98
2 Critical Mass, Efficiency, and Yield
reaches second criticality, (2.95) consequently seriously overestimates the yield.
Some numbers for the 68.8 kg simulation are instructive. The initial core radius in
this case is 9.575 cm, and the initial value of α is 0.3096. The second-criticality radius
is 10.255 cm (r = 0.68 cm), but by the time that the radius has expanded to only
9.607 cm (an increase of only 0.336%), fully 90% of the final yield has already been
realized. By this time, α has dropped by only about 4.5% from its initial value, but the
reaction has already begun shutting down. (It is true that Fig. 2.16 is a tamped-core
simulation, but the behavior of α is very similar for an untamped core.)
Can (2.95) be modified to account for this problem? Here is a possible approach:
When integrating (2.93) to determine the time of second criticality, replace the upper
limit of integration r i + r with r i (1 + f), where f is the fractional increase in the
core radius corresponding to that time at which you think the reaction begins shutting
down; for example, for the above numbers, f = 0.0034 corresponds to 90% energy
release. Carrying out the integral shows that the yield emerges as (2.95) except that
the factor of r in the numerator is replaced with fr i . For the present case of r i =
9.575 cm and f = 0.00336, this modification predicts a yield of 0.597 kt, just twice
the simulation result. There is obviously no preferred value of f to use, but this artifice
removes much of the discrepancy in a straightforward way.
The same program was used to simulate a 6.3 kg core of
239 Pu surrounded by a
230 kg tamper of depleted uranium, with both compressed to densities 2.5 times as
great as their normal densities to account for the implosion nature of this weapon.
(230 kg is the total mass of the Trinity tamper shells as described in the Preamble.)
In this case, the estimated yield is 17.2 kt, which at first glance seems in only fair
agreement with an estimated true yield of 21 kt (Malik 1985). However, since some
30% of Trinity’s yield was contributed by high-energy neutrons inducing fissions in
the DU tamper shell (that is, ~6 kt; Semkow et al. 2006), ~17 kt is reasonable for
the plutonium-only fraction of the yield. The results of these simulations should not
be over-interpreted, but it is encouraging to see that the model gives results of the
correct order of magnitude.
To close this section, a dose of perspective: Do not be too upset that Eq. (2.95) is
not very accurate. It pertains to an untamped core, and any serious bomb-maker will
incorporate a tamper. Ultimately, numerical simulations are what tell the tale of efficiency and yield. Also, treat discrepancies as valuable lessons: Analytic results have
a compelling attractiveness and are powerful for getting a sense of how something
depends on the parameters involved, but always be prepared to question underlying
assumptions.
2.7 History Lesson: Criticality Considered in 1939
In Sect. 2.2, we saw that the criticality condition for threshold criticality (α = 0) for
an untamped core can be expressed as (Eqs. 2.30 and 2.31)
x cot(x) + ε x − 1 = 0,
(2.115)
2 Critical Mass, Efficiency, and Yield
reaches second criticality, (2.95) consequently seriously overestimates the yield.
Some numbers for the 68.8 kg simulation are instructive. The initial core radius in
this case is 9.575 cm, and the initial value of α is 0.3096. The second-criticality radius
is 10.255 cm (r = 0.68 cm), but by the time that the radius has expanded to only
9.607 cm (an increase of only 0.336%), fully 90% of the final yield has already been
realized. By this time, α has dropped by only about 4.5% from its initial value, but the
reaction has already begun shutting down. (It is true that Fig. 2.16 is a tamped-core
simulation, but the behavior of α is very similar for an untamped core.)
Can (2.95) be modified to account for this problem? Here is a possible approach:
When integrating (2.93) to determine the time of second criticality, replace the upper
limit of integration r i + r with r i (1 + f), where f is the fractional increase in the
core radius corresponding to that time at which you think the reaction begins shutting
down; for example, for the above numbers, f = 0.0034 corresponds to 90% energy
release. Carrying out the integral shows that the yield emerges as (2.95) except that
the factor of r in the numerator is replaced with fr i . For the present case of r i =
9.575 cm and f = 0.00336, this modification predicts a yield of 0.597 kt, just twice
the simulation result. There is obviously no preferred value of f to use, but this artifice
removes much of the discrepancy in a straightforward way.
The same program was used to simulate a 6.3 kg core of
239 Pu surrounded by a
230 kg tamper of depleted uranium, with both compressed to densities 2.5 times as
great as their normal densities to account for the implosion nature of this weapon.
(230 kg is the total mass of the Trinity tamper shells as described in the Preamble.)
In this case, the estimated yield is 17.2 kt, which at first glance seems in only fair
agreement with an estimated true yield of 21 kt (Malik 1985). However, since some
30% of Trinity’s yield was contributed by high-energy neutrons inducing fissions in
the DU tamper shell (that is, ~6 kt; Semkow et al. 2006), ~17 kt is reasonable for
the plutonium-only fraction of the yield. The results of these simulations should not
be over-interpreted, but it is encouraging to see that the model gives results of the
correct order of magnitude.
To close this section, a dose of perspective: Do not be too upset that Eq. (2.95) is
not very accurate. It pertains to an untamped core, and any serious bomb-maker will
incorporate a tamper. Ultimately, numerical simulations are what tell the tale of efficiency and yield. Also, treat discrepancies as valuable lessons: Analytic results have
a compelling attractiveness and are powerful for getting a sense of how something
depends on the parameters involved, but always be prepared to question underlying
assumptions.
2.7 History Lesson: Criticality Considered in 1939
In Sect. 2.2, we saw that the criticality condition for threshold criticality (α = 0) for
an untamped core can be expressed as (Eqs. 2.30 and 2.31)
x cot(x) + ε x − 1 = 0,
(2.115)