12.3 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:
CPSD ω
ð Þ / 1= ω
2
þ α
2
0
À
Á
,
ð12:9Þ
where ω ¼ angular frequency. The CPSD in an ADS where the energy and spatial
dependence is considered, however, is much more involved as
CPSD ω
ð Þ ¼ CPSD C ω
ð Þ þ CPSD UN ω
ð Þ þ CPSD CS ω
ð Þ,
ð12:10Þ
CPSD C ω
ð Þ ¼
X 1
‘¼0
X 1
m¼0
X 1
n¼0
S ‘ F ‘!mn D 1, m D 2, n
α ‘ α m þ α n
ð
Þ
α n þ iω
ð
Þ
þ
X 1
‘¼0
X 1
m¼0
X 1
n¼0
S ‘ F ‘!mn D 2, m D 1, n
α ‘ α m þ α n
ð
Þ
α n À iω
ð
Þ
,
ð12:11Þ
CPSD UN ω
ð Þ ¼
2π q
2
T
2
X 1
n¼0
X 1
‘¼0
X 1
m¼À1
D 1, n D 2, ‘ Ψ
Ã
n Ψ
Ã
‘
δ ω À ω m
ð
Þ
α n À iω m
ð
Þα ‘ þ iω m
ð
Þ
,
ð12:12Þ
CPSD CS ω
ð Þ ¼ À
q
T
X 1
n¼0
X 1
‘¼0
D 1, n D 2, ‘ Ψ
Ã
n Ψ
Ã
‘
1
α n þ α ‘
Á
1
α ‘ þ iω
À
q
T
X 1
n¼0
X 1
‘¼0
D 2, n D 1, ‘ Ψ
Ã
n Ψ
Ã
‘
1
α n þ α ‘
Á
1
α ‘ À iω
,
ð12:13Þ
where i ¼
ffiffiffiffiffiffi ffi
À1
p
, ω m ¼ 2πm/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, CPSD UN (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
12 Theory of Power Spectral Density and Feynman-Alpha Method. . .
125
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:
CPSD ω
ð Þ / 1= ω
2
þ α
2
0
À
Á
,
ð12:9Þ
where ω ¼ angular frequency. The CPSD in an ADS where the energy and spatial
dependence is considered, however, is much more involved as
CPSD ω
ð Þ ¼ CPSD C ω
ð Þ þ CPSD UN ω
ð Þ þ CPSD CS ω
ð Þ,
ð12:10Þ
CPSD C ω
ð Þ ¼
X 1
‘¼0
X 1
m¼0
X 1
n¼0
S ‘ F ‘!mn D 1, m D 2, n
α ‘ α m þ α n
ð
Þ
α n þ iω
ð
Þ
þ
X 1
‘¼0
X 1
m¼0
X 1
n¼0
S ‘ F ‘!mn D 2, m D 1, n
α ‘ α m þ α n
ð
Þ
α n À iω
ð
Þ
,
ð12:11Þ
CPSD UN ω
ð Þ ¼
2π q
2
T
2
X 1
n¼0
X 1
‘¼0
X 1
m¼À1
D 1, n D 2, ‘ Ψ
Ã
n Ψ
Ã
‘
δ ω À ω m
ð
Þ
α n À iω m
ð
Þα ‘ þ iω m
ð
Þ
,
ð12:12Þ
CPSD CS ω
ð Þ ¼ À
q
T
X 1
n¼0
X 1
‘¼0
D 1, n D 2, ‘ Ψ
Ã
n Ψ
Ã
‘
1
α n þ α ‘
Á
1
α ‘ þ iω
À
q
T
X 1
n¼0
X 1
‘¼0
D 2, n D 1, ‘ Ψ
Ã
n Ψ
Ã
‘
1
α n þ α ‘
Á
1
α ‘ À iω
,
ð12:13Þ
where i ¼
ffiffiffiffiffiffi ffi
À1
p
, ω m ¼ 2πm/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, CPSD UN (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
12 Theory of Power Spectral Density and Feynman-Alpha Method. . .
125