Neutron Noise Bibliographic Descriptions

[Dates to about 1964]
One method of measuring one or perhaps more of several 'multiplying system parameters involves making measurements of the dynamic characteristics of the system. There are two basic methods for making such measurements. The first involves perturbing the system in some manner and observing the resulting effect on the neutron density. These experiments have been done with periodic perturbations such as oscillation of a neutron absorber in the system, with random perturbations such as the random or "pseudo-random" insertion and removal of a neutron absorber, with step insertion or removal of reactivity, and by introducing pulses of neutrons into the system. The second involves observing the fluctuations in either the neutron density, the number of fissions in a time interval, or quantities proportional to these. The fluctuations result because the processes involved are stochastic in nature. Experiments of the first type are not discussed here as such, but they are related to the results of the second type through the transfer function. Thus the improvements made in terms of a better model reflect information back to the perturbed system experiments.

The fluctuation or "noise" experiments can be divided into two groups. The first contains those which deal with a probability distribution and includes the Rossi- alpha or two counter experiment and lesser known closely related experiments done by Brunson (6) and Moguilner. (16)

The Rossi-alpha concerns the probability of obtaining a count in dt about t after a count at t = 0, and the latter concerns the probability of having a count free interval of length t after a count at t = 0. The second group involves moments of probability distributions. In this group, are the variance of the neutron density, the variance of a measured number of counts in an interval of length t, the auto and cross correlation functions of the neutron density, and the auto and cross spectral density functions. The spectral density functions are Fourier transforms of the correlation functions.

At this time, noise experiments have most frequently obtained

In a second paper, (16) published in 1960, Raievski generalized his model to include delayed neutrons, a choice of thermalization by age or two group theory, and the indication of a detector at a time t. Using the Fermi age theory, fundamental space mode approximation, he obtained the variance of the neutron density, detector current and counts in an interval of length t along with a related quantity, the power spectrum of the current. Only the general procedure was formulated for the two group theory. Also, note that in both of these papers, the diffusion properties of the small cell theory were based upon point theory. Thus, cell size must be limited to small cells, and indeed it seems strange that the problem was not cast in terms of the neutron density entirely.

The results showed that aside from some energy dependent factors the fundamental mode variance is identical to the space independent formulation.

In 1962, Blaquiere(18) published a paper deriving Raievski's fundamental equation by means of an electrical circuit analogy, He assumed that the theory could be applied to macroscopic cells which again implies that the fundamental mode exists in each cell. The theory was applied to two systems for which information could be obtained without obtaining the diffusion transition probabilities between the cells. In one case, a small cell (number 1) in a large reactor (number 2), he found the diffusion effect on the small cell was Poisson by discarding the probably much smaller non-Poisson term. In the second case, that of two large cells each one half of a symmetrical reactor he found the effect on either half to be very much non- Poisson. Quantitatively, he found that the variance was one-half of the average as compared to the average for the Poisson case.

Recently, Matthes (19) published a paper which is an extension of Raievski's basic multigroup theory. Continuing the trend, the equations were only applied to space independent cases. Matthes' formulation of the space dependent noise problem appears to be more directly applicable to computations than Pal's since it is written in multigroup diffusion theory rather than transport theory. Secondly, Matthes' equations are cast directly in terms of the neutron density [forward equation approach] while Pal's involve the above mentioned probability distribution function form and the integrals over weighting functions.

In 1963-64, the neutron density variance equations from the point of view of the quantum Liouville equation were obtained by Osborn, the paper summaries appearing in the ANS Transactions (26,29).

Again, no solutions were obtained since his primary interest was in the approximations that must be made in the exact theory to obtain Matthes' results.

Harris(24) has generalized the physical method of DeHoffman to include space dependent effects. He applied the technique to an idealized one dimensional ring reactor, i. e. space independence or the fundamental mode is assumed to exist in the radial direction of the ring. This requires the ring to be large or have a black center. As a result of either typographical errors or others, his solution is in error as shown by the fact that it is dimensionally incorrect.

Moore(25,28) recently suggested an alternative method which is similar to that followed herein. It was proposed that both the input and output correlation functions be expanded in space eigen functions. The necessary coupling condition between the coefficients of the two expansions was obtained for three functional forms of the input correlation function. These were for point, plane surface, and volumetric input source distributions. No attempt was made to specify in detail the input correlation function or to solve specific problems.

Organization and Scope of Text

In (the full text), the space dependent neutron noise problem is systematically developed using the Langevin (1,book-Wax-6) method. This technique was chosen because it was felt that at least simple problems could be completely solved using this method as opposed to the prior theories which have not been developed beyond the fundamental equations. The reasons for this lack of development are somewhat obscure, but must certainly involve the complexity of the resulting sets of coupled integral and differential equations which arise.

The Langevin method as applied here consists primarily of obtaining the neutron density cross correlation function, or in general an output, in terms of an input noise correlation function and the system Green's function for the mean equation. The procedure is then generalized to give point count rate correlation functions, integrated counts correlation functions, and detector size effects. Through Fourier transforming of either the resulting correlation function or the basic equation, spectral density functions are obtained.

The scope of the reactor model is limited to emphasize the diffusion effects, thus allowing these effects to be studied exclusive of other complications. The omission of energy and transport effects does not mean that they are unimportant or not necessary for a quantitative description including space effects. Further studies will be necessary to ascertain the exact nature of the approximation when only space effects are considered.

Considering briefly the specific approach, the basic theoretical model for obtaining the neutron density cross correlation and spectral density functions in terms of input correlation and spectral density functions is developed

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© 1995-2008, Prof. James R. Sheff, Nuclear and Chemical Engineering Dept. , UMass Lowell