2.5 Estimating Yield—Analytic
91
The orders of magnitude involved in Fig. 2.15 drive home the remarkable conditions that prevail during nuclear explosions, and how the vast majority of the energy
is liberated during only the last few fission generations. With ~10 ns per generation,
the entire timescale the figure is only about one-half of a microsecond. Particularly
striking is the rate of growth of temperature: a mere 10 generations elapse between
when this reaches that of the surface of the Sun and that at its core.
The computations involved in preparing Fig. 2.15 are so trivial that they make it
easy to overlook just how remarkable this plot is. The results are largely independent
of any particular bomb design, core mass, critical mass, fissile material, or number
of neutrons per fission. The graph gets across the orders of magnitude involved in
as simple and general a way as one could desire without undue sacrifice of the
relevant physics. Further, a psychological aspect of the way it is laid out deserves
comment. Usually we plot exponential functions using logarithmic scales so that the
relationships appear as straight lines. Everything in Fig. 2.15 could be presented this
way, but this would have the effect of downplaying the magnitudes of some of the
quantities involved. For example, the greatest neutron density involved is 10 orders of
magnitude greater than that of the greatest pressure; in a logarithmic plot, the line for
the former will always lie well above that of the latter and so draw the lion’s share of
one’s attention. A density of a few quadrillion neutrons per cubic centimeter sounds
fantastic, but in actuality corresponds to fission of less than one part in a million of
the
235 U contained in any one cubic centimeter of the core. In contrast, by the time
that the neutron density has reached this value, the temperature is greater than that
of the surface of the Sun, and the pressure has risen to ~100,000 atmospheres. These
latter numbers are sure to impress anybody.
The values indicated in Fig. 2.15 agree closely with what Serber gives, with
one exception. He states that the gas and radiation pressures will be equal at ~36
generations, but it is not clear how he arrives at this value. For an environment
at absolute temperature T, thermodynamics gives an expression for the radiation
pressure:
P rad =
8π
5 k
4
45c 3 h 3
T
4
=
2.524 × 10
−16 Pa K
−4
T
4
,
(2.104)
where the various symbols have their usual meanings. Adopting n as above for
235 U,
the gas and radiation pressures will be equal at T ~ 1.38 × 10
7 K and P ~ 9.14 ×
10
12 Pa, which (2.102) indicates will occur at t ~ 29 generations (γ = 2/3). This may
simply have been a computational error on Serber’s part in the haste of preparing his
lectures.
To close this section, we compare the efficiency formula derived here to what
was probably the first recorded formulation of the energy expected to be liberated
by a nuclear weapon. This appeared in a document which has come to be known as
the Frisch-Peierls Memorandum. This remarkable 7-page manuscript was prepared
by Otto Frisch and Rudolf Peierls in March, 1940, to alert British government and
military officials to the possibly of creating extremely powerful bombs based on
91
The orders of magnitude involved in Fig. 2.15 drive home the remarkable conditions that prevail during nuclear explosions, and how the vast majority of the energy
is liberated during only the last few fission generations. With ~10 ns per generation,
the entire timescale the figure is only about one-half of a microsecond. Particularly
striking is the rate of growth of temperature: a mere 10 generations elapse between
when this reaches that of the surface of the Sun and that at its core.
The computations involved in preparing Fig. 2.15 are so trivial that they make it
easy to overlook just how remarkable this plot is. The results are largely independent
of any particular bomb design, core mass, critical mass, fissile material, or number
of neutrons per fission. The graph gets across the orders of magnitude involved in
as simple and general a way as one could desire without undue sacrifice of the
relevant physics. Further, a psychological aspect of the way it is laid out deserves
comment. Usually we plot exponential functions using logarithmic scales so that the
relationships appear as straight lines. Everything in Fig. 2.15 could be presented this
way, but this would have the effect of downplaying the magnitudes of some of the
quantities involved. For example, the greatest neutron density involved is 10 orders of
magnitude greater than that of the greatest pressure; in a logarithmic plot, the line for
the former will always lie well above that of the latter and so draw the lion’s share of
one’s attention. A density of a few quadrillion neutrons per cubic centimeter sounds
fantastic, but in actuality corresponds to fission of less than one part in a million of
the
235 U contained in any one cubic centimeter of the core. In contrast, by the time
that the neutron density has reached this value, the temperature is greater than that
of the surface of the Sun, and the pressure has risen to ~100,000 atmospheres. These
latter numbers are sure to impress anybody.
The values indicated in Fig. 2.15 agree closely with what Serber gives, with
one exception. He states that the gas and radiation pressures will be equal at ~36
generations, but it is not clear how he arrives at this value. For an environment
at absolute temperature T, thermodynamics gives an expression for the radiation
pressure:
P rad =
8π
5 k
4
45c 3 h 3
T
4
=
2.524 × 10
−16 Pa K
−4
T
4
,
(2.104)
where the various symbols have their usual meanings. Adopting n as above for
235 U,
the gas and radiation pressures will be equal at T ~ 1.38 × 10
7 K and P ~ 9.14 ×
10
12 Pa, which (2.102) indicates will occur at t ~ 29 generations (γ = 2/3). This may
simply have been a computational error on Serber’s part in the haste of preparing his
lectures.
To close this section, we compare the efficiency formula derived here to what
was probably the first recorded formulation of the energy expected to be liberated
by a nuclear weapon. This appeared in a document which has come to be known as
the Frisch-Peierls Memorandum. This remarkable 7-page manuscript was prepared
by Otto Frisch and Rudolf Peierls in March, 1940, to alert British government and
military officials to the possibly of creating extremely powerful bombs based on