Vol.2, No.3, 139-144 (2010) Natural Science
Copyright © 2010 SciRes. OPEN ACCESS
Student-level numerical simulation of conditions
inside an exploding fission-bomb core
Bruce Cameron Reed
Department of Physics Alma College, Alma, USA; reed@alma.edu
Received 2 December 2009; revised 13 January 2010; accepted 30 January 2010.
This paper describes a freely-available spread-
sheet that has been developed to simulate the
conditions of reaction rate, core acceleration and
velocity, energy generation, and pressure within
a detonating fission-bomb core. When applied
to a model of the Hiroshima Little Boy bomb, the
spreadsheet predicts a yield of 12.7 kilotons, a
figure in reasonable agreement with published
Keywords: Nuclear Weapons; Numerical
Simulation; Fission; Little Boy
The discipline of computational physics is now regarded
as possessing importance equal to the traditional areas of
experimental and theoretical studies. As such, it is im-
portant that our students be introduced early in their ca-
reers to the power of modern desktop computational
tools that can be used to model physically interesting
systems. Computational models can facilitate study of
physical systems where the theory cannot be solved
purely analytically and/or that are not easily realizable
experimentally. Sometimes a single graph can serve to
dramatically make a point about the behavior of a sys-
tem that is “latent” in the mathematics but whose mag-
nitude is not immediately apparent. This paper describes
and makes freely available an Excel spreadsheet to carry
out a student-level simulation of a system that exempli-
fies all of these points: the conditions of energy release,
pressure, fission rate, and expansion inside the core of a
detonating fission bomb. The physics of nuclear weap-
ons has been and will remain a fascinating and timely
subject. These devices are forbidding and mysterious to
students and the general public alike; by helping our
students to understand some of the details of their func-
tioning we can equip them to play constructive roles in
furthering public understanding of them.
The basic physics of criticality conditions in both
“bare” (untamped) and tamped fission cores is described
in two papers previously published by this author [1,2].
Because the time-dependence of conditions within a
fissioning core is highly non-linear, approximate analytic
expressionss had to be developed in those papers in or-
der to arrive at estimates of bomb yields and efficiencies.
The purpose of this paper is to utilize the theoretical
foundations established in those papers to build an ap-
proximate but easy-to-use numerical simulation of a core
whose mass, nuclear properties, and tamper properties
are set by the user. The resulting spreadsheet is made
freely downloadable to interested readers.
The structure of this paper is as follows. In Section 2 I
lay out the theoretical background; while this is adopted
directly from References [1] and [2] it is presented here
for sake of a self-contained discussion. The program-
ming of the simulation itself is described in Section 3,
and in Section 4 I present the results of a simulation of
the Hiroshima Little Boy uranium bomb. A brief sum-
mary is presented in Section 5.
The essential physical quantity within a fissioning bomb
core is the number density of neutrons, N. As described
in References [1] and [2], diffusion theory leads to an
approximate expression for the space and time- depend-
ence of N within a spherically symmetric core of the
eNtrN core
In this expression, No is the initial neutron number den-
sity at the center of the core, and is set by whatever initi-
ating device starts the chain reaction.
is the average
time that a neutron will travel before causing a fission,
fiss v
where vneut is the average neutron speed and core
the mean free path for neutrons between fissions in the
B. C. Reed / Natural Science 2 (2010) 139-144
Copyright © 2010 SciRes. OPEN ACCESS
where n is the nuclear number density and f
the fis-
sion cross-section of the fissile core material. dcore in
Eq.1 has units of length and can be thought of as fun-
damentally setting the size of the critical radius; it is
given by
d (4)
is the number of neutrons emitted per fission
(so-called secondary neutrons) and
core is the trans-
port mean free path for neutrons,
t is the so-called transport cross-section. If non-fission
neutron capture can be ignored (which should be the
case for any sensibly pure fissile material), this is given
by the sum of the fission and elastic-scattering cross-
 (6)
The parameter
in Eqs.1 and 4 arises in a separation
of variables in solving the diffusion equation for the
neutron density, and is itself time-dependent as it de-
pends on the core radius as described in what follows.
First consider the case of an untamped core, also known
as a bare core. As described in References [1] and [2],
application of appropriate boundary conditions to Eq.1
to ensure that there is no way for neutrons which have
passed out through the surface of the core to be reflected
back inside it from the external world leads to the fol-
lowing constraint for the core radius RC:
0 1 /2/3 /cot/ coreC
transcorecoreCcoreC dRddRdR
This is the criticality condition for a bare core. Satis-
faction of this constraint depends on
through its ap-
pearance in dcore. If
= 0, then the neutron number den-
sity is neither growing nor diminishing in time, a condi-
tion known as threshold criticality. (In an implosion
weapon, the moment when this state first occurs during
compression of a subcritical core into a supercritical
mass is known as first criticality. The present work is
not inteneded to model implosion scenarios, which are
much more complex on account of the inward motion of
the core at the moment when fissions are initiated.) In
this case dcore is completely determined by the fissility
parameters, and the value of RC which satisfies Eq.7 is
the bare threshold critical radius, thresh
R; for pure
U-235 this is about 8.4 cm, equivalent to about 46 kg.
As explained in Reference [1], if one starts with a core
of specified radius RC > thresh
R, then Eq.7 can be
solved for
(through dcore); one will find that
> 0,
which means that the neutron density is growing expo-
nentially in time, a condition known as “supercritical-
ity.” As the core rapidly (within a microsecond) expands
due to the extreme rate of release of energy by fissions,
will decline as a function of time until it reaches zero,
at which point the chain reaction will rapidly shut down;
this situation is known as “second criticality” and effec-
tively marks the end of the detonation phase.
In Reference [2], it is shown that if the core is sur-
rounded by a snugly-fitting tamper of non-fissile and
non-neutron-absorbing material, diffusion theory leads
to the following expressions for the neutron density wi-
thin the tamper material:
tamptamp drdr
tamp (8)
is as above and
is again the mean travel time
for neutrons between fissions in the core, A and B are
constants of integration to be determined by boundary
conditions, and dtamp is given by
d (9)
where tamp
is the mean free path for neutron transport
in the tamper material, analogous to Eq.5 except that the
tamper material is presumed to have no fission cross-
trans n
If the core and tamper have outer radii Rcore and Rtamp,
then demanding the continuity of neutron density and
flux at the core/tamper interface and again requiring that
that no neutrons which escape from the tamper to the
external world can be reflected back, one finds that the
criticality conditions emerge as
1 2Rcore
2 Rcore
cot Rcore
 1
0 ,0 
trans (11)
e2xct xt
xccotxc 1
xct 1
Rtamp 2
tamp xt1
B. C. Reed / Natural Science 2 (2010) 139-144
Copyright © 2010 SciRes. OPEN ACCESS
xccotxc 1
Rtamp 2
tamp xt1
Eq.11 corresponds to tamped threshold (
= 0) criti-
cality. Once values for the d’s and
’s are given the only
unknown is Rcore
thresh , the core radius for tamped thresh-
old criticality. In using Eqs.12 and 13 the idea is that one
again specifies the mass and hence radius of the core,
Rcore (>thresh
R), and solves for
. This would presuma-
bly be the value of
when fissions are initiated at t =
0; as the core expands
will subsequently decline until
second criticality is reached. A tamper serves to increase
the efficiency (and hence the yield) of a weapon through
two effects. First, by briefly retarding the expansion of
the core, the tamper causes
to remain greater for a
longer time than it would have otherwise; this is benefic-
ial as the rate of fissions – and hence the energy release – de-
pends exponentially on
. Second, the tamper serves to
reflect neutrons back into the core, effectively decreas-
ing the loss of fission-causing neutrons from within the
core to the outside world. In modern weapons – engi-
neering parlance a tamper is known as a reflector; I re-
tain the historical terminology. The retardation effect is
difficult to model analytically at this level and so is not
accounted for in Eqs.12 and 13; they do, however, in-
clude the reflection effect in the boundary conditions
used to establish them. The retardation effect is treated
approximately in the simulation as described following
Eq.17 below.
We can now consider the time-dependence of various
quantities in order to begin formulating a simulation. At
a moment when the core has volume Vcore, the rate of
fissions R(t) (fission/sec) is given by
coreo e
If each fission releases energy Efiss (typically ~ 180
MeV), then the rate of energy release within the core is
fisscoreo e
The total energy liberated to a given time can be
tracked by numerically integrating Eq.15; this deter-
mines the pressure within the core as a function of time.
This follows from the thermodynamic pressure – energy
density relationship
 
The choice of the parameter
depends on whether gas
pressure (
= 2/3) or radiation pressure (
= 1/3) is
dominant; the latter dominates for per-particle energies
greater than about 2 keV and will presumably be the
case for the later, more energetic stages of the reaction. I
use the core volume in Eq.16 on the rationale that the
fission products which cause the gas/radiation pressure
will likely largely remain within the core.
Following Reference [1], I model the core as an ex-
panding sphere of radius r(t) with all parts of the sphere
moving at speed v(t), driven by the energy release from
fissions. Do not confuse this velocity with the average
neutron speed, which does not directly come into this
part of the development (it does enter implicitly, how-
ever, through
). Invoking the work-energy theorem in
its thermodynamic formulation, W = P(t) dV, I equate
the work done by the gas (or radiation) pressure in
changing the core volume by dV over time dt to the
change in the core’s kinetic energy over that time:
P (17)
For simplicity in developing the simulation, I treat the
tamper as remaining of constant density. Now, it is de-
sirable to make some effort to account for the retarding
effect of the tamper on the core. To do this, I treat the
dK/dt term in Eq.17 as involving the velocity of the core
expansion but with the mass involved being that of the
core plus that of the tamper. The dV/dt term is taken to
apply to the core. With r as the radius and v the expan-
sion velocity of the core, we have
core total
Et dV dK
Vt dtdt
Vcore t
 4
 1
With dr/dt = v
tccore MV
Numerically integrating this result will give v(t) =
dr/dt, which can be integrated to give the core radius as a
function of time.
I have developed an Excel spreadsheet to carry out the
calculations described above. This is freely available at
http://othello.alma.edu/~reed/FissionCore.xls. In this Se-
ction I describe the general layout of this spreadsheet;
some results are described in Section 4 below.
B. C. Reed / Natural Science 2 (2010) 139-144
Copyright © 2010 SciRes. OPEN ACCESS
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
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
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
norm RCC
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
(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.
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
B. C. Reed / Natural Science 2 (2010) 139-144
Copyright © 2010 SciRes. OPEN ACCESS
tungsten-carbide tamper of outer radius 18 cm (mass 311 kg),
Sheet 1 of the author’s spreadsheet indicates that the
tamped threshold critical mass of U-235 in this configu-
ration is 18.4 kg, about a 60% reduction from the bare
threshold critical mass of 45.9 kg. Figure 1 shows that
the run of
(r) for this situation is quite linear out to the
computed second criticality radius of 12.04 cm. Upon
Table 1. Parameters for U-235 core/tungsten-carbide tamper
Figure 1. Criticality parameter
as a function of radius for the
simulation of the Little Boy 64-kg U-235 core plus 311-kg
tungsten-carbide tamper.
(r) is approximately linear, with
slope -20.35 m-1 and intercept 2.45 m.
Figure 2. Criticality parameter
(thin line, left scale) and
cumulative energy yield in kilotons (thick ascending solid line,
right scale) for the Little Boy simulation.
Figure 3. Logarithmic plots of pressure in Pa (thin line, right
scale) and rate of fissions per second (thick descending solid
line, left scale) for the Little Boy simulation.
= 1/3, an equivalence of 4.2 × 1012 Joule/kt
and a single initial neutron to start the reaction (No = 292
m-3), Figures 2 and 3 show the results of the simulation.
The brevity and violence of the detonation are aston-
ishing. The vast majority of the energy is liberated
within an interval of about 0.1
s. The pressure peaks at
close to 5 × 1015 Pa, or about 50 billion atmospheres,
equivalent to about one-fifth of that at the center of the
Sun, and the fission rate peaks at about 4 × 1031 per sec-
ond. The core acceleration peaks at about 1.5 × 1012 m/s2
at t – 0.9
s, and second criticality occurs at t – 1.05
at which time the core expansion velocity is about 270
km/s. These graphs dramatically illustrate what Robert
Serber wrote in The Los Alamos Primer : “Since only the
last few generations will release enough energy to pro-
duce much expansion, it is just possible for the reaction
to occur to an interesting extent before it is stopped by
the spreading of the active material” [4]. The predicted
yield of Little Boy from the present model is 12.7 kt.
This result is in surprisingly good agreement with the
estimated – 12 kt yield published by Penney, et al. [5]. At
a fission yield of 17.59 kt per kg of pure U-235, this
represents an efficiency of only about 1.1% for the 64 kg
core. Some of this agreement must be fortuitous, how-
ever, in view of the approximations incorporated in the
present model. That the yield estimate needs to be taken
with some skepticism is demonstrated by the fact that
increasing the initial number of neutrons to 10 increases
the yield to 18.7 kt. However, this change does not much
affect the timescale or the peak pressure and fission rates.
Users who download FissionCore.xls will find this ex-
ample pre-loaded.
This paper describes the development of a spreadsheet
for simulating the conditions within a detonating fis-
sion-bomb core. The simulation is straightforward
enough to be used with students, and for a simulation of
B. C. Reed / Natural Science 2 (2010) 139-144
Copyright © 2010 SciRes. OPEN ACCESS
the Hiroshima Little Boy bomb predicts a yield in rea-
sonable accord with published values. This type of
simulation can help students grasp some of the underly-
ing physics of and get a sense of the extreme physical
conditions that briefly occur within such devices.
[1] Reed, C. (2007) Arthur Compton’s 1941 report on explo-
sive fission of U-235: A look at the physics. American
Journal of Physics, 75(12), 1065-1072.
[2] Reed, C. (2009) A brief primer on tamped fission-bomb
cores. American Journal of Physics, 77(8), 730-733.
[3] http://nuclearweaponarchive.org/Nwfaq/Nfaq8.html
[4] Serber, R. (1992) The Los Alamos primer: The first lec-
tures on how to build an atomic bomb. University of
California, Berkeley, CA, 12.
[5] Penney, L., Samuels, D.E.J. and Scorgie, G.C. (1970) The
Nuclear Explosive Yields at Hiroshima and Nagasaki.
Philosophical Transactions of the Royal Society of Lon-
don, A266, 357-424.