2.5 Estimating Yield—Analytic
81
2.5 Estimating Yield—Analytic
Material in this section is adopted from Reed (2007).
In the preceding sections, we examined how to estimate critical masses for bare,
tamped, and tamped composite cores of fissile material. The analysis in Sect. 2.2
revealed that the threshold bare critical mass of
235 U is about 46 kg. In Sect. 1.6, we
saw that complete fission of 1 kg of
235 U liberates energy equivalent to that of about
17 kilotons of TNT. Given that the Little Boy uranium bomb that was dropped on
Hiroshima used about 53 kg of
235 U and is estimated to have had an explosive yield
of only about 13 kilotons, we can infer that it must have been rather inefficient. The
purpose of this section is to explore what factors dictate the efficiency of a fission
weapon and to show how one can estimate that efficiency.
This section is the first of several in this Chapter and in Chap. 4 devoted to the
questions of weapon efficiency and yield. In this section, these issues are examined
purely analytically. The advantage of an analytic approach is that it is helpful for
establishing a sense of how the efficiency depends on the parameters involved: The
mass and density of the core and the values of various nuclear constants. However,
conditions inside an exploding bomb core evolve very rapidly as a function of time,
and this evolution cannot be fully captured with analytic approximations. To get a
sense of the time-evolution of the process, it is necessary to numerically integrate the
core conditions as a function of time, tracking core size, expansion rate, pressure,
neutron density, and energy release along the way. Such an analysis is the subject
of the next section; these two sections therefore closely complement each other and
should be read as a unit. Bomb efficiency and yield can also be affected by various
phenomena that can trigger the chain-reaction before the weapon core has reached
its fully assembled state; these issues are explored in Chap. 4.
In the present section we consider only untamped cores for sake of simplicity; a
tamped core is simulated numerically in Sect. 2.6.
To begin, it is helpful to appreciate that the efficiency of a nuclear weapon involves
three distinct time scales. The first is mechanical in nature: The time required to
assemble the subcritical fissile components into a critical assembly before fission
is initiated. In principle, this time can be as long as is desired, but in practice it is
constrained by the occurrence of spontaneous fissions, which could lead to reactiontriggering stray neutrons during the assembly period.
What is the order of magnitude of the assembly time? This was discussed in the
Preamble, where it was shown that for a “gun-type” bomb where a “projectile” piece
of fissile material is fired as a shell inside an artillery barrel toward a mating “target”
piece of fissile material, about 200 microseconds will be required to achieve full
core assembly from the moment when the leading edge of the projectile meets the
target piece. As we will see in Sect. 4.2, the rate of spontaneous fission was not an
issue for a uranium core over this time, but was such a problem for plutonium as to
necessitate development of the implosion mechanism for triggering those weapons.
So far as the present section is concerned, however, the essential idea is that if the
spontaneous fission probability can be kept negligible during the assembly time
81
2.5 Estimating Yield—Analytic
Material in this section is adopted from Reed (2007).
In the preceding sections, we examined how to estimate critical masses for bare,
tamped, and tamped composite cores of fissile material. The analysis in Sect. 2.2
revealed that the threshold bare critical mass of
235 U is about 46 kg. In Sect. 1.6, we
saw that complete fission of 1 kg of
235 U liberates energy equivalent to that of about
17 kilotons of TNT. Given that the Little Boy uranium bomb that was dropped on
Hiroshima used about 53 kg of
235 U and is estimated to have had an explosive yield
of only about 13 kilotons, we can infer that it must have been rather inefficient. The
purpose of this section is to explore what factors dictate the efficiency of a fission
weapon and to show how one can estimate that efficiency.
This section is the first of several in this Chapter and in Chap. 4 devoted to the
questions of weapon efficiency and yield. In this section, these issues are examined
purely analytically. The advantage of an analytic approach is that it is helpful for
establishing a sense of how the efficiency depends on the parameters involved: The
mass and density of the core and the values of various nuclear constants. However,
conditions inside an exploding bomb core evolve very rapidly as a function of time,
and this evolution cannot be fully captured with analytic approximations. To get a
sense of the time-evolution of the process, it is necessary to numerically integrate the
core conditions as a function of time, tracking core size, expansion rate, pressure,
neutron density, and energy release along the way. Such an analysis is the subject
of the next section; these two sections therefore closely complement each other and
should be read as a unit. Bomb efficiency and yield can also be affected by various
phenomena that can trigger the chain-reaction before the weapon core has reached
its fully assembled state; these issues are explored in Chap. 4.
In the present section we consider only untamped cores for sake of simplicity; a
tamped core is simulated numerically in Sect. 2.6.
To begin, it is helpful to appreciate that the efficiency of a nuclear weapon involves
three distinct time scales. The first is mechanical in nature: The time required to
assemble the subcritical fissile components into a critical assembly before fission
is initiated. In principle, this time can be as long as is desired, but in practice it is
constrained by the occurrence of spontaneous fissions, which could lead to reactiontriggering stray neutrons during the assembly period.
What is the order of magnitude of the assembly time? This was discussed in the
Preamble, where it was shown that for a “gun-type” bomb where a “projectile” piece
of fissile material is fired as a shell inside an artillery barrel toward a mating “target”
piece of fissile material, about 200 microseconds will be required to achieve full
core assembly from the moment when the leading edge of the projectile meets the
target piece. As we will see in Sect. 4.2, the rate of spontaneous fission was not an
issue for a uranium core over this time, but was such a problem for plutonium as to
necessitate development of the implosion mechanism for triggering those weapons.
So far as the present section is concerned, however, the essential idea is that if the
spontaneous fission probability can be kept negligible during the assembly time