theoretical value of the sum of Y C (Δ) and Y CS (Δ) calculated with Eqs. (12.3)
and (12.4). “Theory total” shows “Theory (correlated)” plus the theoretical value
of the uncorrelated component Y UN (Δ), calculated with Eq. (12.5). The neutron flux
and α m , which are needed to calculate the theoretical values of Eqs. (12.3), (12.4),
and (12.5), are calculated with the Monte Carlo method up to the third-order mode
[16]. Beyond the third order, those are approximated with the diffusion theory:
α m ¼ υ Σ f þ Σ c þ DB
2
m À νΣ f
À
Á
, m ! 4
ð12:6Þ
ψ m x
ð Þ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
2υ
H þ 2d
r
sin B m x þ d
ð
Þ, m ! 4
ð12:7Þ
B m ¼
m þ 1
ð
Þπ
H þ 2d
:
ð12:8Þ
where d ¼ extrapolated length(¼0.7104/Σ t ). The summation in Eqs. (12.3), (12.4),
and (12.5) is taken up to the 250th mode. As shown in Fig. 12.2, there is good
agreement between the Monte Carlo simulation and the theory, which shows
verification of the theoretical formula of Eq. (12.2). Using Eqs. (12.3), (12.4), and
(12.5), the Feynman Y function is decomposed into mode components. Figures 12.3
and 12.4 show the mode components of the correlated component and of the
uncorrelated component, respectively. In these figures, each mode component
includes the cross terms with the lower-order mode components. For example,
“1st higher” includes the cross terms between the fundamental mode and the first
higher-order mode as well as the first higher-order mode itself. Figure 12.1 shows
that detector 1 is located at the bottom of the first higher-order mode. Thus, the first
higher-order mode has a significant effect on the Feynman Y function of detector
1. Especially, the higher-order mode is more remarkable in the uncorrelated
component, as shown in Fig. 12.4.
0.0
0.2
0.4
0.6
0.8
1.0
0
0.01
0.02
0.03
Fundamental
1st higher
2nd higher
3rd higher
Total
Y (D)
Counting gate width D (s)
Fig. 12.3 Mode components of the correlated component in the Feynman Y function
12 Theory of Power Spectral Density and Feynman-Alpha Method. . .
123
and (12.4). “Theory total” shows “Theory (correlated)” plus the theoretical value
of the uncorrelated component Y UN (Δ), calculated with Eq. (12.5). The neutron flux
and α m , which are needed to calculate the theoretical values of Eqs. (12.3), (12.4),
and (12.5), are calculated with the Monte Carlo method up to the third-order mode
[16]. Beyond the third order, those are approximated with the diffusion theory:
α m ¼ υ Σ f þ Σ c þ DB
2
m À νΣ f
À
Á
, m ! 4
ð12:6Þ
ψ m x
ð Þ ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
2υ
H þ 2d
r
sin B m x þ d
ð
Þ, m ! 4
ð12:7Þ
B m ¼
m þ 1
ð
Þπ
H þ 2d
:
ð12:8Þ
where d ¼ extrapolated length(¼0.7104/Σ t ). The summation in Eqs. (12.3), (12.4),
and (12.5) is taken up to the 250th mode. As shown in Fig. 12.2, there is good
agreement between the Monte Carlo simulation and the theory, which shows
verification of the theoretical formula of Eq. (12.2). Using Eqs. (12.3), (12.4), and
(12.5), the Feynman Y function is decomposed into mode components. Figures 12.3
and 12.4 show the mode components of the correlated component and of the
uncorrelated component, respectively. In these figures, each mode component
includes the cross terms with the lower-order mode components. For example,
“1st higher” includes the cross terms between the fundamental mode and the first
higher-order mode as well as the first higher-order mode itself. Figure 12.1 shows
that detector 1 is located at the bottom of the first higher-order mode. Thus, the first
higher-order mode has a significant effect on the Feynman Y function of detector
1. Especially, the higher-order mode is more remarkable in the uncorrelated
component, as shown in Fig. 12.4.
0.0
0.2
0.4
0.6
0.8
1.0
0
0.01
0.02
0.03
Fundamental
1st higher
2nd higher
3rd higher
Total
Y (D)
Counting gate width D (s)
Fig. 12.3 Mode components of the correlated component in the Feynman Y function
12 Theory of Power Spectral Density and Feynman-Alpha Method. . .
123