160
4 Complicating Factors
(t init ) f ull ≥ t O
1 −
2 τ F
α O t O
.
(4.20)
For the above parameters, this evaluates to about 9.5 μs.
In reality, one will most likely have (t O ) f ull < t init < (t init ) f ull , in which case
a partial-yield explosion will occur. To examine this situation, take the solution of
(4.16) for the time at which e
F fissions have occurred, and use (4.15) and (4.17) to
determine the yield:
Y
Y O
=
t init
t O
3
1 +
2 τ t O F
α O t
2
init
3/2
.
(4.21)
You should be able to show that this expression gives the correct minimum yield
of (4.18) for t init → 0.
To estimate the probability of achieving the yield predicted by (4.21) requires a
model for the spontaneous-fission (SF) predetonation-probability characteristics of
the core. As remarked earlier, the premise here is that if P is the probability that a
predetonation does not occur over the time interval (0, t init ), then one can state that
the chance of obtaining at least the yield predicted by (4.21) is 100P percent. By
investigating the situation for various values of t init , we can build up a plot of the
probability of achieving a given fractional yield as a function of fractional yield.
The predetonation model was developed in the preceding section. As applied to
the current situation, the average number of SFs during some span of time (0, t init )
can be written from (4.5) as
μ = M Rt init ,
(4.22)
where M is the mass of spontaneously fissioning material and R the rate of SFs.
[R is used here to designate the rate of SFs as opposed to F, as the latter symbol
is already taken in (4.16). Do not confuse R with the core radius R core , which is
introduced below.] If each SF releases on average ν neutrons, the probability of not
experiencing a predetonation during the selected time interval is, from (4.6) and
(4.7),
Pno
predet
= e
−μ
k=0
μ
k
k!
P escape
kν ,
(4.23)
where P escape is the probability that an individual neutron will escape from the core
without causing a fission. Reminder: As in Chap. 2, we avoid the issue of a spectrum
of neutron-number emission by using an average “effective” value ν.
As described in the preceding section, P escape depends on the maximum number of
times S that one is willing to allow neutrons to scatter before escaping the core. The
case which results in the highest probability of predetonation (that is, the worst-case
4 Complicating Factors
(t init ) f ull ≥ t O
1 −
2 τ F
α O t O
.
(4.20)
For the above parameters, this evaluates to about 9.5 μs.
In reality, one will most likely have (t O ) f ull < t init < (t init ) f ull , in which case
a partial-yield explosion will occur. To examine this situation, take the solution of
(4.16) for the time at which e
F fissions have occurred, and use (4.15) and (4.17) to
determine the yield:
Y
Y O
=
t init
t O
3
1 +
2 τ t O F
α O t
2
init
3/2
.
(4.21)
You should be able to show that this expression gives the correct minimum yield
of (4.18) for t init → 0.
To estimate the probability of achieving the yield predicted by (4.21) requires a
model for the spontaneous-fission (SF) predetonation-probability characteristics of
the core. As remarked earlier, the premise here is that if P is the probability that a
predetonation does not occur over the time interval (0, t init ), then one can state that
the chance of obtaining at least the yield predicted by (4.21) is 100P percent. By
investigating the situation for various values of t init , we can build up a plot of the
probability of achieving a given fractional yield as a function of fractional yield.
The predetonation model was developed in the preceding section. As applied to
the current situation, the average number of SFs during some span of time (0, t init )
can be written from (4.5) as
μ = M Rt init ,
(4.22)
where M is the mass of spontaneously fissioning material and R the rate of SFs.
[R is used here to designate the rate of SFs as opposed to F, as the latter symbol
is already taken in (4.16). Do not confuse R with the core radius R core , which is
introduced below.] If each SF releases on average ν neutrons, the probability of not
experiencing a predetonation during the selected time interval is, from (4.6) and
(4.7),
Pno
predet
= e
−μ
k=0
μ
k
k!
P escape
kν ,
(4.23)
where P escape is the probability that an individual neutron will escape from the core
without causing a fission. Reminder: As in Chap. 2, we avoid the issue of a spectrum
of neutron-number emission by using an average “effective” value ν.
As described in the preceding section, P escape depends on the maximum number of
times S that one is willing to allow neutrons to scatter before escaping the core. The
case which results in the highest probability of predetonation (that is, the worst-case