4.3 Predetonation Yield
161
estimate) was found to be S = 0, and this value is adopted here (Users can change
this in the corresponding spreadsheet if desired; see below.) In this case, P escape is
directly equal to the spherical-escape probability (4.9) derived in Appendix F:
P escape =
3
8x 3
2x
2
+ e
−2x
(2x + 1) − 1
,
(4.24)
where x = σ total nR core .
The spreadsheet FissionYield.xls carries out the foregoing calculations. A spherical core is assumed. The user enters values for the mass and density of the core material, the mass of spontaneously-fissioning material, values for the cross-sections and
number of neutrons per spontaneous fission, and values for τ, t O , F, and α O . Upon
entering a value for t init , the non-predetonation probability and fractional yield are
calculated. As with the pre-detonation calculations, the spreadsheets take an upper
limit of k = 20 when computing (4.23). τ, t O , F and α O are not used in the calculation
of the probability, but are needed to compute the yield according to (4.21).
Figure 4.7 shows results obtained for a 6-kg Pu core contaminated with 1%, 6%,
and 20% Pu-240 by mass for (τ , F, α O , t O ) = (10
–8 s, 45, 1, 10 μs). Computations
were run for t init from 0.5 to 9.5 μs in steps of 0.5 μs. The probability of achieving
full yield with 1% contamination is about 80%; for 6% contamination the full-yield
chance falls to about 27%.
The curve for 20% contamination corresponds to what one would expect for
reactor-grade plutonium. The chance of achieving any sensible fraction of the design
yield with such contamination is abysmal. While this might seem comforting when
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
Probability of realizing yield fraction
Yield fraction
6 %
20 %
1%
Fig. 4.7 Probability of achieving a given fractional yield as a function of fractional yield for a 6-kg
core of 239 Pu of normal density (15.6 gr cm −3 ) contaminated with 1%, 6%, and 20% 240 Pu (top
to bottom curves). (τ , F, α O , t O ) = (10 –8 s, 45, 1, 10 –5 s). The values of the nuclear constants for
239 Pu are (σ f , σ el ) = (1.800 bn, 4.394 bn); for Pu-240, (ν, t 1/2 ) = (2.257, 1.14 × 10 11 year)
161
estimate) was found to be S = 0, and this value is adopted here (Users can change
this in the corresponding spreadsheet if desired; see below.) In this case, P escape is
directly equal to the spherical-escape probability (4.9) derived in Appendix F:
P escape =
3
8x 3
2x
2
+ e
−2x
(2x + 1) − 1
,
(4.24)
where x = σ total nR core .
The spreadsheet FissionYield.xls carries out the foregoing calculations. A spherical core is assumed. The user enters values for the mass and density of the core material, the mass of spontaneously-fissioning material, values for the cross-sections and
number of neutrons per spontaneous fission, and values for τ, t O , F, and α O . Upon
entering a value for t init , the non-predetonation probability and fractional yield are
calculated. As with the pre-detonation calculations, the spreadsheets take an upper
limit of k = 20 when computing (4.23). τ, t O , F and α O are not used in the calculation
of the probability, but are needed to compute the yield according to (4.21).
Figure 4.7 shows results obtained for a 6-kg Pu core contaminated with 1%, 6%,
and 20% Pu-240 by mass for (τ , F, α O , t O ) = (10
–8 s, 45, 1, 10 μs). Computations
were run for t init from 0.5 to 9.5 μs in steps of 0.5 μs. The probability of achieving
full yield with 1% contamination is about 80%; for 6% contamination the full-yield
chance falls to about 27%.
The curve for 20% contamination corresponds to what one would expect for
reactor-grade plutonium. The chance of achieving any sensible fraction of the design
yield with such contamination is abysmal. While this might seem comforting when
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
Probability of realizing yield fraction
Yield fraction
6 %
20 %
1%
Fig. 4.7 Probability of achieving a given fractional yield as a function of fractional yield for a 6-kg
core of 239 Pu of normal density (15.6 gr cm −3 ) contaminated with 1%, 6%, and 20% 240 Pu (top
to bottom curves). (τ , F, α O , t O ) = (10 –8 s, 45, 1, 10 –5 s). The values of the nuclear constants for
239 Pu are (σ f , σ el ) = (1.800 bn, 4.394 bn); for Pu-240, (ν, t 1/2 ) = (2.257, 1.14 × 10 11 year)