156
4 Complicating Factors
Given this situation, a corollary question arises: Is it possible to predict what fraction
of the design yield of a weapon might be realized in the case of such an event?
Historically, concern with the yield-fraction probability was motivated not only by
the desire to have some idea of what yield might be expected, but also by the desire
to ensure an explosion violent enough to destroy the bomb and disperse the fissile
material even if a minimum-yield explosion occurred. The rationale for this is that if
a bomb fails to operate properly but still destroys itself, an adversary would be unable
to recover the fissile material and reverse-engineer the weapon. A minimum-yield
explosion is known to weapons engineers as a “fizzle.”
A yield-fraction model comprises two separate components which are then linked
together. The first is a model for the yield Y (t init ) one might expect to realize from
a weapon if a spontaneous fission initiates the chain reaction at some time t init after
the core first achieves a critical state during its assembly but prior to assembly being
completed. The moment at which the criticality parameter α achieves a value of zero
is taken to define t = 0. (Recall that α = 0 for a core of threshold critical mass, and α
> 0 for a supercritical core.) The second is a probabilistic model for the chance that
the reaction will not be initiated by time t init ; this is based on the material developed
in the preceding section. By combining these, one can then make the statement that
if P is the probability that a predetonation does not occur during the time interval
(0, t init ), then the chance of obtaining at least yield Y (t init ) is 100P percent. There
are some subtleties to this argument as discussed in what follows, but this is the
fundamental idea.
This yield model is adopted from one developed by J. Carson Mark in collaboration
with Frank von Hippel and Edwin Lyman (Mark et al. 2009), although the analysis
given here is somewhat more general than theirs. In what follows, I refer to their paper
as MvHL. As in the previous section, the development here assumes an untamped
core.
When operation of a weapon is triggered, the core will initially be subcritical, but
as it is assembled by either an implosion or a gun mechanism, it will reach a condition
where α = 0, “first criticality.” Subsequently, α will increase until the core is fully
assembled. The most desirable situation is that the chain reaction not be initiated
until the core reaches its fully-assembled state, as this would result in the most
efficient explosion. The value of α in the fully-assembled condition is designated as
α O , and the time at which this happens after first criticality is designated as t O ; see
Fig. 4.5. As soon as the chain reaction starts, the core will begin to expand and α
will begin to decline. When α reaches zero, “second criticality” occurs, after which
the reaction essentially shuts down. α O is thus the maximum possible value that α
can have, and can be thought of as the design or “nominal” value of the weapon’s
criticality parameter. On the other hand, the worst circumstance would be that the
reaction is initiated via a spontaneous fission at just the moment when first criticality
is achieved. In this case the bomb will (likely) blow itself apart and generate only
the minimum possible “fizzle” yield. The important point for the moment, however,
is that the yield of a weapon depends essentially on the value of α when the chain
reaction begins.
4 Complicating Factors
Given this situation, a corollary question arises: Is it possible to predict what fraction
of the design yield of a weapon might be realized in the case of such an event?
Historically, concern with the yield-fraction probability was motivated not only by
the desire to have some idea of what yield might be expected, but also by the desire
to ensure an explosion violent enough to destroy the bomb and disperse the fissile
material even if a minimum-yield explosion occurred. The rationale for this is that if
a bomb fails to operate properly but still destroys itself, an adversary would be unable
to recover the fissile material and reverse-engineer the weapon. A minimum-yield
explosion is known to weapons engineers as a “fizzle.”
A yield-fraction model comprises two separate components which are then linked
together. The first is a model for the yield Y (t init ) one might expect to realize from
a weapon if a spontaneous fission initiates the chain reaction at some time t init after
the core first achieves a critical state during its assembly but prior to assembly being
completed. The moment at which the criticality parameter α achieves a value of zero
is taken to define t = 0. (Recall that α = 0 for a core of threshold critical mass, and α
> 0 for a supercritical core.) The second is a probabilistic model for the chance that
the reaction will not be initiated by time t init ; this is based on the material developed
in the preceding section. By combining these, one can then make the statement that
if P is the probability that a predetonation does not occur during the time interval
(0, t init ), then the chance of obtaining at least yield Y (t init ) is 100P percent. There
are some subtleties to this argument as discussed in what follows, but this is the
fundamental idea.
This yield model is adopted from one developed by J. Carson Mark in collaboration
with Frank von Hippel and Edwin Lyman (Mark et al. 2009), although the analysis
given here is somewhat more general than theirs. In what follows, I refer to their paper
as MvHL. As in the previous section, the development here assumes an untamped
core.
When operation of a weapon is triggered, the core will initially be subcritical, but
as it is assembled by either an implosion or a gun mechanism, it will reach a condition
where α = 0, “first criticality.” Subsequently, α will increase until the core is fully
assembled. The most desirable situation is that the chain reaction not be initiated
until the core reaches its fully-assembled state, as this would result in the most
efficient explosion. The value of α in the fully-assembled condition is designated as
α O , and the time at which this happens after first criticality is designated as t O ; see
Fig. 4.5. As soon as the chain reaction starts, the core will begin to expand and α
will begin to decline. When α reaches zero, “second criticality” occurs, after which
the reaction essentially shuts down. α O is thus the maximum possible value that α
can have, and can be thought of as the design or “nominal” value of the weapon’s
criticality parameter. On the other hand, the worst circumstance would be that the
reaction is initiated via a spontaneous fission at just the moment when first criticality
is achieved. In this case the bomb will (likely) blow itself apart and generate only
the minimum possible “fizzle” yield. The important point for the moment, however,
is that the yield of a weapon depends essentially on the value of α when the chain
reaction begins.