2 Subcriticality
19
p (τ ) =
λ d ¯
ν
2
2
7
i = 1
α i Y i e
− α i τ
λ f +
m 2 − m
2
1
(−ρ)
m 1 ν (ν − 1) Λ
+
λ d m 1 ¯
ν
2
(−ρ)
2 ν (ν − 1) Λ
7
i = 1
α i Y i
⎧
⎨
⎩
e
− α i
τ −
[ f τ ]
f
+ e
−
α i
f e
α i
τ −
[ f τ ]
f
1 − e
−
α i
f
⎫
⎬
⎭
. (2.18)
The conditional counting probability p(τ )τ is a probability that, given a neutron
count at a time, there is a subsequent count in τ around time τ later. First, the time
interval τ of the Rossi-α analysis is restricted within the following range:
τ
1
α i
i = 1, 2, . . . , 6.
(2.19)
Under the above time-interval range, the following Maclaurin expansions can be
done:
e
−α i τ
1 − α i τ,
(2.20)
e
± α i
τ −
[ f τ ]
f
1 ± α i
τ −
[ f τ ]
f
+
1
2
α
2
i
τ −
[ f τ ]
f
2
.
(2.21)
We substitute the above equations into Eq. (2.18) to obtain the final form:
p(τ ) )τ = C 5 e
− α τ
+ C 6
⎧
⎨
⎩
e
− α
τ −
[ f τ ]
f
+ e
α
τ −
[ f τ ] + 1
f
1 − e
−
α
f
⎫
⎬
⎭
+ C 7 − C 8 τ,
(2.22)
where
C 5 =
λ d ¯
ν
2
2
α 7 Y 7
λ f +
m 2 − m
2
1
(−ρ)
m 1 ν (ν − 1) Λ
τ,
(2.23)
C 6 =
λ d m 1 ¯
ν
2
(−ρ)
2 ν (ν − 1) Λ
α 7 Y 7 τ,
(2.24)
C 7 =
λ d ¯
ν
2
2
6
i = 1
α i Y i
λ f +
m 2 − m
2
1
(−ρ)
m 1 ν (ν − 1) Λ
τ
+
f λ d m 1 ¯
ν
2
(−ρ)
ν (ν − 1) Λ
6
i = 1
Y i τ,
(2.25)