2.3 Critical Mass: Tamped Core
73
1 +
λ
ε
d core
R thresh
R thresh
R tamp
2
−
R thresh
R tamp
R thresh
d core
cot
R thresh
d core
− 1
+ λ = 0,
(2.49)
where ε is as defined in (2.31) and where
λ =
λ
tamp
trans
λ
core
trans
.
(2.50)
Once the core material is chosen, the value of ε is fixed; values for
235 U and
239 Pu are given in Table 2.1. For chosen values of ε and λ, Eq. (2.49) can be solved
numerically for R thresh /d core for a given value of R tamp /R thresh ; this latter quantity must
by definition be > 1. A graph of R thresh /d core versus R tamp /R thresh can then be built up.
Figures 2.9 and 2.10 show such graphs for the same tamper materials as Figs. 2.7 and
2.8 for
235 U and
239 Pu, respectively. In Fig. 2.9, the curves converge to R thresh /d core =
2.379 at R tamp /R thresh = 1, which corresponds to a bare
235 U core; similarly for
239 Pu
in Fig. 2.10, where they converge at R thresh /d core = 2.125.
The Little Boy example above can be reconstructed from Fig. 2.9, except that the
question is now posed in reverse to ask: What tamper mass will be needed to render
a core of mass 16.17 kg just critical? A
235 U core of mass 16.17 kg will have a radius
R thresh of ~5.91 cm, or, from the value of d core in Table 2.1, R thresh /d core ~ 5.91/3.52
~ 1.68. Eyeballing across the graph shows that a horizontal line at R thresh /d core ~
1.68 will cross the curve for WC at R tamp /R thresh ~ 3.5. This corresponds to R thresh ~
20.7 cm, which would give a tamper thickness of ~14.8 cm and a mass of ~ 570 kg,
close to that described above. Readers are cautioned that small changes in input
parameters can have significant effects on results.
A converse approach to using Figs. 2.9 and 2.10 would be to decide on a value of
R tamp /R thresh in advance, and then determine R core /d for a given value of λ. There are
various ways to use all of these plots.
An important aspect of Figs. 2.9 and 2.10 is that, like Fig. 2.5, they will also work
for cases where the core and tamper have been compressed; this is not the case for
Figs. 2.7 and 2.8. If the density of both materials is increased by the same factor, λ
will not be affected. (If the density changes are different, which is likely to be the
case because materials have different bulk moduli, it is easy to compute the new λ:
Individual λ’s are simply inversely proportional to their densities). As with Fig. 2.5,
the only factor that will be different will be d core .
The curve for aluminum in Fig. 2.10 shows a slight maximum for small values of
R tamp /R thresh , which would indicate a value of R thresh /d core greater than that for a bare
core, a clearly nonsensical result. The reason for this is that such a core would be
very thin in comparison to its transport mean free path, for which a diffusion analysis
is not accurate.
Here is a hypothetical exercise using Fig. 2.10 and Tables 2.1 and 2.2. Suppose that
terrorists have stolen 12 kg of
239 Pu. They also have 150 kg of Aluminum with which
they can fashion a tamper. They have no implosion technology, so both materials are
at normal density. Can they make a threshold-critical device, assuming spherical
73
1 +
λ
ε
d core
R thresh
R thresh
R tamp
2
−
R thresh
R tamp
R thresh
d core
cot
R thresh
d core
− 1
+ λ = 0,
(2.49)
where ε is as defined in (2.31) and where
λ =
λ
tamp
trans
λ
core
trans
.
(2.50)
Once the core material is chosen, the value of ε is fixed; values for
235 U and
239 Pu are given in Table 2.1. For chosen values of ε and λ, Eq. (2.49) can be solved
numerically for R thresh /d core for a given value of R tamp /R thresh ; this latter quantity must
by definition be > 1. A graph of R thresh /d core versus R tamp /R thresh can then be built up.
Figures 2.9 and 2.10 show such graphs for the same tamper materials as Figs. 2.7 and
2.8 for
235 U and
239 Pu, respectively. In Fig. 2.9, the curves converge to R thresh /d core =
2.379 at R tamp /R thresh = 1, which corresponds to a bare
235 U core; similarly for
239 Pu
in Fig. 2.10, where they converge at R thresh /d core = 2.125.
The Little Boy example above can be reconstructed from Fig. 2.9, except that the
question is now posed in reverse to ask: What tamper mass will be needed to render
a core of mass 16.17 kg just critical? A
235 U core of mass 16.17 kg will have a radius
R thresh of ~5.91 cm, or, from the value of d core in Table 2.1, R thresh /d core ~ 5.91/3.52
~ 1.68. Eyeballing across the graph shows that a horizontal line at R thresh /d core ~
1.68 will cross the curve for WC at R tamp /R thresh ~ 3.5. This corresponds to R thresh ~
20.7 cm, which would give a tamper thickness of ~14.8 cm and a mass of ~ 570 kg,
close to that described above. Readers are cautioned that small changes in input
parameters can have significant effects on results.
A converse approach to using Figs. 2.9 and 2.10 would be to decide on a value of
R tamp /R thresh in advance, and then determine R core /d for a given value of λ. There are
various ways to use all of these plots.
An important aspect of Figs. 2.9 and 2.10 is that, like Fig. 2.5, they will also work
for cases where the core and tamper have been compressed; this is not the case for
Figs. 2.7 and 2.8. If the density of both materials is increased by the same factor, λ
will not be affected. (If the density changes are different, which is likely to be the
case because materials have different bulk moduli, it is easy to compute the new λ:
Individual λ’s are simply inversely proportional to their densities). As with Fig. 2.5,
the only factor that will be different will be d core .
The curve for aluminum in Fig. 2.10 shows a slight maximum for small values of
R tamp /R thresh , which would indicate a value of R thresh /d core greater than that for a bare
core, a clearly nonsensical result. The reason for this is that such a core would be
very thin in comparison to its transport mean free path, for which a diffusion analysis
is not accurate.
Here is a hypothetical exercise using Fig. 2.10 and Tables 2.1 and 2.2. Suppose that
terrorists have stolen 12 kg of
239 Pu. They also have 150 kg of Aluminum with which
they can fashion a tamper. They have no implosion technology, so both materials are
at normal density. Can they make a threshold-critical device, assuming spherical