Chapter 2
Critical Mass, Efficiency, and Yield
This chapter forms the heart of this book. Every gram of enriched uranium or synthesized plutonium produced in the Manhattan Project was obtained at great cost and
with great difficulty, so estimating the amount of fissile material needed to make
a workable nuclear weapon—the so-called critical mass—was a crucial issue for
the developers of Little Boy and Fat Man. Equally important was to estimate what
efficiency one might expect for a bomb. For various reasons, not all of the material in a bomb core undergoes fission during a nuclear explosion; if the expected
efficiency were to prove so low that one might just as well use a few conventional
bombs to achieve the same energy release, there would be no point in taking on the
massive engineering challenges involved in making nuclear weapons. In this chapter
we investigate these issues.
The concept of critical mass involves two competing effects. As nuclei fission, they
emit secondary neutrons. A fundamental empirical law of nuclear physics, derived
in Sect. 2.1, shows that while some neutrons will cause other fissions, the remainder
will reach the surface of the mass and escape. If, however, more than one neutron is
emitted per fission, we can afford to let some escape since only one is required to
initiate a subsequent fission. For a small sample of material the escape probability
is high; as the size of the sample increases, the escape probability declines and at
some point will reach a value such that the number of neutrons that fail to escape will
be sufficient to fission every nucleus in the mass—in theory, at least. Thus, there is
a minimum size (hence mass) of material for which every nucleus will in principle
fission even while some neutrons escape.
The above description of critical mass should be regarded as being only qualitative. Technically, the important issue is known as criticality. Criticality is said to
obtain when the number of free neutrons inside a bomb core is increasing with time.
A full understanding of criticality demands familiarity with time-dependent diffusion theory. Application of diffusion theory to this problem requires understanding
a concept known as the mean free path (MFP) for neutron travel, so this is also
developed in Sect. 2.1. Section 2.2 takes up a time-dependent diffusion theory treatment of criticality for the simplest possible configuration: a sphere of fissile material
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
B. C. Reed, The Physics of the Manhattan Project,
https://doi.org/10.1007/978-3-030-61373-0_2
51
Critical Mass, Efficiency, and Yield
This chapter forms the heart of this book. Every gram of enriched uranium or synthesized plutonium produced in the Manhattan Project was obtained at great cost and
with great difficulty, so estimating the amount of fissile material needed to make
a workable nuclear weapon—the so-called critical mass—was a crucial issue for
the developers of Little Boy and Fat Man. Equally important was to estimate what
efficiency one might expect for a bomb. For various reasons, not all of the material in a bomb core undergoes fission during a nuclear explosion; if the expected
efficiency were to prove so low that one might just as well use a few conventional
bombs to achieve the same energy release, there would be no point in taking on the
massive engineering challenges involved in making nuclear weapons. In this chapter
we investigate these issues.
The concept of critical mass involves two competing effects. As nuclei fission, they
emit secondary neutrons. A fundamental empirical law of nuclear physics, derived
in Sect. 2.1, shows that while some neutrons will cause other fissions, the remainder
will reach the surface of the mass and escape. If, however, more than one neutron is
emitted per fission, we can afford to let some escape since only one is required to
initiate a subsequent fission. For a small sample of material the escape probability
is high; as the size of the sample increases, the escape probability declines and at
some point will reach a value such that the number of neutrons that fail to escape will
be sufficient to fission every nucleus in the mass—in theory, at least. Thus, there is
a minimum size (hence mass) of material for which every nucleus will in principle
fission even while some neutrons escape.
The above description of critical mass should be regarded as being only qualitative. Technically, the important issue is known as criticality. Criticality is said to
obtain when the number of free neutrons inside a bomb core is increasing with time.
A full understanding of criticality demands familiarity with time-dependent diffusion theory. Application of diffusion theory to this problem requires understanding
a concept known as the mean free path (MFP) for neutron travel, so this is also
developed in Sect. 2.1. Section 2.2 takes up a time-dependent diffusion theory treatment of criticality for the simplest possible configuration: a sphere of fissile material
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
B. C. Reed, The Physics of the Manhattan Project,
https://doi.org/10.1007/978-3-030-61373-0_2
51