2.2 Critical Mass: Bare Core
63
Fig. 2.5 Solution of (2.30)
for α = 0
0
0.5
1
1.5
2
2.5
3
0
0 . 5
1
1 . 5
2
x ( = 0)
over the fission-energy spectra of the two nuclides. The ν values were adopted from
the Evaluated Nuclear Data Files (ENDF) maintained by the National Nuclear Data
Center at Brookhaven National Laboratory (www.nndc.bnl.gov), and are for neutrons
of energy 2 MeV, about the average energy of fission neutrons. The density for
235 U is
(235/238) times the density of natural uranium, 18.95 gr/cm
3 . It is worth noting that
the timescales involved in fission-bomb phenomena are remarkably brief: Neutrons
travel for only τ ~ 1/100 of a microsecond (= 10 ns) between fissions!
Figure 2.5 shows the solution of (2.30) for x when α = 0 over the range of ε likely
to be of practical interest. Readers should check that the values of ε, d, and radii in
Table 2.1 are consistent with this plot. In Sect. 2.7, a similar plot is developed using
a parameter akin to ε but of interesting historical provenance.
An important aspect of Fig. 2.5 is that it also applies in cases where the core has
been compressed, as in an implosion bomb. For a given core material, ε is independent
of density, so the value of x will not change. However, d core is inversely proportional
to density, so the critical radius R crit = xd core , will be reduced. If the compression
factor (the factor by which the density is increased) is C, then R crit will be reduced
to R o /C, and the critical mass, which is proportional to ρR
3 , will consequently be
reduced by a factor of C
2 . For example, consider
239 Pu, which has an uncompressed
bare critical mass of ~16.7 kg. If C = 2.5 [about what was achieved in the Trinity
test as indicated by analysis of fallout products; Semkow et al. (2006)], the critical
mass would reduce to ~2.67 kg. If one were not aware of the bare critical mass but
knew that ε = 1.09 for this isotope, Fig. 2.5 indicates x ~ 2.1 (the precise value is
2.125). The normal (uncompressed value) of d is 2.99 cm, so the compressed R crit
becomes ~(2.1)(2.99/2.5) ~ 2.51 cm. With the density increased to (15.6 g cm
−3 )(2.5)
= 39 g cm
−3 , this radius gives a mass of ~2.6 kg, close enough to the precise value.
The 6.3 kg core of the Trinity device, when compressed, was thus more than critical.
The second way of solving (2.30) begins with assuming that one has a core of
some radius r > R o . In this case one will find that (2.30) will be satisfied by some
value of α > 0, with α increasing as r increases. The rationale here is that since the
middle term in (2.30), εx = 3r
2λ t , is independent of α, we can set r to some
63
Fig. 2.5 Solution of (2.30)
for α = 0
0
0.5
1
1.5
2
2.5
3
0
0 . 5
1
1 . 5
2
x ( = 0)
over the fission-energy spectra of the two nuclides. The ν values were adopted from
the Evaluated Nuclear Data Files (ENDF) maintained by the National Nuclear Data
Center at Brookhaven National Laboratory (www.nndc.bnl.gov), and are for neutrons
of energy 2 MeV, about the average energy of fission neutrons. The density for
235 U is
(235/238) times the density of natural uranium, 18.95 gr/cm
3 . It is worth noting that
the timescales involved in fission-bomb phenomena are remarkably brief: Neutrons
travel for only τ ~ 1/100 of a microsecond (= 10 ns) between fissions!
Figure 2.5 shows the solution of (2.30) for x when α = 0 over the range of ε likely
to be of practical interest. Readers should check that the values of ε, d, and radii in
Table 2.1 are consistent with this plot. In Sect. 2.7, a similar plot is developed using
a parameter akin to ε but of interesting historical provenance.
An important aspect of Fig. 2.5 is that it also applies in cases where the core has
been compressed, as in an implosion bomb. For a given core material, ε is independent
of density, so the value of x will not change. However, d core is inversely proportional
to density, so the critical radius R crit = xd core , will be reduced. If the compression
factor (the factor by which the density is increased) is C, then R crit will be reduced
to R o /C, and the critical mass, which is proportional to ρR
3 , will consequently be
reduced by a factor of C
2 . For example, consider
239 Pu, which has an uncompressed
bare critical mass of ~16.7 kg. If C = 2.5 [about what was achieved in the Trinity
test as indicated by analysis of fallout products; Semkow et al. (2006)], the critical
mass would reduce to ~2.67 kg. If one were not aware of the bare critical mass but
knew that ε = 1.09 for this isotope, Fig. 2.5 indicates x ~ 2.1 (the precise value is
2.125). The normal (uncompressed value) of d is 2.99 cm, so the compressed R crit
becomes ~(2.1)(2.99/2.5) ~ 2.51 cm. With the density increased to (15.6 g cm
−3 )(2.5)
= 39 g cm
−3 , this radius gives a mass of ~2.6 kg, close enough to the precise value.
The 6.3 kg core of the Trinity device, when compressed, was thus more than critical.
The second way of solving (2.30) begins with assuming that one has a core of
some radius r > R o . In this case one will find that (2.30) will be satisfied by some
value of α > 0, with α increasing as r increases. The rationale here is that since the
middle term in (2.30), εx = 3r
2λ t , is independent of α, we can set r to some