2.6 Estimating Yield—Numerical
95
With this, the expansion speed of the system core and the core and tamper
radii are updated according as v(t) = v(t) + v and r(t) = r(t) + v(t)t.
(viii) Increment time according as t = t + t, and return to step (iii) to begin the
next timestep; continue until second criticality is reached when α = 0. At the
beginning of each timestep, the core and tamper densities must be updated,
as well as their nuclear number densities, mean free paths, and the neutron
travel time between fissions.
The assumption that the core and tamper experience the same expansion speed
is a quite arbitrary one for sake of simplicity of the programming. Other assumptions could be made (such as, perhaps, having the tamper retain a constant density),
none of which are likely to be particularly realistic. Nuclear engineers speak of
the “snowplow” effect, where high-density tamper material piles up outside the
expanding core/tamper interface. But the point here is an order-of-magnitude pedagogical model. Also, this same assumption is made in a simplified yield model
developed in Sect. 2.8.4; this allows for some cross-checking of results.
What of the timestep t? In setting this, it is helpful to appreciate that it is not
necessary to start a simulation at t = 0. From (2.111), little energy will be released
while (α/τ )t is small. An example using
235 U will help make this clear. With τ ~
8.64 × 10
−9 s (Table 2.2). and, say, α ~ 0.5, then (α/τ ) ~ 5.8 × 10
7 s
−1 . Starting a
simulation at t = 10
−8 s should thus sacrifice no accuracy. However, the choice of a
timestep t is a sensitive issue, since the rate of energy release grows exponentially at
later times. For a function of the form y = exp[(α/τ )t], the fractional change in y over
a time t will be dy/y = (α/τ ) t; to have dy/y be small suggests adopting a value
of t no larger than the inverse of (α/τ ), which is about 1.7 × 10
−8 s. Consequently,
all of the results described in what follows utilized a starting time of 10
−8 s, and a
timestep of t = 5 × 10
−10 s; a run to a final time of 1.1 microseconds will involve
nearly 2,200 timesteps. With this value of t, dy/y ~ 0.029.
This author has developed a FORTRAN program for carrying out this simulation;
the code and an accompanying user manual are available upon request.
2.6.1 A Simulation of the Hiroshima Little Boy Bomb
Figures 2.16 and 2.17 show the results of a simulation of an idealized Little Boy
configuration: a 53 kg
235 U core plus a 550 kg WC tamper. The initial core radius is
8.78 cm, and the initial outer radius of the tamper is 20.86 cm. The initial number
of neutrons was set to be one. Clearly, the exponential parameter α remains largely
unchanged during the time over which the bulk of the energy is emitted. Second
criticality occurs when the core has expanded to a radius of 11.08 cm at a time
of 1.18 μs, an expansion distance of r = 2.3 cm. As remarked earlier, a tamper
significantly affects the expansion distance over which criticality holds. The total
yield is estimated at 11.95 kt; this is discussed further below.
95
With this, the expansion speed of the system core and the core and tamper
radii are updated according as v(t) = v(t) + v and r(t) = r(t) + v(t)t.
(viii) Increment time according as t = t + t, and return to step (iii) to begin the
next timestep; continue until second criticality is reached when α = 0. At the
beginning of each timestep, the core and tamper densities must be updated,
as well as their nuclear number densities, mean free paths, and the neutron
travel time between fissions.
The assumption that the core and tamper experience the same expansion speed
is a quite arbitrary one for sake of simplicity of the programming. Other assumptions could be made (such as, perhaps, having the tamper retain a constant density),
none of which are likely to be particularly realistic. Nuclear engineers speak of
the “snowplow” effect, where high-density tamper material piles up outside the
expanding core/tamper interface. But the point here is an order-of-magnitude pedagogical model. Also, this same assumption is made in a simplified yield model
developed in Sect. 2.8.4; this allows for some cross-checking of results.
What of the timestep t? In setting this, it is helpful to appreciate that it is not
necessary to start a simulation at t = 0. From (2.111), little energy will be released
while (α/τ )t is small. An example using
235 U will help make this clear. With τ ~
8.64 × 10
−9 s (Table 2.2). and, say, α ~ 0.5, then (α/τ ) ~ 5.8 × 10
7 s
−1 . Starting a
simulation at t = 10
−8 s should thus sacrifice no accuracy. However, the choice of a
timestep t is a sensitive issue, since the rate of energy release grows exponentially at
later times. For a function of the form y = exp[(α/τ )t], the fractional change in y over
a time t will be dy/y = (α/τ ) t; to have dy/y be small suggests adopting a value
of t no larger than the inverse of (α/τ ), which is about 1.7 × 10
−8 s. Consequently,
all of the results described in what follows utilized a starting time of 10
−8 s, and a
timestep of t = 5 × 10
−10 s; a run to a final time of 1.1 microseconds will involve
nearly 2,200 timesteps. With this value of t, dy/y ~ 0.029.
This author has developed a FORTRAN program for carrying out this simulation;
the code and an accompanying user manual are available upon request.
2.6.1 A Simulation of the Hiroshima Little Boy Bomb
Figures 2.16 and 2.17 show the results of a simulation of an idealized Little Boy
configuration: a 53 kg
235 U core plus a 550 kg WC tamper. The initial core radius is
8.78 cm, and the initial outer radius of the tamper is 20.86 cm. The initial number
of neutrons was set to be one. Clearly, the exponential parameter α remains largely
unchanged during the time over which the bulk of the energy is emitted. Second
criticality occurs when the core has expanded to a radius of 11.08 cm at a time
of 1.18 μs, an expansion distance of r = 2.3 cm. As remarked earlier, a tamper
significantly affects the expansion distance over which criticality holds. The total
yield is estimated at 11.95 kt; this is discussed further below.