# IV Basic Research on Reactor Physics of ADS: Basic Theoretical Studies for Reactor Physics in ADS

# Theory of Power Spectral Density and Feynman-Alpha Method in AcceleratorDriven System and Their Higher-Order Mode Effects

Abstract This chapter discusses the theory of higher-order modes in the Feynman *Y* function and cross-power spectral density (CPSD) in an accelerator-driven system (ADS) where pulsed spallation neutrons are injected at a constant time interval. Theoretical formulae that consider the higher-order modes of the correlated and uncorrelated components in the Feynman *Y* function and CPSD for an ADS were recently derived in a paper published by the author. These formulae for the Feynman *Y* function and CPSD are applied to a subcritical multiplying system with a one-dimensional infinite slab geometry in this chapter. The Feynman *Y* functions and CPSD calculated with the theoretical formulae are compared with the Monte Carlo simulations of these noise techniques. The theoretical formulae reproduce the Monte Carlo simulations very well, thereby substantiating the theoretical formulae derived in this chapter. The correlated and uncorrelated components of the Feynman *Y* functions and CPSD are decomposed into the sum of the fundamental mode and higher-order modes. This chapter discusses the effect of subcriticality on the higher-order mode effects.

Keywords ADS • Feynman-α method • Higher-order mode • Monte Carlo • Neutron noise • Power spectral density

## Introduction

In accelerator-driven systems (ADS), fission chain reactions are driven by spallation neutrons emitted from a proton beam target. An ADS is quite different from an ordinary nuclear reactor in that it is always operated at a subcritical state. Thus, the safety requirements for reactivity control can be eased in ADSs. The subcriticality of an ADS, however, needs to be continuously monitored to maintain its criticality safety. A reactor noise technique such as the Feynman-α method and the power spectral density method can be a potential candidate for monitoring the subcriticality of ADSs. The noise theory in ADSs is different from the classical reactor noise theory in that multiple neutrons are injected from the proton beam target at a single spallation event and pulsed neutrons are emitted deterministically at a constant period. Many theoretical and experimental studies on the noise theory in ADSs have been performed thus far. The theoretical formula for the Feynman-α method or Rossi-α method in ADSs was studied by, for example, Pa´zsit et al. [1], Pa´zsit et al. [2], Kitamura et al. [3], and Mun˜oz-Cobo et al. [4]. Another technique that uses the auto-power spectral density (APSD) or cross-power spectral density (CPSD) was studied by, for example, Mun˜ oz-Cobo et al. [5], Rugama et al. [6], Ballester and Mun˜oz-Cobo [7], and Degweker and Rana [8]. Sakon et al. recently carried out a series of power spectral analyses in a thermal subcritical reactor system driven by a periodically pulsed 14 MeV neutron source at the Kyoto University Critical Assembly (KUCA) [9].

Both the Feynman-α method and the power spectral density method are intended to measure a prompt neuron time-decay constant α of the fundamental mode because the subcriticality is directly related to the fundamental mode α. The measured results, however, are inevitably contaminated by the higher-order mode components. To obtain an accurate knowledge of the subcriticality, the effect of the higher-order modes needs to be quantified in detail.

Endo et al. [10] derived a theoretical formula of the Feynman *Y* function that considers the higher order modes. Mun˜oz-Cobo et al. [11] also derived a similar theoretical formula from a different approach. Using these formulae, Yamamoto [12, 13] demonstrated quantitative analyses of the spatialand energy-higher order modes in Feynman *Y* functions, respectively. In these two works, the Feynman *Y* functions were successfully resolved into spatialor energy-higher order modes. These discussions, however, involved subcritical multiplying systems driven by a neutron source with Poisson character. They did not account for either a periodically pulsed neutron source or its non-Poisson character. Some previous work that considered the higher-order modes in the noise techniques for ADSs has been published (e.g., [6], [7]). In these previous publications, however, the effects of the higher-order modes have not been quantitatively investigated. Yamamoto [14, 15] presented the formulae of the Feynman *Y* function and CPSD for ADSs that consider the higher-order mode effects. Yamamoto [15] resolved the Feynman *Y* functions and power spectral densities into the mode components. Verification of the formulae was demonstrated by comparing the theoretical predictions with the Monte Carlo simulations of the subcriticality measurement in an ADS.

The purpose of the present chapter is to investigate how the subcriticality would affect Feynman *Y* function and power spectral density. The subcriticality of an ADS differs from design to design. The smaller the subcriticality, the larger the neutron multiplication that can be gained, which, on the other hand, decreases the margin of criticality safety. The subcriticality undergoes a gradual change as the fuel burn-up proceeds. The Feynman *Y* function and power spectral density emerge differently as the subcriticality changes. This chapter shows the dependence of subcriticality measurement on its subcriticality, which will contribute to the design of ADSs and planning of subcriticality measurements in the future.