82
2 Critical Mass, Efficiency, and Yield
(which we assume), the efficiency of the weapon is dictated by the two other time
scales.
The first of these other time scales is nuclear in nature. Once fission has been
initiated, some time will be required for all of the fissile material to be consumed.
This time we call t fiss . The other time scale is again mechanical. As soon as fissions
have been initiated, the core will begin to expand due to the extreme gas pressure of
the fission fragments. This expansion will lead after a time t crit to loss of criticality,
after which the reaction rate will diminish. Weapon efficiency will depend on how
these times compare: If t crit > t fiss , then in principle all of the core material will
undergo fission and the efficiency will be 100%. In reality, this is difficult to achieve.
Before proceeding with the detailed analysis, we pause to make a rough estimate
of how much time is required to fission the entire core once the chain reaction has
been initiated. In Sect. 2.2 we saw that once a neutron is emitted in a fission, it will
travel for only about 10 ns before causing another fission. Suppose that we have a
core of mass M kilograms of fissile material of atomic weight A grams per mole. The
number of nuclei N in the mass will be N = 10
3 M N A
A. If we start with ν o neutrons
at the start of “generation 1” and each generation liberates ν neutrons, then the algebra
of a geometric progression shows that after G generations, the number of fissions F G
that will have occurred, if there is no neutron loss, is F G = ν o
ν
G
− 1
(ν − 1).
For any sensible value of G, the numerator can be approximated as ν
G , and it follows
that the number of generations necessary to have fissioned each nucleus is G =
ln
N (ν − 1)
ν o
ln(ν). At τ seconds per generation, it follows that the time to
fission the entire mass will be t fiss ~ τ G. For M = 50 kilograms of
235 U, N ~ 1.28
× 10
26 . If we set ν o = 1 and ν = 2.6, the number of generations evaluates as G
~ 63, which gives t fiss ~ 0.6 μs, an incredibly brief time. Even if only half of the
neutrons cause fissions (ν = 1.3), G ~ 230, and t fiss ~2.3 µs. Such are the timescales
of nuclear-weapon physics.
Once a chain reaction has been initiated, a bomb core will rapidly (within about
a microsecond) heat up, melt, vaporize, and thereafter behave as an expanding gas
with the expansion driven by the gas pressure in a thermodynamic PΔV manner; we
assume that the vast majority of energy liberated in fission reactions can be assumed
to go into the form of kinetic energy of the fission products. In what follows, the
approach to estimating yield and efficiency will be to use these concepts to establish
the range of radius (and hence time) over which the core can expand before the
expansion lowers the density of the fissile material to subcriticality. Some fissions
will continue to happen after this time, but it is this “criticality shutdown timescale”
that fundamentally dictates the efficiency of the weapon.
As above, let τ represent the time for which a neutron will travel before causing
a fission; see (2.21). Inverting this, we can say that a single neutron will lead to a
subsequent fission at a rate of 1/τ per second:
rate of f issions per neutron =
1
τ
.
(2.82)
2 Critical Mass, Efficiency, and Yield
(which we assume), the efficiency of the weapon is dictated by the two other time
scales.
The first of these other time scales is nuclear in nature. Once fission has been
initiated, some time will be required for all of the fissile material to be consumed.
This time we call t fiss . The other time scale is again mechanical. As soon as fissions
have been initiated, the core will begin to expand due to the extreme gas pressure of
the fission fragments. This expansion will lead after a time t crit to loss of criticality,
after which the reaction rate will diminish. Weapon efficiency will depend on how
these times compare: If t crit > t fiss , then in principle all of the core material will
undergo fission and the efficiency will be 100%. In reality, this is difficult to achieve.
Before proceeding with the detailed analysis, we pause to make a rough estimate
of how much time is required to fission the entire core once the chain reaction has
been initiated. In Sect. 2.2 we saw that once a neutron is emitted in a fission, it will
travel for only about 10 ns before causing another fission. Suppose that we have a
core of mass M kilograms of fissile material of atomic weight A grams per mole. The
number of nuclei N in the mass will be N = 10
3 M N A
A. If we start with ν o neutrons
at the start of “generation 1” and each generation liberates ν neutrons, then the algebra
of a geometric progression shows that after G generations, the number of fissions F G
that will have occurred, if there is no neutron loss, is F G = ν o
ν
G
− 1
(ν − 1).
For any sensible value of G, the numerator can be approximated as ν
G , and it follows
that the number of generations necessary to have fissioned each nucleus is G =
ln
N (ν − 1)
ν o
ln(ν). At τ seconds per generation, it follows that the time to
fission the entire mass will be t fiss ~ τ G. For M = 50 kilograms of
235 U, N ~ 1.28
× 10
26 . If we set ν o = 1 and ν = 2.6, the number of generations evaluates as G
~ 63, which gives t fiss ~ 0.6 μs, an incredibly brief time. Even if only half of the
neutrons cause fissions (ν = 1.3), G ~ 230, and t fiss ~2.3 µs. Such are the timescales
of nuclear-weapon physics.
Once a chain reaction has been initiated, a bomb core will rapidly (within about
a microsecond) heat up, melt, vaporize, and thereafter behave as an expanding gas
with the expansion driven by the gas pressure in a thermodynamic PΔV manner; we
assume that the vast majority of energy liberated in fission reactions can be assumed
to go into the form of kinetic energy of the fission products. In what follows, the
approach to estimating yield and efficiency will be to use these concepts to establish
the range of radius (and hence time) over which the core can expand before the
expansion lowers the density of the fissile material to subcriticality. Some fissions
will continue to happen after this time, but it is this “criticality shutdown timescale”
that fundamentally dictates the efficiency of the weapon.
As above, let τ represent the time for which a neutron will travel before causing
a fission; see (2.21). Inverting this, we can say that a single neutron will lead to a
subsequent fission at a rate of 1/τ per second:
rate of f issions per neutron =
1
τ
.
(2.82)