
B. C. Reed / Natural Science 2 (2010) 139-144
Copyright © 2010 SciRes. OPEN ACCESS
142
This spreadsheet consists of three interlinked sheets.
On the first, the user inputs fundamental data such as
core and tamper material densities, atomic weights,
cross-sections, the secondary neutron number, the aver-
age secondary-neutron energy, values for Ef and
, the
desired core mass, the outer radius of the tamper, and the
number of “initial neutrons” in the core at t = 0. These
are entered in convenient units such as g/cm3, barns, and
MeV; the spreadsheet subsequently carries out all calcu-
lations in MKS units. The Excel “Goal Seek” function is
then run three times, to establish values for 1) the bare
threshold critical radius, 2) the tamped threshold critical
radius, and 3) the value of
corresponding to the chosen
core mass. The masses in 1) and 2) are computed for
reference and for the fact that they are needed for some
calculations involving the expansion of the core as de-
scribed below. The chosen core mass should exceed that
corresponding to thresh
tamp
R.
A significant complexity in carrying out this simula-
tion is that one apparently needs to solve Eq.12 for the
value of
corresponding to each time-stepped core ra-
dius between first and second criticality: the fission rate,
energy generation rate, and pressure all depend on
as a
function of time. I have found, however, that
is usually
quite linear as a function of core radius. This behavior
greatly simplifies the actual time-dependent simulation.
Sheet 2 of the spreadsheet allows one to establish pa-
rameters for this linear behavior for the values of the
various parameters that the user inputs on Sheet 1. Here,
the user solves (again using the Goal Seek function) for
the value of
for 25 values of the radius. These start at
the initial core radius and proceed to 1.25 times the
value of the second-criticality radius for a bare core of
the mass chosen by the user on Sheet 1; this range ap-
pears to be suitable to establish the behavior of
. The
rationale for this arrangement is as follows. As shown in
Reference [2], if the chosen core mass is equal to C bare
threshold critical masses, criticality will hold over a
range of radii given by
1/ 21/ 3
thresh
bare
rC CR (19)
The presence of a tamper means that the core will ex-
pand somewhat beyond
r before second criticality is
reached, but Eq.19 sets the essential length scale of the
expansion. For convenience, Sheet 2 utilizes a “normal-
ized” radius defined as
thresh
tamp
thresh
tamp
norm RCC
RCr
r3/12/1
3/1
(20)
where C is now defined as the number of tamped thresh-
old critical masses. rnorm = 1 corresponds to the second
criticality radius one would compute from Eq.19 if it
applied as well to a tamped core. Sheet 2 tracks the
changing mass density, nuclear number densities, and
mean-free-paths within the core as a function of r. By
running the Goal Seek function on each of the 25 radii,
the user adjusts
in each case to render Eq.12 equal to
zero. The behavior of
(r) is then displayed in an au-
tomatically-generated graph. On a separate line with
fixed to a value very near zero (10-10 is built-in), the
user adjusts the radius to once again render Eq.12 equal
to zero, thus establishing the radius of second criticality
for his or her parameters. The slope and intercept of a
linear
(r) fit are then automatically computed in prepa-
ration for the next step.
The actual time-dependent simulation occurs on Sheet
3. The simulation is set up to involve 500 timesteps, one
per row. The initial core radius is transferred from Sheet
1 for t = 0. Because much of the energy release in a nu-
clear weapon occurs during the last few generation of
fissions before second criticality, this Sheet allows the
user to set up two different timescales: an “initial” one
(dtinit) intended for use in the first few rows of the Sheet
when a larger timestep can be tolerated without much
loss of accuracy, and a later one (dtlate), to be chosen
considerably smaller and used for the majority of the
rows. In this way a user can optimize the 500 rows to
both capture sufficient accuracy in the last few fission
generations and arrange that
(r) is just approaching
zero at the last steps of the process. Typical choices for
dtinit and dtlate might be a few tenths of a microsecond
and a few tenths of a nanosecond, respectively. At each
radius, the Sheet computes the value of
(r) from the
linear approximation of Sheet 2, the core volume, mass
density, nuclear number densities and mean free paths
within the core,
, rates of fission and energy generation,
pressure, and total energy liberated to that time. The ac-
celeration of the core is computed from Eq.18, and the
core velocity and radius are updated depending upon the
timestep in play; the new radius is transferred to the
subsequent row to seed the next step. The user is auto-
matically presented with graphs of
(r), the fission rate,
pressure, and total energy liberated (in kilotons equiva-
lent) as functions of time.
4. A SIMULATION OF THE HIROSHIMA
LITTLE BOY BOMB
As described in References [2] and particularly [3], the
Hiroshima Little Boy core comprised about 64 kg of en-
riched U-235 in a cylindrical configuration surrounded
by a cylindrical tungsten-carbide tamper of diameter and
length 13 inches, mass approximately 310 kg, and den-
sity 14.8 g /cm3. Values for the various core and tamper
parameters are given in Table 1; these are adopted from
Reference [2]. Assuming these values and taking the
core to be spherical (radius 9.35 cm at a density of 18.71
g/cm3; this figure is 235/238 times the density of natural
uranium, 18.95 g/cm3) and surrounded by a spherical