4.3 Predetonation Yield
157
Fig. 4.5 Sketch of time-evolution of predetonation. The core achieves first criticality (α = 0) at
t = 0, and t O is the time when α reaches its nominal design value α O . If the chain reaction starts
at t init , some time is required for e F fissions to occur, after which the nuclear explosion proper is
underway (see the text)
Here is the main subtlety: This crucial value of α is not that at t init , because some
additional time is required for the reaction to build up to the point where the pressure
exerted by the fission fragments is great enough to begin causing a sensible expansion
of the bomb core against any remaining force of the assembly mechanism; that is,
the nuclear reaction proper requires some time to become established. Even if the
first fission occurs before assembly is complete, it may well be that this build-up
time could be great enough to allow completion of the core assembly, resulting in
achieving the full design yield. The time after first criticality at which the nuclear
reaction proper begins is designated as t F , so it is the value of α(t F ) that dictates the
yield.
In view of the above considerations, we need to pin down three concepts in order
to establish an expression for the expected yield. These are (i) A model for the growth
of α during the time between first criticality and full assembly; this is needed in order
to be able to eventually estimate α(t F ); (ii) A procedure for estimating t F ; and (iii)
A model for how the efficiency of the explosion depends on α(t F ). These factors are
addressed individually in the following paragraphs.
From Sect. 2.2, the growth rate of the number of neutrons N (or of the neutron
density) within a bomb core is described by the differential equation
d N
dt
=
α
τ
N ,
(4.14)
where τ is the mean time that a neutron will travel before causing a fission, ~10
–8 s.
For the time-dependence of α, I follow MvHL and adopt a simple model: That α
grows linearly from zero to α O over time t O :
α(t) =
α O
t O
t.
(4.15)
In computing actual numbers, I will take t O = 10 μs (characteristic of an implosion
weapon), but this quantity is left as a general variable in the analysis.
157
Fig. 4.5 Sketch of time-evolution of predetonation. The core achieves first criticality (α = 0) at
t = 0, and t O is the time when α reaches its nominal design value α O . If the chain reaction starts
at t init , some time is required for e F fissions to occur, after which the nuclear explosion proper is
underway (see the text)
Here is the main subtlety: This crucial value of α is not that at t init , because some
additional time is required for the reaction to build up to the point where the pressure
exerted by the fission fragments is great enough to begin causing a sensible expansion
of the bomb core against any remaining force of the assembly mechanism; that is,
the nuclear reaction proper requires some time to become established. Even if the
first fission occurs before assembly is complete, it may well be that this build-up
time could be great enough to allow completion of the core assembly, resulting in
achieving the full design yield. The time after first criticality at which the nuclear
reaction proper begins is designated as t F , so it is the value of α(t F ) that dictates the
yield.
In view of the above considerations, we need to pin down three concepts in order
to establish an expression for the expected yield. These are (i) A model for the growth
of α during the time between first criticality and full assembly; this is needed in order
to be able to eventually estimate α(t F ); (ii) A procedure for estimating t F ; and (iii)
A model for how the efficiency of the explosion depends on α(t F ). These factors are
addressed individually in the following paragraphs.
From Sect. 2.2, the growth rate of the number of neutrons N (or of the neutron
density) within a bomb core is described by the differential equation
d N
dt
=
α
τ
N ,
(4.14)
where τ is the mean time that a neutron will travel before causing a fission, ~10
–8 s.
For the time-dependence of α, I follow MvHL and adopt a simple model: That α
grows linearly from zero to α O over time t O :
α(t) =
α O
t O
t.
(4.15)
In computing actual numbers, I will take t O = 10 μs (characteristic of an implosion
weapon), but this quantity is left as a general variable in the analysis.