66
2 Critical Mass, Efficiency, and Yield
As an example of how good an estimate (2.34) provides, we consider the
Hiroshima Little Boy bomb core. It is described in the Preamble that this core had
a total mass of ~66 kg. If we take this to be pure
235 U of density 18.71 gr/cm
3 for
sake of argument, this would correspond to a radius R core = 9.443 cm. With R o =
8.366 cm and ν = 2.637 from Table 2.1, (2.34) gives α core ~ 0.352. The true value
for α for such a core (if pure
235 U) is 0.277. The approximation is about 27% high:
not terribly accurate, but certainly in the ballpark. (The Little Boy core was actually
cylindrical, so we have taken some liberty in this example for sake of simplicity.)
To close this section, it is interesting to look briefly at a famous miscalculation
of critical mass on the part of Werner Heisenberg. At the end of World War II, a
number of prominent German physicists including Heisenberg were interned for six
months in England and their conversations secretly recorded. This story is detailed
in Bernstein (2001); see also Logan (1996) and Bernstein (2002). On the evening of
August 6, 1945, the internees were informed that an atomic bomb had been dropped
on Hiroshima, and that the energy released was equivalent to about 20,000 tons of
TNT. (In actuality, the yield was about 13,000 tons, but this is not the problem with
Heisenberg’s calculation.) Heisenberg then estimated the critical mass based on this
number and a subtly erroneous model of the fission process.
We saw in Sect. 1.6 that complete fission of 1 kg of
235 U liberates energy equivalent
to about 17 kilotons of TNT. Heisenberg predicated his estimate of the critical mass
on assuming that about 1 kg of material did in fact fission. One kilogram of
235 U
corresponds to about Ω ~ 2.56 × 10
24 nuclei. Assuming that on average ν = 2
neutrons are liberated per fission, then the number of generations G necessary to
fission the entire kilogram would be ν
G
= Ω. Solving for G gives G = ln(Ω)/ln(ν) ~
81, which Heisenberg rounded to 80. So far, this calculation is fine. He then argued
that as neutrons fly around in the bomb core, they will randomly bounce between
nuclei, traveling a mean distance λ f before causing fissions; λ f is the mean free path
between fissions as in (2.15) above. From Table 2.1, λ f ~ 17 cm for
235 U, but, at the
time, Heisenberg took λ f ~ 6 cm. Since a random walk of G steps where each is of
length λ f will take one a distance r ∼ λ f
√
G from the starting point, he estimated a
critical radius of r ∼ (6 cm)
√
80 ~ 54 cm. This would correspond to a mass of some
12,500 kg, roughly 13 tons! Given that only one kilogram of uranium fissioned, this
would be a fantastically inefficient weapon. Such a bomb and its associated tamper,
casing, and instrumentation would represent an unbearably heavy load for a World
War II-era bomber.
The problem with Heisenberg’s calculation was that he imagined the fission
process to be created by a single neutron that randomly bounced throughout the
bomb core, begetting secondary neutrons along the way. Further, his model is too
stringent; there is no need for every neutron to cause a fission; many neutrons escape.
In the days following August 6, Heisenberg revised his model, arriving at the diffusion
theory approach described in this section.
2 Critical Mass, Efficiency, and Yield
As an example of how good an estimate (2.34) provides, we consider the
Hiroshima Little Boy bomb core. It is described in the Preamble that this core had
a total mass of ~66 kg. If we take this to be pure
235 U of density 18.71 gr/cm
3 for
sake of argument, this would correspond to a radius R core = 9.443 cm. With R o =
8.366 cm and ν = 2.637 from Table 2.1, (2.34) gives α core ~ 0.352. The true value
for α for such a core (if pure
235 U) is 0.277. The approximation is about 27% high:
not terribly accurate, but certainly in the ballpark. (The Little Boy core was actually
cylindrical, so we have taken some liberty in this example for sake of simplicity.)
To close this section, it is interesting to look briefly at a famous miscalculation
of critical mass on the part of Werner Heisenberg. At the end of World War II, a
number of prominent German physicists including Heisenberg were interned for six
months in England and their conversations secretly recorded. This story is detailed
in Bernstein (2001); see also Logan (1996) and Bernstein (2002). On the evening of
August 6, 1945, the internees were informed that an atomic bomb had been dropped
on Hiroshima, and that the energy released was equivalent to about 20,000 tons of
TNT. (In actuality, the yield was about 13,000 tons, but this is not the problem with
Heisenberg’s calculation.) Heisenberg then estimated the critical mass based on this
number and a subtly erroneous model of the fission process.
We saw in Sect. 1.6 that complete fission of 1 kg of
235 U liberates energy equivalent
to about 17 kilotons of TNT. Heisenberg predicated his estimate of the critical mass
on assuming that about 1 kg of material did in fact fission. One kilogram of
235 U
corresponds to about Ω ~ 2.56 × 10
24 nuclei. Assuming that on average ν = 2
neutrons are liberated per fission, then the number of generations G necessary to
fission the entire kilogram would be ν
G
= Ω. Solving for G gives G = ln(Ω)/ln(ν) ~
81, which Heisenberg rounded to 80. So far, this calculation is fine. He then argued
that as neutrons fly around in the bomb core, they will randomly bounce between
nuclei, traveling a mean distance λ f before causing fissions; λ f is the mean free path
between fissions as in (2.15) above. From Table 2.1, λ f ~ 17 cm for
235 U, but, at the
time, Heisenberg took λ f ~ 6 cm. Since a random walk of G steps where each is of
length λ f will take one a distance r ∼ λ f
√
G from the starting point, he estimated a
critical radius of r ∼ (6 cm)
√
80 ~ 54 cm. This would correspond to a mass of some
12,500 kg, roughly 13 tons! Given that only one kilogram of uranium fissioned, this
would be a fantastically inefficient weapon. Such a bomb and its associated tamper,
casing, and instrumentation would represent an unbearably heavy load for a World
War II-era bomber.
The problem with Heisenberg’s calculation was that he imagined the fission
process to be created by a single neutron that randomly bounced throughout the
bomb core, begetting secondary neutrons along the way. Further, his model is too
stringent; there is no need for every neutron to cause a fission; many neutrons escape.
In the days following August 6, Heisenberg revised his model, arriving at the diffusion
theory approach described in this section.