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The theory of the Feynman-alpha method, which is used to determine the subcritical reactivity of systems driven by an external source such as an ADS, is extended to two energy groups with the inclusion of delayed neutrons. This paper presents a full derivation of the variance to mean formula with the inclusion of two energy groups and delayed neutrons. The results are illustrated quantitatively and discussed in physical terms.

Methods of online measurement of subcritical reactivity, in connection with ADS, have been studied over a decade by now. Both deterministic methods, such as the area ratio or Sjöstrand method [

In this paper, we will discuss another aspect of stochastic reactivity measurement methods, which is related more to the system properties than those of the source. The new aspect is to take into account the energy dependence of the neutrons by the use of a two-group approach. All work so far in which compact analytical results could be obtained in this area (calculation of the Feynman- and Rossi-alpha formulae) was made by the use of one-group theory. This is justified by the fact that the methods were used in thermal systems, where the neutron population and hence its dynamics is dominated by thermal neutrons. However, many of the planned ADS concepts will use a core with a fast spectrum, in which the dominance of thermal neutrons will be significantly reduced. In terms of a two-group approach, unlike in a thermal system where there is one time or decay constant in a pulsed experiment, there will be two components in the temporal response with two different time constants and with comparable amplitudes. One indication for this possibility comes from the area of nuclear safeguards, where such effects have already been investigated, as it will be described below.

Actually there have been experiments, such as in the EU-supported project MUSE [

The purpose of the present paper is the elaboration of the two-group theory of the Feynman-alpha method, based on the backward master equation approach. Such an energy-dependent extension of the existing theory might have a relevance also to on-going and future ADS experiments, such as the European FP7 project FREYA. This work has actually been started already by the present authors, although in a different setting. In nuclear safeguards, identification of fissile material can be achieved by detecting the temporal decay of fast neutrons from an unknown sample, following irradiation by a pulse of fast neutrons. Appearance of a second decay constant is an indication of the presence of fissile material. This is the so-called differential die-away analysis (DDAA) method. A stochastic generalisation of this method, to the use of a stationary (intrinsic) source and the measurement of the time correlations, the so-called DDSI (differential die-away self interrogation) method, was recently suggested by Menlove et al. [

In the present paper, we extend the treatment by the inclusion of delayed neutrons in the formalism. Understandably, the calculations get quite involved. Although a full analytical treatment is possible, many expressions in the final results become too extensive to be quoted explicitly. Hence these will not be given and analysed in this paper. Likewise, the question of how to extract the subcritical reactivity from the measurement of the two time constants will not be discussed here, rather it will be given in a subsequent communication. Instead, here we focus on the formulation of the problem and the full derivation sequence until arriving to the final result. The intermediate and final results will be discussed in physical terms.

As usual in the context, we shall assume a homogeneous infinite medium with properties constant in space. We shall use a two-group theory model, that is, describe the neutron population with two type of neutrons: fast and thermal. One group of delayed neutron precursors will be assumed. The possible neutron reactions are absorption of both the fast and thermal neutrons, downscattering (“removal’’) of neutrons from the fast group to the thermal, and thermal fission, which produces a random number of fast neutrons according to a probability distribution function. At such a thermal fission also at most one delayed neutron precursor can be generated with a certain probability. The decay of this precursor will lead to the appearance of one fast neutron. For simplicity, fast fission will be neglected. It can be easily incorporated into the model, at the expense of some further complication of the calculations, but without any essential problem. Figure

The possible reactions which the three different particles can undergo. Fast neutrons (red circle) can be detected (yellow), absorbed (white), and thermalized (green). Thermal neutrons (green) can be absorbed, and through fission lead to a random number of delayed neutron precursors (grey) and fast neutrons (red). Delayed neutron precursors decay into a fast neutron.

A word on the notations used is in order here. In a two-group model of reactor physics, the indices 1 and 2 are used to denote the fast and the thermal group, respectively. This notation will be used also in this paper, whenever it will not lead to confusion. For practical reasons, the delayed neutron precursors will be taken as group 3. However, often it will be simpler and more practical to refer to the fast and thermal neutrons and the precursors with the notations

However, in this paper we will keep a more general mathematical system of notations for the factorial moments and transition intensities. This is partly because the purpose of this paper is to describe the formalism used in a general setting, and partly in order to relate the work reported here to our preceding paper, where the two-group generalisation of the Feynman- and Rossi-alpha methods was introduced [

The final quantity we need in order to formulate a probability balance equation is the number distribution of fast neutrons and delayed neutron precursors in a thermal fission event. Denote by

Since we are going to use the backward master equation approach, first we will need the neutron and precursor distributions, generated by one single starting particle (a fast or thermal neutron or a delayed neutron precursor). First the number distribution of generated particles will be studied and explicit results derived for the first two moments. In the next section the detection process will also be accounted for, and in the last section the variance and the mean of the number of neutrons detected in a time interval, induced by a stationary external source of fast neutrons, will be calculated from the results of the preceding two sections.

Let us introduce the random functions

By using (

The arising integral equation system can be readily solved by Laplace transform methods. Introduce the Laplace transforms:

Clearly, the system is subcritical, if none of the

Characteristic function

The algebraic solutions for the Laplace transforms

The expectations can be obtained in a rather simple way by inversion of these Laplace transforms. All solutions consist of the sum of three exponential functions, namely, of

Dependence of the expectations of the numbers of particles of types

It is to be mentioned that the above results could also be obtained directly from deterministic equations, namely, from the two-group point kinetic equations with one group of delayed neutrons.

As it follows from the definitions and formulae in the previous section, the variance of the numbers of, say, fast neutrons at the time

It is seen that for the determination of variances, one needs the second factorial moments which can be obtained from the generating function (

Analogous definitions can be introduced for

For the solution of the arising system of integral equations, again the method of the Laplace transform is used. Before performing the transforms, it is practical to introduce short-hand notations for the functions appearing in (

Based on the form of the solutions in the Laplace domain, one notices that the solutions in the time domain can be written in the form a convolution as follows:

The convolution integrals over these known functions can be performed analytically and closed form analytical solutions can be obtained for the second factorial moments. However, in the present case these explicit forms are extremely long and complicated expressions containing the exponential functions

Although not needed explicitly for the calculation of the two-group version of the Feynman-alpha formula, it might give some insight to calculate the covariances of the numbers of particles of different types at a given time moment, provided that at the time instant

In order to determine the covariance between two different random functions at a given time moment, one should calculate first the mixed second moments. In the present case, one needs the following moments:

The calculations are straightforward and similar to the calculation of the second factorial moments of the previous section, but rather involved and lengthy. Hence the details of the calculations will not be given here. We only note that similar to the case of the second factorial moments, it is practical to introduce a shorthand notation for the functions

By using the formulae (

Figure

Dependence of the covariances between the numbers of particles of types

The reason for the initially negative value of the covariances can be explained in the same way as for the covariance between the fast and thermal neutrons without the presence of delayed neutrons, as was discussed in [

In order to calculate the variance to mean of detected particles, induced by a stationary extraneous source, two steps remain. One is the introduction of the detection process, which is treated in this section. The second is the introduction of a stationary source, which will be treated in the next section.

For brevity, in the forthcoming only the detection of the fast neutrons will be discussed. For the application of the Feynman-alpha method, even in a fast system, presumably the detection of the thermal neutrons will be more practical. However, the calculation goes exactly along the same lines, hence for illustration it is sufficient to discuss the detection of fast neutrons.

Denote by

Define the probabilities:

The expectations of detected number of

After simple considerations one obtains the Laplace transforms of the expectations given by

For the characterization of the detecting process, one needs the variances of the number of the fast neutrons, counted in the time interval

With the help of these expressions, the variances of the number of fast neutron detections for the three starting particle types are given by

Since the detection in the time interval

For later use, let us determine the expectation:

The calculation of the second factorial moment:

The last step in the derivation of the variance to mean or Feynman-alpha formula is to calculate the first two factorial moments of the detected fast neutrons, induced in a subcritical reactor by a stationary source of fast neutrons. Suppose that at the time instant

In this work we assume that the source events constitute a Poisson point process, that is, that the random time interval between two consecutive injections of fast neutrons follows an exponential distribution with parameter

For the case of a Poisson source, it can be easily shown that the generating function of the probability

By using the logarithm of the generating function, one can write the formulae:

In the subcritical state, that is, if

For the determination of variances of the number of fast neutrons, generated by the three different possible starting particles, one obtains the expressions:

If the injection of fast neutrons into the medium is performed by a stationary source emitting particles according to a Poisson process defined by the intensity parameter

By using the method described in [

In the stationary case the expectation of the number of the detection of fast neutrons in the time interval

By using the logarithmic generation function

Figure

Dependence of the ratio of the variance to mean of the number of

In view of the fact that there are three decay constants

However, the Feynman-alpha formula will show the different plateaus if the decay constants are sufficiently different and differ by orders of magnitude from each other. Even the traditional (one-group) Feynman formula demonstrates this, since even if six different delayed neutron precursor types are accounted for, there are only two plateaus distinguishable: one corresponding to the prompt neutrons and only one more for all the six delayed neutron groups [

The variance to mean formula was derived in a two-group treatment with one group of delayed neutrons with the use of the backward master equation technique. The temporal behaviour of both the first and second factorial moment of the detected particles is determined by three exponentials. The various factorial moments can be fully determined analytically; however, the expressions in most cases are too lengthy to write them out. Qualitatively, the form of the solutions is the same as in the traditional case, but the two decay constants associated with the prompt response of the system, are not distinguishable in the plots. The relationship between the decay constants and the subcritical reactivity of the system will be investigated in future communications.

This work was supported by the Hungarian Academy of Sciences and the FP7 EU Collaborative Research Project FREYA, Grant Agreement no. FP7-269665.