B. C. Reed / Natural Science 2 (2010) 139-144

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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