# Theory of Power Spectral Density in ADS

Another subcriticality measurement technique is the power spectral density method. This chapter focuses on the cross-power spectral density (CPSD), which is the Fourier transformation of a cross-correlation function between two neutron detector signals. In an infinite and homogeneous subcritical system where the energy and spatial dependence of the neutron is neglected, the CPSD is simply expressed as a function of frequency:

where ω ¼ angular frequency. The CPSD in an ADS where the energy and spatial dependence is considered, however, is much more involved as

-1, ω*m* ¼ 2π*m*/*T*, and the subscripts *C*, UN, and CS have the same

meanings as in the previous section. For the two systems in the previous section (large subcritical system and nearly critical system), Monte Carlo simulations were performed to obtain CPSDs between detectors 1 and 2 in Fig. 12.1. In the simulations, the pulse period is *T* ¼ 0.05 s (20 Hz). The simulation result for the nearly critical system is compared with the theoretical one in Fig. 12.6. The results of the large subcritical system are shown by Yamamoto [15]. The theoretical results agree well with the Monte Carlo simulations. The uncorrelated component, CPSDUN(ω), emerges only at the integer multiples of the pulse frequency as the Delta-functionlike peaks. Thus, either of the correlated and uncorrelated components can be easily discriminated from the CPSD. Using Eq. (12.12), the uncorrelated and correlated components of the CPSD in the nearly critical system is decomposed into the mode components, shown in Figs. 12.7 and 12.8, respectively. In the correlated

**Fig. 12.6 Amplitude of cross-power spectral density (CPSD) in the nearly critical system by Monte Carlo simulation and theoretical value**

**Fig. 12.7 Mode components of the uncorrelated component in the CPSD in the nearly critical system (real part)**

component, the higher-order modes are negligibly small, and almost the whole of the CPSD is made up of the fundamental mode. The same condition holds for the large subcritical system. The higher-order mode effect in the correlated component is minor even in the large subcritical system. In the uncorrelated component, the higher-order mode effect is significant even in the nearly critical system. In the large subcritical system, the higher-order mode effect is much more significant. Thus, fitting the uncorrelated component to Eq. (12.9) yields an inaccurate α value unless the system is nearly critical. For example, in the large subcritical system we obtain α ¼ 789 (s-1) for the true fundamental mode α value of 940 (s-1) [15] from the uncorrelated component. On the other hand, we obtain α ¼ 900 (s-1) from the

**Fig. 12.8 Mode components of the correlated component in the CPSD in the nearly critical system (real part)**

correlated component. In the nearly critical system, we obtain α ¼ 172 (s-1) and 178 (s-1) from the uncorrelated and correlated component, respectively, for the true fundamental mode α value of 179 (s-1) [15].

# Conclusions

In a subcriticality measurement for an ADS, the Feynman *Y* function in general appears as the sum of the correlated and uncorrelated components. The highermode effect in the correlated component is less significant than in the uncorrelated component. Thus, a relatively good approximation of the true fundamental mode α can be obtained by using the correlated component. However, it is not necessarily easy to separate the correlated component from the measured Feynman *Y* function. Considering the difficulty of separating the correlated component, the Feynman-α method is not always suitable as a subcriticality measurement technique for ADSs. In an ADS that is nearly critical, the uncorrelated component is very minor. Thus, by fitting the measured Feynman *Y* function to the correlated component, the fundamental mode α can be accurately estimated.

In a subcriticality measurement using the power spectral density method, the uncorrelated component emerges at the integer multiples of the pulse frequency as delta-function-like peaks. Thus, the uncorrelated component can be easily discriminated from the correlated component. The correlated component is less contaminated by the higher-order modes. An approximate fundamental mode α can be obtained by fitting the Feynman *Y* function to the correlated component of the power spectral density. The use of the uncorrelated component is not always recommended, because the higher-order modes are more significant in the uncorrelated component.