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 spatial- and energy-higher order
modes in Feynman Y functions, respectively. In these two works, the Feynman
Y functions were successfully resolved into spatial- or 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
120
T. Yamamoto
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 spatial- and energy-higher order
modes in Feynman Y functions, respectively. In these two works, the Feynman
Y functions were successfully resolved into spatial- or 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
120
T. Yamamoto