feynman-alpha formula with stationary multiple emission sources [Pázsit, I., Yamane, Y., 1998. Nuclear Instruments and Methods A403, 431–441], a series of experiments were performed by using three different neutron sources, i.e., an Am–Be, a 252Cf, and a randomly triggered D–T pulsed neutron generator. As a result, it was found that the prompt neutron decay constant evaluated from the measured Y-function was hardly affected by the difference of neutron sources that have respective source multiplicities. On the other hand, it was demonstrated that the amplitude of Y-function, i.e., the correlation amplitude, depended on both the subcriticality and the neutron source multiplicity. These experimental facts agreed with the theoretical prospects that were deduced from the Feynman-alpha formula with multiple emission sources. Therefore, it was concluded that the validity of this formula was experimentally shown.

06/00418 Calculation of the pulsed Feynman-alpha formulae and their experimental verification: Pázsit, I. et al. Annals of Nuclear Energy, 2005, 32, (9), 986–1007

06/00418 Calculation of the pulsed Feynman-alpha formulae and their experimental verification: Pázsit, I. et al. Annals of Nuclear Energy, 2005, 32, (9), 986–1007

05/01304 Analytical solution for the Feynman-Alpha formula for ADS with pulsed neutron sources

05/01304 Analytical solution for the Feynman-Alpha formula for ADS with pulsed neutron sources

Calculation of the pulsed Feynman-alpha formulae and their experimental verification

An effective method of calculating the pulsed Feynman-alpha formula for finite width pulses is introduced and applied in this paper. The method is suitable for calculating both the deterministic and the stochastic Feynman-alpha formulae, while also being capable of treating various pulse shapes through very similar steps and partly identical formulae. In the paper both the deterministic and the stochastic cases are treated for square and Gaussian pulses. The solutions show a very good agreement with the results of currently performed experiments by some of the authors at the Kyoto University Critical Assembly (KUCA). The formulae obtained are also used for a quantitative evaluation of the prompt neutron decay constant from a large number of experiments made at the KUCA for a wide range of parameters such as subcritical reactivity, pulse repetition frequency and pulse width. The suitability of the formulae to determine the prompt neutron decay constant α by curve fitting to the measured data was investigated. It was found that, despite the larger deviation from the traditional Feynman Y( T)-curves from the traditional ones with a constant source (i.e., larger ripples superimposed on a smooth curve), the stochastic pulsing method is superior to the deterministic one in that it yields the correct α value for all subcriticalities. The deterministic method also works fine for most cases, but its application is not so straightforward.

Calculation of the pulsed Feynman- and Rossi-alpha formulae with delayed neutrons

In previous works, the authors have developed an effective solution technique for calculating the pulsed Feynman- and Rossi-alpha formulae. Through derivation of these formulae, it was shown that the technique can easily handle various pulse shapes of the pulsed neutron source. Furthermore, it was also shown that both the deterministic (i.e., synchronizing with the pulsing of neutron source) and stochastic (non-synchronizing) Feynman-alpha formulae can be obtained with this solution technique. However, for mathematical simplicity and the sake of insight, the formal derivation was performed in a model without delayed neutrons. In this paper, to demonstrate the robustness of the technique, the pulsed Feynman- and Rossi-alpha formulae were re-derived by taking one group of delayed neutrons into account. The results show that the advantages of this technique are retained even by inclusion of the delayed neutrons. Compact explicit formulae are derived for the Feynman- and Rossi-alpha methods for various pulse shapes and pulsing methods.

04/02120 Analytical solution for the Feynman-Alpha formula for ADS with pulsed neutron sources: Ceder, M. and Pázsit, I. Progress in Nuclear Energy, 2004, 43, (1–4), 429–436

04/02120 Analytical solution for the Feynman-Alpha formula for ADS with pulsed neutron sources: Ceder, M. and Pázsit, I. Progress in Nuclear Energy, 2004, 43, (1–4), 429–436

Analytical solution for the Feynman-Alpha formula for ADS with pulsed neutron sources

The theory of Feynman-alpha measurements is elaborated for the case of a “stochastically pulsed” subcritical system. The corresponding physical situation is when a pulsed neutron source is used, and no synchronisation between the start of the measurement time gate and the pulsing is made. This is the case in the European Community supported research project MUSE.The solution to the Feynman-alpha formula was obtained for such a case through complex function techniques in an analytical form by Laplace transform and residue calculus. The final expression is a smoothly regular function with a simple periodic modulation. It consists of a Feynman-curve corresponding to a stationary source, plus an infinite sum of periodic sine functions squared. The series converges as 1/n6 with the summation index n, thus in practice two or three terms are sufficient for a high accuracy quantitative result. This few-term representation amounts to a compact closed form analysis solution. Such a solution is well suitable for use in the determination of the subcritical reactivity from measurements, in contrast to the case of deterministic pulsing (measurement start synchronized with pulsing), where no simple solution is available, and where no explicit relationship between the continuous and pulsed forms of the Feynman-alpha exists.

The general backward theory of neutron fluctuations in subcritical systems with multiple emission sources

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.

Heuristic derivation of rossi-alpha formula with delayed neutrons and correlated source

Formulae of neutron correlation measurements, such as Feynman-and Rossi-alpha, have been derived in the past by the master equation method or the joint detection probability method. Generally speaking, the master equation method is rigorous in the sense, that all elementary processes can be taken into account, but its manipulations are complicated and elaborate. On the other hand, the method of joint detection probability is heuristic, but several kinds of formulae can be easily derived from it. It seems that these methods are complementary. Recently in order to develop a method for evaluating subcriticality in the accelerator driven subcritical reactors, we derived the Feynman-alpha formula with a correlated neutron source and delayed neutrons based on the master equation method. In this paper, based on a heuristic method, we have derived the joint detection probability including one group of delayed neutrons and a correlated neutron source, which is valid in deep subcritical systems. Then, as an application of this joint probability, the Rossi-alpha formula with delayed neutrons and a correlated neutron source has been derived.

Theory of neutron fluctuations in source-driven subcritical systems

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.