2.5 Estimating Yield—Analytic
89
untamped case involves the density and radius of the core in the combination ρ r; in
the tamped case the criticality condition admits no such combination of parameters,
so the subsequent calculations of criticality timescale and efficiency do not transform
unaltered to using a tamped core. Efficiency in the case of a tamped core can only
be established numerically.
The conditions that exist within a nuclear explosion are so fantastic by everyday
standards that simply manipulating formulae or running specific calculations cannot
really give an overall sense of the orders of magnitude involved. To this end, I
develop here an approximate graphical representation of the evolution of neutron
density, energy density, pressure, and temperature within an exploding bomb core
as a function of time. This approach is based on a hand-drawn graph prepared in
early 1943 by Robert Serber for his Los Alamos Primer. Serber’s goal was not a
precise analysis—many nuclear parameters were only approximately known at the
time—but rather to try to get across to his colleagues a sense the extreme conditions
that they would be dealing with. Serber’s plot still stands as a remarkable example
of effective graphical display of information. The development given here is adapted
from previous publications (Reed 2016, 2020c).
From Eqs. (2.22), (2.86), and (2.87), the neutron density N, energy density U, and
pressure P within an exploding bomb core can be written as
N (t) = N o e
(α/ τ ) t
,
(2.100)
U =
E f
α
N o e
(α/ τ ) t
,
(2.101)
and
P = γ U.
(2.102)
In early 1943, Serber had no idea what value α would take for an eventual bomb
design, but knew it would likely be on the order of a few tenths by virtue of Eq. (2.34).
Similarly, the neutron travel time τ depends on the fission cross-section; while this
was still being pinned down experimentally, it was already estimated to be on the
order of ~10 ns. For simplicity, Serber took α = 1, and used τ as his unit of time. The
neutron density is of course also a function of position within the core, but Serber’s
concern was the order of magnitude of the time- dependence of the various quantities
involved. Even though radiation pressure dominates over gas pressure in the later
stages of the explosion, I follow Serber and adopt γ = 2/3; for the purpose intended
here, a factor of two is of no real consequence.
Serber did not include any indication of temperature in his graph; I do so by
assuming that the vaporized fissile material acts as an ideal gas, in which case
temperature can be determined through the pressure via
P gas = n k T.
(2.103)
89
untamped case involves the density and radius of the core in the combination ρ r; in
the tamped case the criticality condition admits no such combination of parameters,
so the subsequent calculations of criticality timescale and efficiency do not transform
unaltered to using a tamped core. Efficiency in the case of a tamped core can only
be established numerically.
The conditions that exist within a nuclear explosion are so fantastic by everyday
standards that simply manipulating formulae or running specific calculations cannot
really give an overall sense of the orders of magnitude involved. To this end, I
develop here an approximate graphical representation of the evolution of neutron
density, energy density, pressure, and temperature within an exploding bomb core
as a function of time. This approach is based on a hand-drawn graph prepared in
early 1943 by Robert Serber for his Los Alamos Primer. Serber’s goal was not a
precise analysis—many nuclear parameters were only approximately known at the
time—but rather to try to get across to his colleagues a sense the extreme conditions
that they would be dealing with. Serber’s plot still stands as a remarkable example
of effective graphical display of information. The development given here is adapted
from previous publications (Reed 2016, 2020c).
From Eqs. (2.22), (2.86), and (2.87), the neutron density N, energy density U, and
pressure P within an exploding bomb core can be written as
N (t) = N o e
(α/ τ ) t
,
(2.100)
U =
E f
α
N o e
(α/ τ ) t
,
(2.101)
and
P = γ U.
(2.102)
In early 1943, Serber had no idea what value α would take for an eventual bomb
design, but knew it would likely be on the order of a few tenths by virtue of Eq. (2.34).
Similarly, the neutron travel time τ depends on the fission cross-section; while this
was still being pinned down experimentally, it was already estimated to be on the
order of ~10 ns. For simplicity, Serber took α = 1, and used τ as his unit of time. The
neutron density is of course also a function of position within the core, but Serber’s
concern was the order of magnitude of the time- dependence of the various quantities
involved. Even though radiation pressure dominates over gas pressure in the later
stages of the explosion, I follow Serber and adopt γ = 2/3; for the purpose intended
here, a factor of two is of no real consequence.
Serber did not include any indication of temperature in his graph; I do so by
assuming that the vaporized fissile material acts as an ideal gas, in which case
temperature can be determined through the pressure via
P gas = n k T.
(2.103)