152
4 Complicating Factors
spontaneously fissile material, the maximum number of scatterings to be considered,
and the assembly timescale. To calculate the sum in (4.13), the spreadsheets take
an upper limit of k = 20, which is entirely sufficient for any reasonable situation.
Results of such calculations are described in the following two subsections.
4.2.1 Little Boy Predetonation Probability
As described in the Preamble, the Hiroshima Little Boy core comprised about 66 kg
of uranium in a cylindrical configuration, of which about 80% was
235 U and 20%
(13.2 kg) was
238 U. The half-life of
238 U for spontaneous fission, 8.2 × 10
15 year,
is about 1200 times shorter than that of
235 U, rendering the latter isotope almost
negligible as far as the predetonation probability is concerned. As in Sect. 2.3, I
model the (bare) core of Little Boy as being spherical; a 66 kg sphere of density
18.71 gm cm
−3 has a radius of 9.44 cm. For a 200 μs assembly time, an average
of only 0.018 spontaneous fissions will occur; the probability that no spontaneous
fissions will occur at all is 98.2%. The non-predetonation probability evaluates as
about 98.3% for S max = 0, and as 98.8% for S max = 5 (probably too large). At worst,
fizzles could be expected to occur in about two such bombs out of every one-hundred.
The spherically-averaged direct escape probability
P sph
for this 66 kg core is 0.265;
for S max = 5, P escape of (4.13) is 0.606. For a 100 μs assembly time, the mean number
of spontaneous fissions is only about 0.009, and the probability of no pre-detonation
for S max = 0 rises to 99.2% (S max = 0).
4.2.2 Fat Man Predetonation Probability
The untamped critical mass of
239 Pu is about 17 kg. However, the Trinity and Fat
Man bombs used cores of mass about 6.3 kg due to the greater efficiency afforded
by implosion. For a 6.3 kg core of pure
239 Pu, an assembly time of 200 μs yields
a no-predetonation probability of 99.2% (S max = 0). In reality, however, the cores
contained about 1.2% Pu-240 (0.0756 kg), which makes the outcome very different.
Figure 4.1 shows the S max = 0 non-predetonation probability for this case as a function
of the assembly time. For S max = 0 and a time of 100 μs, the non-predetonation
probability is only 5.5% (11.5% for S max = 1); there is consequently no realistic
hope of successfully assembling the core in a time scale characteristic of a gun
mechanism. Here
P sph
= 0.496, and, for S max = 5, P escape = 0.770. Although these
numbers do not differ much from those of the Little Boy calculation, the mean number
of spontaneous fissions is enormously greater in the case of the
240 Pu-contaminated
Fat Man device: 3.7 in comparison to 0.009 over 100 μs.
We can make a rough estimate of the imploded Trinity core non-predetonation
probability as follows. Neglecting the neutron initiator housed at its center, a 6.3-kg
core would have a radius of about 4.59 cm for a density of 15.6 gm cm
−3 . If the
4 Complicating Factors
spontaneously fissile material, the maximum number of scatterings to be considered,
and the assembly timescale. To calculate the sum in (4.13), the spreadsheets take
an upper limit of k = 20, which is entirely sufficient for any reasonable situation.
Results of such calculations are described in the following two subsections.
4.2.1 Little Boy Predetonation Probability
As described in the Preamble, the Hiroshima Little Boy core comprised about 66 kg
of uranium in a cylindrical configuration, of which about 80% was
235 U and 20%
(13.2 kg) was
238 U. The half-life of
238 U for spontaneous fission, 8.2 × 10
15 year,
is about 1200 times shorter than that of
235 U, rendering the latter isotope almost
negligible as far as the predetonation probability is concerned. As in Sect. 2.3, I
model the (bare) core of Little Boy as being spherical; a 66 kg sphere of density
18.71 gm cm
−3 has a radius of 9.44 cm. For a 200 μs assembly time, an average
of only 0.018 spontaneous fissions will occur; the probability that no spontaneous
fissions will occur at all is 98.2%. The non-predetonation probability evaluates as
about 98.3% for S max = 0, and as 98.8% for S max = 5 (probably too large). At worst,
fizzles could be expected to occur in about two such bombs out of every one-hundred.
The spherically-averaged direct escape probability
P sph
for this 66 kg core is 0.265;
for S max = 5, P escape of (4.13) is 0.606. For a 100 μs assembly time, the mean number
of spontaneous fissions is only about 0.009, and the probability of no pre-detonation
for S max = 0 rises to 99.2% (S max = 0).
4.2.2 Fat Man Predetonation Probability
The untamped critical mass of
239 Pu is about 17 kg. However, the Trinity and Fat
Man bombs used cores of mass about 6.3 kg due to the greater efficiency afforded
by implosion. For a 6.3 kg core of pure
239 Pu, an assembly time of 200 μs yields
a no-predetonation probability of 99.2% (S max = 0). In reality, however, the cores
contained about 1.2% Pu-240 (0.0756 kg), which makes the outcome very different.
Figure 4.1 shows the S max = 0 non-predetonation probability for this case as a function
of the assembly time. For S max = 0 and a time of 100 μs, the non-predetonation
probability is only 5.5% (11.5% for S max = 1); there is consequently no realistic
hope of successfully assembling the core in a time scale characteristic of a gun
mechanism. Here
P sph
= 0.496, and, for S max = 5, P escape = 0.770. Although these
numbers do not differ much from those of the Little Boy calculation, the mean number
of spontaneous fissions is enormously greater in the case of the
240 Pu-contaminated
Fat Man device: 3.7 in comparison to 0.009 over 100 μs.
We can make a rough estimate of the imploded Trinity core non-predetonation
probability as follows. Neglecting the neutron initiator housed at its center, a 6.3-kg
core would have a radius of about 4.59 cm for a density of 15.6 gm cm
−3 . If the