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Neutron noise spectra in nuclear reactors are a convolution of multiple effects. For the IBR-2M pulsed reactor (JINR, Dubna), one part of these is represented by the reactivities induced by the two moving auxiliary reflectors and another part of these by other sources that are moderately stable. The study of neutron noise involves, foremostly, knowing its frequency spectral distribution, hence Fourier transforms of the noise. Traditional methods compute the Fourier transform of the autocorrelation function. We show in the present study that this is neither natural nor real-time adapted, for both the autocorrelation function and the Fourier transform are highly CPU intensive. We present flash algorithms for processing the Fourier-like transforms of the noise spectra.

Neutronic processes in nuclear reactors have a probabilistic character due to the quantum mechanics of scattering and the stochastic of propagation in materials. Design is mostly performed on the equations of neutron flux transport (in energy and space) and associated effects (fission, thermal fluxes, etc.). Statistical deviations from average quantities give however the complete image of neutron physics in the reactor—the so-termed neutron noise, described by large using Markov-chain theories. The theories associated with the underlying stochasticity that produces the said fluctuations are actually century old [

On top of said, neutron stochastic behavior is the modulation of the neutron flux by various reactivities: some due to 2-phase liquid flow (bubbling), or fuel embrittlement, mechanically induced reactivities, and so forth. Any addition to the spectrum may be detected and classified, issuing a specific warning. In this respect, neutron noise spectrum analysis is a very far reaching tool in nuclear safety.

To analyze the noise, usually a Fourier transform of the autocorrelation function is performed. This is basically the same as the square of the Fourier transformed signal. In practice, one also needs a time window, given by ad hoc apodisation functions. This is rather arbitrary and nonnatural. We show a more natural approach passing the noise through a resonant analyzing instrument together with flash algorithms for processing the Fourier-like transforms of the noise spectra.

The IBR-2M [_{2} fuel elements. The reactor coolant is liquid sodium. The pulsed mode operation is enabled by a reactivity modulator consisting of two rotating parts—a main movable reflector (OPO, at 1500 rpm) and an auxiliary movable reflector (DPO, at 300 rpm)—shown in Figure

Details of the IBR-2M reactor showing the active core and the two movable reflectors.

Power versus time between two subsequent pulses. Data is normalized to the maximum pulse.

The pulsed operation mode of the reactor is established when the prompt neutron supercriticality ^{18}O (part of the oxide fuel);

For a power noise analysis [

Reactor power during power up (red line) with fitted baseline (blue curve of spline order

The concept of Fourier transform in engineering can be quite different from its mathematical analogue. In mathematics, a given signal (function of time) is transformed to another function of frequency, basically the coefficients of the signal in the canonically conjugate image.

In engineering, a signal is long, and we need an “instantaneous” Fourier Transform at each given time, thus a function that provides a time—window-termed apodisation function. The question is, for discrete Fourier transforms (DFFT-[

Fourier spectrum for various apodising functions. The Parzen function shows best the sidelobes that occur, while the SRS function provides a relatively flattop spectrum with good side-rejection of the lobes.

Traditionally, the Fourier transform of the power noise is computed from the autocorrelation function:

Beyond the above considerations, special tools are needed to compute—what is mathematically improper termed—Fourier Transform of noise signals. The basic image is that of sound noise, perceived as higher or lower pitch. In categorizing noise as such, the ear for instance (as one example of analysis instrument) does not work with orthogonal wave-vectors, or mathematically complete spaces, nor does it obtain coefficients for said wave-vectors. What such an analysis instrument does is to use a set of resonance phenomena, tuned on various frequencies. The instantaneous mechanical quantity producing measurable effect is (in the case of sound) the air velocity, or in electrical circuits, the current:

As shown above, this is a natural choice—with a definite model in sight—for an apodising function, containing what is natural as

The remaining problem is that such an instrument is highly computationally the intensive.

It can be recognized that the sine function can be written as sum of exponentials and that the mathematical construct to be computed is principally

There are two cases for how the summation padding is advantageous, depending on how

As mentioned in the introduction, there are beneficial aspects in detecting and subtracting the baseline. Apart from extracting from the low frequency end reactor dynamics features, also technical aspects are achieved, such as rejecting aliasing. In general, such unwanted phenomenon is due to the fact that some frequency is much higher than the Nyquist sampling frequency; in our case,

Fourier transform of reactor power, baseline, and noise. It can be seen that the raw power and baseline both are affected by aliasing in the same amount. The strong peaks at 1 Hz and 2 Hz disappear tracelessly in the difference (power noise spectrum), proving the efficiency of the method.

A toolkit with spectrum processing various utility codes has been created to help do all the steps required in a modular fashion.

For low frequency power noise monitoring, it is in certain contexts impossible to remove aliasing frequency components, presampling. Although aliasing collision is not present in the high end of the spectrum, close to

The method presented is feasible in real time, via a flash updating procedure of the data on the fly, allowing a physically credible estimate of noise spectral density.

It is interesting to note that FFT of noise is a very useful tool also in monitoring the electrical grid, for instance in observing hysteresis noise in transformer cores (leading to magnetic embrittlement and anomalous heating of the core, resp., transformer power losses). Other types of losses (dielectric polarization losses in areas with high humidity) can also be well monitored with this method.

Other nuclear reactor types could also benefit from the method, in principle power reactors which have significant water cooling systems, as these reactors produce vacuum reactivity noise if boiling occurs in the pipes.

One of the authors (M. Dima) acknowledges the support by a grant from the Romanian National Authority for Scientific Research ANCS Grant PN09370104 and two of the authors (Y. N. Pepelyshev and L. Tayibov) acknowledge the support by a grant from JINR, Dubna, Order No. 71, Item 21/2012.