88
2 Critical Mass, Efficiency, and Yield
Table 2.3 Criticality and efficiency parameters for C = 1.5, E f = 180 MeV, γ = 1/3; no tamper.
Masses are 68.8 kg for 235 U and 25.04 kg for 239 Pu
Quantity
Unit
Physical meaning
235 U
239 Pu
r initial
cm
Initial core radius
9.58
7.26
n
10 22 cm −3
Nuclear number density
4.794
3.930
α
—
Criticality parameter α
0.307
0.376
R O
cm
Threshold critical radius
8.37
6.345
r
cm
Expansion distance to crit shutdown
0.67
0.51
Efficiency
%
Efficiency
1.02
1.29
P(t crit )
10 15 Pa
Pressure at crit shutdown
4.72
4.87
Yield
kt
Explosive yield
12.4
5.6
t fiss
μs
Time to fission all nuclei
1.67
1.12
t crit
μs
Time to crit shutdown
1.54
1.04
N o
Neutron m −3
Initial neutron density
271.8
622.9
Initial number of neutrons = 1
Secondary neutron energy = 2 MeV
spreadsheet then computes and displays quantities such as the expansion distance
to second criticality, the fission and criticality timescales, the pressure within and
velocity of the core at second criticality, and the efficiency and yield.
When applied to a bare 53 kg core of
235 U (C = 1.155), CriticalityAnalytic.xls
indicates that the yield will be about 0.4 kilotons, with a core-expansion distance to
second criticality r of only 2.1 mm. This yield figure is not directly comparable
to the true ~13 kt yield of Little Boy, however, as that device was tamped; a more
realistic simulation of Little Boy that incorporates a tamper is discussed in the next
section. The sensitivity of calculations to input parameters is indicated by the fact
that if the core mass is raised to 66 kg (the full mass of the Little Boy core), the
estimated yield rises to ~8.5 kilotons!
How drastically does this analysis tend to overestimate efficiency? In Sect. 2.6
a program is described which carries out a time-dependent simulation of a tamped
core. Applying this program to a
235 U core of mass 68.8 kg as in Table 2.3 with no
tamper gives a predicted yield of only about 0.3 kt, about 1/40 of the analytical result
of ~12.4 kt. The reason for this drastic discrepancy is explored further at the end of
Sect. 2.6. In the meantime, there is a moral here: Beware of the danger of blindly
applying an impressive-looking formula.
It is important to emphasize that the above calculations cannot be applied to a
tamped core; that is, one cannot simply solve (2.47) and (2.48) for a core of some
specified mass and tamper of some outer radius and use the value of α so obtained
in the time and efficiency expressions established above. The reason for this has to
do with the distance r through which the core expands before second criticality,
Eq. (2.91). This expression derived from the fact that the criticality equation for the
2 Critical Mass, Efficiency, and Yield
Table 2.3 Criticality and efficiency parameters for C = 1.5, E f = 180 MeV, γ = 1/3; no tamper.
Masses are 68.8 kg for 235 U and 25.04 kg for 239 Pu
Quantity
Unit
Physical meaning
235 U
239 Pu
r initial
cm
Initial core radius
9.58
7.26
n
10 22 cm −3
Nuclear number density
4.794
3.930
α
—
Criticality parameter α
0.307
0.376
R O
cm
Threshold critical radius
8.37
6.345
r
cm
Expansion distance to crit shutdown
0.67
0.51
Efficiency
%
Efficiency
1.02
1.29
P(t crit )
10 15 Pa
Pressure at crit shutdown
4.72
4.87
Yield
kt
Explosive yield
12.4
5.6
t fiss
μs
Time to fission all nuclei
1.67
1.12
t crit
μs
Time to crit shutdown
1.54
1.04
N o
Neutron m −3
Initial neutron density
271.8
622.9
Initial number of neutrons = 1
Secondary neutron energy = 2 MeV
spreadsheet then computes and displays quantities such as the expansion distance
to second criticality, the fission and criticality timescales, the pressure within and
velocity of the core at second criticality, and the efficiency and yield.
When applied to a bare 53 kg core of
235 U (C = 1.155), CriticalityAnalytic.xls
indicates that the yield will be about 0.4 kilotons, with a core-expansion distance to
second criticality r of only 2.1 mm. This yield figure is not directly comparable
to the true ~13 kt yield of Little Boy, however, as that device was tamped; a more
realistic simulation of Little Boy that incorporates a tamper is discussed in the next
section. The sensitivity of calculations to input parameters is indicated by the fact
that if the core mass is raised to 66 kg (the full mass of the Little Boy core), the
estimated yield rises to ~8.5 kilotons!
How drastically does this analysis tend to overestimate efficiency? In Sect. 2.6
a program is described which carries out a time-dependent simulation of a tamped
core. Applying this program to a
235 U core of mass 68.8 kg as in Table 2.3 with no
tamper gives a predicted yield of only about 0.3 kt, about 1/40 of the analytical result
of ~12.4 kt. The reason for this drastic discrepancy is explored further at the end of
Sect. 2.6. In the meantime, there is a moral here: Beware of the danger of blindly
applying an impressive-looking formula.
It is important to emphasize that the above calculations cannot be applied to a
tamped core; that is, one cannot simply solve (2.47) and (2.48) for a core of some
specified mass and tamper of some outer radius and use the value of α so obtained
in the time and efficiency expressions established above. The reason for this has to
do with the distance r through which the core expands before second criticality,
Eq. (2.91). This expression derived from the fact that the criticality equation for the