2.5 Estimating Yield—Analytic
83
The total number of fissions per second would be this rate times the number of
neutrons in the core. The latter will be the product of the number density N (t) =
N o e
(α/τ ) t from (2.22) times the volume V of the core. Hence we have
f issions/sec =
N o V
τ
e
(α/τ ) t
.
(2.83)
In this expression, α is given by solving (2.25), (2.30), and (2.31) for the core at
hand, and N o is the neutron density at the center of the core at t = 0; this will be
set by the number of neutrons released by some “initiator” device. Recall that α = 0
for threshold criticality, whereas α > 0 for a core of more than one critical mass, an
issue to which we will return shortly.
Equation (2.83) is actually more complicated than it appears, because α and τ are
functions of time. To appreciate this, consider a core of some general radius r and
density ρ. As the core expands, r will increase while ρ decreases. The decreasing
density will cause τ to increase; simultaneously, the discussion following (2.31)
indicates that we can expect α to decrease. For sake of simplicity, we assume that α
and τ remain constant; not accounting for changes in them will lead to overestimating
the fission rate in (2.83). Since an exponential function is involved, the overestimate
could be serious; indeed, we will see in Sect. 2.6 that direct use of our resulting yield
formula, Eq. (2.95), can easily result in overestimating the efficiency by an order of
magnitude. For the present, however, we will stick with the assumption of constant
α and τ values since the purpose here is to get a sense of how the expected yield
and efficiency depend on the various factors involved. Section 2.6 discusses a simple
refinement to (2.83) that eliminates much of the overestimate.
The time required to fission the entire core can be computed by demanding that
the integral of (2.83) from time zero to time t fiss to be equal to the total number of
nuclei within the core, nV:
nV =
N o V
τ
t f iss
0
e
(α/τ ) t dt ⇒ t f iss =
τ
α
ln
α n
N o
,
(2.84)
where it has been assumed that e
(α/τ )t
>> 1 for the timescale of interest, an
assumption to be investigated a posteriori.
What happens as the exploding core expands? Recall from Sect. 2.2 that the condition for criticality can be expressed as ρ r > K, where K is a constant characteristic
of the material being used. We also saw that for a core of some mass M, ρ r ∝ M/r
2 .
As the core expands, the value of ρ r must drop, and will eventually fall below the
level needed to maintain criticality. This “criticality shutdown” situation will obtain
at time t crit , and is technically known as second criticality.
For a single critical mass of normal-density material, second criticality will occur
as soon as the expansion begins. One way to circumvent this is to provide a tamper
to momentarily retard the expansion and so to give the reaction time to build up to a
83
The total number of fissions per second would be this rate times the number of
neutrons in the core. The latter will be the product of the number density N (t) =
N o e
(α/τ ) t from (2.22) times the volume V of the core. Hence we have
f issions/sec =
N o V
τ
e
(α/τ ) t
.
(2.83)
In this expression, α is given by solving (2.25), (2.30), and (2.31) for the core at
hand, and N o is the neutron density at the center of the core at t = 0; this will be
set by the number of neutrons released by some “initiator” device. Recall that α = 0
for threshold criticality, whereas α > 0 for a core of more than one critical mass, an
issue to which we will return shortly.
Equation (2.83) is actually more complicated than it appears, because α and τ are
functions of time. To appreciate this, consider a core of some general radius r and
density ρ. As the core expands, r will increase while ρ decreases. The decreasing
density will cause τ to increase; simultaneously, the discussion following (2.31)
indicates that we can expect α to decrease. For sake of simplicity, we assume that α
and τ remain constant; not accounting for changes in them will lead to overestimating
the fission rate in (2.83). Since an exponential function is involved, the overestimate
could be serious; indeed, we will see in Sect. 2.6 that direct use of our resulting yield
formula, Eq. (2.95), can easily result in overestimating the efficiency by an order of
magnitude. For the present, however, we will stick with the assumption of constant
α and τ values since the purpose here is to get a sense of how the expected yield
and efficiency depend on the various factors involved. Section 2.6 discusses a simple
refinement to (2.83) that eliminates much of the overestimate.
The time required to fission the entire core can be computed by demanding that
the integral of (2.83) from time zero to time t fiss to be equal to the total number of
nuclei within the core, nV:
nV =
N o V
τ
t f iss
0
e
(α/τ ) t dt ⇒ t f iss =
τ
α
ln
α n
N o
,
(2.84)
where it has been assumed that e
(α/τ )t
>> 1 for the timescale of interest, an
assumption to be investigated a posteriori.
What happens as the exploding core expands? Recall from Sect. 2.2 that the condition for criticality can be expressed as ρ r > K, where K is a constant characteristic
of the material being used. We also saw that for a core of some mass M, ρ r ∝ M/r
2 .
As the core expands, the value of ρ r must drop, and will eventually fall below the
level needed to maintain criticality. This “criticality shutdown” situation will obtain
at time t crit , and is technically known as second criticality.
For a single critical mass of normal-density material, second criticality will occur
as soon as the expansion begins. One way to circumvent this is to provide a tamper
to momentarily retard the expansion and so to give the reaction time to build up to a