2 Subcriticality
17
Y (T ) =
¯
ν
2
λ d
m 1
7
i=1
Y i
1 −
1 − e
−α i T
α i T
m 1 λ f +
m 2 − m
2
1
(−ρ)
ν(ν − 1)Λ
+
¯
ν
2
λ d m 1 (−ρ)
ν(ν − 1)Λ
7
i=1
Y i
⎧
⎨
⎩
e
−α i
T −
[ f T ]
f
+ e
−
α i
f e
α i
T −
[ f T ]
f
− 1 − e
−
α i
f
T α i
1 − e
−
α i
f
⎫
⎬
⎭
,
+
λ d m 1 Λ
(−ρ)
1 + 2[ f T ] −
[ f T ]
f T
([ f T ] + 1) − f T
(2.4)
where f is pulse repetition frequency and [f T] represents largest integer less or
equal to f T. The largest α 7 is a prompt-neutron decay constant to be determined
and the other α i is a decay constant of each delayed-neutron mode. Other notations
are conventional except for m 1 and m 2 , which are first and second factorial moment
of source multiplicity distribution and are defined by the following equations [8],
respectively:
m 1 = ¯
N ν sp ,
(2.5)
m 2 = N (N − 1) ν sp
2
+ ¯
N ν sp
ν sp − 1
,
(2.6)
where N and ν sp are number of protons in a pulsed bunch and number of neutrons
produced by each spallation event, respectively. These definitions lead to the
following expression:
m 2 − m
2
1 =
N 2 − ¯
N − ¯
N
2
ν sp
2
+ ¯
N ν sp
ν sp − 1
.
(2.7)
The above quantity gives an expression for non-Poisson character of a neutron
source and is included in the first term of Eq. (2.4). When the proton number N
follows the Poisson distribution, the first term of Eq. (2.7) disappears. The second
term is expected to increase with an increase in proton energy but the present energy
100 MeV may lead to a small positive value.
Equation (2.4) is not available for least-squares fitting to the Y data because of a
complexity of the delayed neutron terms and many unknown parameters included in
the terms. Here, we reduce the rigorous equation to obtain a practical fitting formula.
First, the gate-time T of the Feynman-α analysis is restricted within the following
range:
T
1
α i
i = 1, 2, . . . , 6.
(2.8)
In the above range, the following Maclaurin expansions can be done:
e
− α i T
1 − α i T +
(− α i T )
2
2
,
(2.9)
17
Y (T ) =
¯
ν
2
λ d
m 1
7
i=1
Y i
1 −
1 − e
−α i T
α i T
m 1 λ f +
m 2 − m
2
1
(−ρ)
ν(ν − 1)Λ
+
¯
ν
2
λ d m 1 (−ρ)
ν(ν − 1)Λ
7
i=1
Y i
⎧
⎨
⎩
e
−α i
T −
[ f T ]
f
+ e
−
α i
f e
α i
T −
[ f T ]
f
− 1 − e
−
α i
f
T α i
1 − e
−
α i
f
⎫
⎬
⎭
,
+
λ d m 1 Λ
(−ρ)
1 + 2[ f T ] −
[ f T ]
f T
([ f T ] + 1) − f T
(2.4)
where f is pulse repetition frequency and [f T] represents largest integer less or
equal to f T. The largest α 7 is a prompt-neutron decay constant to be determined
and the other α i is a decay constant of each delayed-neutron mode. Other notations
are conventional except for m 1 and m 2 , which are first and second factorial moment
of source multiplicity distribution and are defined by the following equations [8],
respectively:
m 1 = ¯
N ν sp ,
(2.5)
m 2 = N (N − 1) ν sp
2
+ ¯
N ν sp
ν sp − 1
,
(2.6)
where N and ν sp are number of protons in a pulsed bunch and number of neutrons
produced by each spallation event, respectively. These definitions lead to the
following expression:
m 2 − m
2
1 =
N 2 − ¯
N − ¯
N
2
ν sp
2
+ ¯
N ν sp
ν sp − 1
.
(2.7)
The above quantity gives an expression for non-Poisson character of a neutron
source and is included in the first term of Eq. (2.4). When the proton number N
follows the Poisson distribution, the first term of Eq. (2.7) disappears. The second
term is expected to increase with an increase in proton energy but the present energy
100 MeV may lead to a small positive value.
Equation (2.4) is not available for least-squares fitting to the Y data because of a
complexity of the delayed neutron terms and many unknown parameters included in
the terms. Here, we reduce the rigorous equation to obtain a practical fitting formula.
First, the gate-time T of the Feynman-α analysis is restricted within the following
range:
T
1
α i
i = 1, 2, . . . , 6.
(2.8)
In the above range, the following Maclaurin expansions can be done:
e
− α i T
1 − α i T +
(− α i T )
2
2
,
(2.9)