52
2 Critical Mass, Efficiency, and Yield
not surrounded by a tamper, that is, a so-called bare or naked core. A tamper is a
heavy metal casing which enhances weapon efficiency in two ways: By reflecting
escaped neutrons back into the core and hence giving them fresh chances at causing
fissions, and by briefly retarding the violent expansion of the core in order to give the
chain reaction more time over which to operate. Providing a tamper can significantly
increase the efficiency of a weapon at a very low cost, and tampers were employed in
both the Little Boy and Fat Man bombs of the Manhattan Project. Tamped criticality is
taken up in Sect. 2.3. Another modification to bomb design which was contemplated
during the Manhattan Project but not utilized is that of constructing a composite core,
that is, one comprising nested shells of two different fissile materials. The point of
this is to optimize bomb production: If, for example, you are producing
235 U and
239 Pu at the same rate, you can in principle produce bombs more steadily if they are
designed with cores having equal amounts of each material, as opposed to waiting
until you have enough for one of each type. This description is loose in that the exact
mass ratio of the two materials that is necessary to achieve criticality will depend on
their individual fissility characteristics, but makes the point that bomb productivity
can be improved in this way. Tamped composite cores are analyzed in Sect. 2.4.
Sections 2.5 and 2.6 take up the issue of bomb efficiency and yield through analytic
approximations and a numerical simulation. Section 2.7 presents an alternate treatment of untamped criticality that has an interesting historical connection. Section 2.8,
which is new in this edition of this book, presents several approximate methods for
analyzing critical mass and bomb efficiency. The point of these less rigorous analyses
is to provide treatments that can be used to make quick estimates or which might be
suitable for classroom discussions when time is at a premium. It must be remembered,
however, that they are approximate treatments, and that for more precise answers one
should return to the fuller diffusion-theory treatments. Finally, Sect. 2.9 presents an
approximate treatment of criticality for cylindrical bomb cores.
For readers interested in further sources, an excellent account of the concept of
critical mass appears in Logan (1996); see also Bernstein (2002).
2.1 Cross-Sections, Mean Free Path, and the Diffusion
Equation
See Fig. 2.1. A thin slab of material of thickness s (ideally, one atomic layer) and
cross-sectional area Σ is bombarded by incoming neutrons at a rate R o neutrons per
square meter per second. Such an areal-specified rate is known as the neutron flux.
Let the bulk density of the material be ρ gr cm
−3 . In nuclear reaction calculations,
however, density is usually expressed as a number density of nuclei in the material,
that is, as the number of nuclei per cubic meter. In terms of ρ, this is given by
2 Critical Mass, Efficiency, and Yield
not surrounded by a tamper, that is, a so-called bare or naked core. A tamper is a
heavy metal casing which enhances weapon efficiency in two ways: By reflecting
escaped neutrons back into the core and hence giving them fresh chances at causing
fissions, and by briefly retarding the violent expansion of the core in order to give the
chain reaction more time over which to operate. Providing a tamper can significantly
increase the efficiency of a weapon at a very low cost, and tampers were employed in
both the Little Boy and Fat Man bombs of the Manhattan Project. Tamped criticality is
taken up in Sect. 2.3. Another modification to bomb design which was contemplated
during the Manhattan Project but not utilized is that of constructing a composite core,
that is, one comprising nested shells of two different fissile materials. The point of
this is to optimize bomb production: If, for example, you are producing
235 U and
239 Pu at the same rate, you can in principle produce bombs more steadily if they are
designed with cores having equal amounts of each material, as opposed to waiting
until you have enough for one of each type. This description is loose in that the exact
mass ratio of the two materials that is necessary to achieve criticality will depend on
their individual fissility characteristics, but makes the point that bomb productivity
can be improved in this way. Tamped composite cores are analyzed in Sect. 2.4.
Sections 2.5 and 2.6 take up the issue of bomb efficiency and yield through analytic
approximations and a numerical simulation. Section 2.7 presents an alternate treatment of untamped criticality that has an interesting historical connection. Section 2.8,
which is new in this edition of this book, presents several approximate methods for
analyzing critical mass and bomb efficiency. The point of these less rigorous analyses
is to provide treatments that can be used to make quick estimates or which might be
suitable for classroom discussions when time is at a premium. It must be remembered,
however, that they are approximate treatments, and that for more precise answers one
should return to the fuller diffusion-theory treatments. Finally, Sect. 2.9 presents an
approximate treatment of criticality for cylindrical bomb cores.
For readers interested in further sources, an excellent account of the concept of
critical mass appears in Logan (1996); see also Bernstein (2002).
2.1 Cross-Sections, Mean Free Path, and the Diffusion
Equation
See Fig. 2.1. A thin slab of material of thickness s (ideally, one atomic layer) and
cross-sectional area Σ is bombarded by incoming neutrons at a rate R o neutrons per
square meter per second. Such an areal-specified rate is known as the neutron flux.
Let the bulk density of the material be ρ gr cm
−3 . In nuclear reaction calculations,
however, density is usually expressed as a number density of nuclei in the material,
that is, as the number of nuclei per cubic meter. In terms of ρ, this is given by