158
4 Complicating Factors
If the reaction starts at some time t init (≤ t O ), then the neutron population at some
later time t will be given by integrating (4.14) from t init to t after substituting the
α-growth model of (4.15) into (4.14). It proves to be more useful, however, to speak
in terms of the number of fissions that have occurred between these time limits. Since
the number of neutrons created is proportional to the number of fissions that have
occurred and since the constant of proportionality will cancel from both sides in
(4.14), that equation also dictates the number of fissions that will have taken place.
Let the number of fissions that occur over the interval (t init , t) be e
F . Upon integrating
we find
F =
α O
2 τ t O
t
2
− t
2
init
.
(4.16)
This result is central to the developments that follow.
To address point (ii) above, if we had available a value for F react corresponding to
the time t F at which the nuclear explosion can be considered to have become fully
established, we could compute that time simply by solving (4.16) for t on setting F
= F react . To pin this value down, MvHL offer the following argument. Consider a
plutonium core of mass 10 kg. The specific heat of Pu is about 130 J/(kg K), and
a typical fission releases about 180 MeV of energy. To raise the temperature of the
core by a modest 30° would require F ~ 35. To melt the core would require F ~
38. Beyond this, to liberate by fissions an amount of energy per gram of material
equivalent to that of detonating TNT (~1 kcal/gr) would require F ~ 42. But as
MvHL point out, by this time the plutonium will have vaporized and begun to exert a
pressure on its surroundings in the megabar range, a pressure which will overwhelm
any remaining force of the assembly mechanism. Thus, we are justified in taking the
nuclear explosion to have started by, say, the time that F react ~ 45. This is the value
which MvHL adopted, but sensible changes make little difference to the results.
F react is left as a general parameter in the development that follows, and is hereafter
abbreviated as F.
Now to point (iii). In Sections 17 and 18 of his Los Alamos Primer, Serber (1992)
develops an argument to show that if α has the value α F when the nuclear explosion
begins (that is, after e
F fission have occurred), then the yield Y of the weapon will
behave approximately as.
Y
Y O
∼
α F
α O
3
,
(4.17)
where Y O is the nominal design yield. This dependence can be understood from
the analysis of efficiency in Sect. 2.5. From (2.96), the efficiency is proportional
to α
2
ρr . But numerical solution of the formal criticality conditions with cores of
from one to two critical masses shows that ρr is roughly linearly proportional to
α, which leads to the conclusion that the efficiency must be proportional to α
3 . The
proportionality for cores of one to two critical masses of
235 U is illustrated in Fig. 4.6.
4 Complicating Factors
If the reaction starts at some time t init (≤ t O ), then the neutron population at some
later time t will be given by integrating (4.14) from t init to t after substituting the
α-growth model of (4.15) into (4.14). It proves to be more useful, however, to speak
in terms of the number of fissions that have occurred between these time limits. Since
the number of neutrons created is proportional to the number of fissions that have
occurred and since the constant of proportionality will cancel from both sides in
(4.14), that equation also dictates the number of fissions that will have taken place.
Let the number of fissions that occur over the interval (t init , t) be e
F . Upon integrating
we find
F =
α O
2 τ t O
t
2
− t
2
init
.
(4.16)
This result is central to the developments that follow.
To address point (ii) above, if we had available a value for F react corresponding to
the time t F at which the nuclear explosion can be considered to have become fully
established, we could compute that time simply by solving (4.16) for t on setting F
= F react . To pin this value down, MvHL offer the following argument. Consider a
plutonium core of mass 10 kg. The specific heat of Pu is about 130 J/(kg K), and
a typical fission releases about 180 MeV of energy. To raise the temperature of the
core by a modest 30° would require F ~ 35. To melt the core would require F ~
38. Beyond this, to liberate by fissions an amount of energy per gram of material
equivalent to that of detonating TNT (~1 kcal/gr) would require F ~ 42. But as
MvHL point out, by this time the plutonium will have vaporized and begun to exert a
pressure on its surroundings in the megabar range, a pressure which will overwhelm
any remaining force of the assembly mechanism. Thus, we are justified in taking the
nuclear explosion to have started by, say, the time that F react ~ 45. This is the value
which MvHL adopted, but sensible changes make little difference to the results.
F react is left as a general parameter in the development that follows, and is hereafter
abbreviated as F.
Now to point (iii). In Sections 17 and 18 of his Los Alamos Primer, Serber (1992)
develops an argument to show that if α has the value α F when the nuclear explosion
begins (that is, after e
F fission have occurred), then the yield Y of the weapon will
behave approximately as.
Y
Y O
∼
α F
α O
3
,
(4.17)
where Y O is the nominal design yield. This dependence can be understood from
the analysis of efficiency in Sect. 2.5. From (2.96), the efficiency is proportional
to α
2
ρr . But numerical solution of the formal criticality conditions with cores of
from one to two critical masses shows that ρr is roughly linearly proportional to
α, which leads to the conclusion that the efficiency must be proportional to α
3 . The
proportionality for cores of one to two critical masses of
235 U is illustrated in Fig. 4.6.