64
2 Critical Mass, Efficiency, and Yield
desired value; (2.30) can then be solved for x, which gives d from (2.26) and hence
α from (2.25). If α > 0, the reaction will in principle grow exponentially in time until
all of the fissile material is used up, a situation known as “supercriticality” as in the
Trinity core discussed above.
To see why increasing the radius demands that α must increase, implicitly differentiate (2.30) to show that dε
dx = −
1
x
2
1 − x
2
sin
2 x
. This expression
demands dε/dx > 0 for all values of x. From the definition of x, an increase in r (and/or
in the density, for that matter) will cause x to increase. To keep (2.30) satisfied means
that ε must increase, which, from (2.30), can happen only if α increases.
We come now to a very important point. This is that the condition for threshold
criticality can in general be expressed as a constraint on the product ρ r, where ρ is
the mass density of the material and r is the core radius. Recall that ε in (2.30) is
independent of the density. Hence, for α = 0, (2.30) will be satisfied by some unique
value of x which will be characteristic of the material being considered (Fig. 2.5).
Since x = r/d and d itself is proportional to 1/ρ [see (2.25)], we can equivalently say
that the solution of (2.30) demands a unique value of ρ r for a given combination of
values of σ and ν. If R o is the bare threshold critical radius for material of normal
density ρ o , then any combination of r and ρ such that ρ r = ρ o R o will also be
threshold critical, and any combination such that ρ r > ρ o R o will be supercritical.
For a sphere of material of mass M, the mass, density, and radius relate as M ∝ ρ
r
3 , which means that the “criticality product” ρ r can be written as ρ r ∝ M/r
2 .
This relationship underlies the concept of implosion weapons. If a sufficiently strong
implosion can be achieved, then one can get away with having less than a “normal”
critical mass by starting with a sphere of material of normal density and crushing
it to high density by implosion; such weapons inherently make more efficient use
of available fissile material than those that depend on a non-implosive mechanism
to assemble subcritical components. As described in Sect. 4.2, the implosion technique also helps to overcome predetonation issues with spontaneous fission. The key
message here is that there is no unique critical mass for a given fissile material.
Lest you think that publishing estimates of critical masses is engaging in revealing
classified data, do not be alarmed; such estimates have been available in the public
domain for decades. In a review article on fast-neutron reactors, Koch and Paxton
(1959) quote a value of 48.7 kg for a spherical assembly of highly enriched uranium
(93.9%
235 U), and 16.6 kg for a sphere of
239 Pu. A 1963 publication of the United
States Atomic Energy Commission, “Reactor Physics Constants,” a compilation of
data for nuclear engineers, lists the experimentally determined bare critical mass for
93.9%
235 U as 48.8 kg, and that for
239 Pu as 16.3 kg. Both values are close to those
listed in Table 2.1. Estimating a critical mass is one of the least difficult parts of
making a nuclear weapon.
Spreadsheet CriticalityAnalytic.xls allows users to carry out the above calculations for themselves. This spreadsheet is also used for calculations developed in
Sects. 2.3 and 2.5. In its simplest use—corresponding to this section—the user enters
five parameters: the density, atomic weight, fission and scattering cross-sections of
the core material, and the number of secondary neutrons per fission. The “Goal Seek”
2 Critical Mass, Efficiency, and Yield
desired value; (2.30) can then be solved for x, which gives d from (2.26) and hence
α from (2.25). If α > 0, the reaction will in principle grow exponentially in time until
all of the fissile material is used up, a situation known as “supercriticality” as in the
Trinity core discussed above.
To see why increasing the radius demands that α must increase, implicitly differentiate (2.30) to show that dε
dx = −
1
x
2
1 − x
2
sin
2 x
. This expression
demands dε/dx > 0 for all values of x. From the definition of x, an increase in r (and/or
in the density, for that matter) will cause x to increase. To keep (2.30) satisfied means
that ε must increase, which, from (2.30), can happen only if α increases.
We come now to a very important point. This is that the condition for threshold
criticality can in general be expressed as a constraint on the product ρ r, where ρ is
the mass density of the material and r is the core radius. Recall that ε in (2.30) is
independent of the density. Hence, for α = 0, (2.30) will be satisfied by some unique
value of x which will be characteristic of the material being considered (Fig. 2.5).
Since x = r/d and d itself is proportional to 1/ρ [see (2.25)], we can equivalently say
that the solution of (2.30) demands a unique value of ρ r for a given combination of
values of σ and ν. If R o is the bare threshold critical radius for material of normal
density ρ o , then any combination of r and ρ such that ρ r = ρ o R o will also be
threshold critical, and any combination such that ρ r > ρ o R o will be supercritical.
For a sphere of material of mass M, the mass, density, and radius relate as M ∝ ρ
r
3 , which means that the “criticality product” ρ r can be written as ρ r ∝ M/r
2 .
This relationship underlies the concept of implosion weapons. If a sufficiently strong
implosion can be achieved, then one can get away with having less than a “normal”
critical mass by starting with a sphere of material of normal density and crushing
it to high density by implosion; such weapons inherently make more efficient use
of available fissile material than those that depend on a non-implosive mechanism
to assemble subcritical components. As described in Sect. 4.2, the implosion technique also helps to overcome predetonation issues with spontaneous fission. The key
message here is that there is no unique critical mass for a given fissile material.
Lest you think that publishing estimates of critical masses is engaging in revealing
classified data, do not be alarmed; such estimates have been available in the public
domain for decades. In a review article on fast-neutron reactors, Koch and Paxton
(1959) quote a value of 48.7 kg for a spherical assembly of highly enriched uranium
(93.9%
235 U), and 16.6 kg for a sphere of
239 Pu. A 1963 publication of the United
States Atomic Energy Commission, “Reactor Physics Constants,” a compilation of
data for nuclear engineers, lists the experimentally determined bare critical mass for
93.9%
235 U as 48.8 kg, and that for
239 Pu as 16.3 kg. Both values are close to those
listed in Table 2.1. Estimating a critical mass is one of the least difficult parts of
making a nuclear weapon.
Spreadsheet CriticalityAnalytic.xls allows users to carry out the above calculations for themselves. This spreadsheet is also used for calculations developed in
Sects. 2.3 and 2.5. In its simplest use—corresponding to this section—the user enters
five parameters: the density, atomic weight, fission and scattering cross-sections of
the core material, and the number of secondary neutrons per fission. The “Goal Seek”