30
K. Hashimoto
repetition frequency and the beam width were set to 30 Hz and 100 ns, respectively.
The proton beam intensity of an accelerator was set to 30 pA.
2.2.2 Formula for Power Spectral Analyses
2.2.2.1 Degweker’s Formula
Degweker and Rana [8] formulated the auto- and cross-power spectral densities for
a periodically pulsed non-Poisson source, where delayed neutron contribution was
neglected and each pulse was assumed to be a delta function. The neglect of delayed
neutron contribution is acceptable as mentioned in the above Sect. 2.2.1. Since the
pulse width 100 ns of our accelerator is much shorter than the time scale of the
present analysis, the assumption of the delta function is also acceptable. Here, their
formulae are rewritten as a function of not angular frequency ω s
−1 but frequency f
Hz. Then, their cross-power spectral density between neutron detector 1 and 2 can
be described as follows:
Φ 12 ( f ) =
C 1 ( f )
(2 π f )
2
+ α 2 + C 2 ( f )
∞
n = −∞
δ ( f − n f R )
(2 π f )
2
+ α 2 ,
(2.37)
where
C 1 ( f ) = H ( f ) q
2 f R λ d1 λ d2
m 2 − m
2
1 + 2m 1 Y 1
,
(2.38)
C 2 ( f ) = H ( f ) q
2 f
2
R λ d1 λ d2 m
2
1 ,
(2.39)
Y 1 = λ f
ν (ν − 1)
2 α
.
(2.40)
Their auto-power spectral density of neutron detector 1 can be also described as
follows:
Φ 11 ( f ) = C 3 ( f ) +
C 4 ( f )
(2 π f )
2
+ α 2 + C 5 ( f )
∞
n = −∞
δ( f − n f R )
(2 π f )
2
+ α 2 , (2.41)
where
C 3 ( f ) =
H ( f ) q
2 f R λ d1 m 1
α
,
(2.42)
C 4 ( f ) = H ( f ) q
2 f R λ
2
d1
m 2 − m
2
1 + 2 m 1 Y 1
,
(2.43)