B. C. Reed / Natural Science 3 (2011) 323-327

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326

2

2

23 15

16

C

Total SSC

S

aZ

UaA aAN

. (14)

With aS = 18 MeV and ao = 1.2 fm, 15aC/16aS ~ 0.0375.

These values are assumed in what follows.

3. PROGRAMS AND RESULTS

Two double-precision FORTRAN programs, ANGLE

and FISSION, have been developed to carry out the cal-

culations described above. Since the geometry of the

process depends only on the emergence angle θ, it is

convenient to have one program (ANGLE) generate a

tabulation of R, d, and S

as functions of θ, and a sec-

ond one (FISSION) carry out the numerical integration

for C

. The programs need only be run once, after

which the run of total energy for any (A, Z) as a function

of θ can be obtained by scaling S

and C

according

as Eq.14. ANGLE is written to step

in increments of

−0.01 radians from π/2 at the start of the process down

to π/6 at the moment of fission.

FISSION has been written to take advantage of the

axial symmetry of the model. The internal and inter-pair

potentials of the N

angular wedges need not be com-

puted for all individual and pairs of wedges since the

internal potentials of all wedges will be identical and the

potentials between all possible pairs of wedges can be

determined by computing the potential between any one

and half of all the others. The equatorial symmetry of the

model was not built into the program in the anticipation

that it might later be used to simulate non-symmetric

fissions. After some experimentation, it was found that

choosing (NZ, NR, N

) = (200, 100, 100) provided for

both reasonably expedient run times and sensible accu-

racy; a complete run requires about 12 hours on a Mac-

intosh G5 and typically returns a starting-configuration

energy accurate to better than 0.1% (see below).

Figure 2 shows results for (A, Z) = (90, 40) and (235,

92), that is, nuclides of zirconium and uranium, respec-

tively. The ordinate is the change in total configuration

energy in the sense (deformed nucleus – original nu-

cleus). Zirconium is used as an example because for real

nuclei, much more sophisticated models, such as that of

Myers and Swiatecki [6], reveal that fission barriers

peak at a value of ~ 55 MeV for nuclei with A ~ 90.

While there is some noise in the simulation due to the

finite number of lattice cells, the computed maximum of

~ 47 MeV for zirconium is in good agreement with this

peak value. For uranium, the computation gives a barrier

of ~ 13 MeV, somewhat high compared to the true value

of ~ 6 MeV but still respectably accurate considering the

simplicity of the model. At the moment of fission, ∆E in

the case of uranium has dropped to about −45 MeV. One

Figure 2. Evolution of (deformed - initial) configuration en-

ergy for nuclides with (A, Z) = (90, 40) (zirconium; upper

curve) and (235, 92) (uranium; lower curve). (aS, aC, ao) =

(0.72 MeV, 18 MeV, 1.2 fm). The limiting value of d/RO is

about 1.3685.

feature to note in Figure 1 is that in the case of uranium

the maximum distortion energy is reached at about the

middle of the fission process, whereas for zirconium the

maximum occurs at very near the end of the process.

The numerical precision achieved with the model can

be judged by examining how well it reproduces the con-

figuration energy for the initial undeformed nucleus.

This can be computed analytically as

2

23

1C

start S

S

aZ

UaA aA

. (15)

For (A, Z) = (235, 92) and (aC, aS) = (0.72 MeV, 18

MeV), this gives Ustart = 1673.0 MeV; the simulation

gives 1673.74 MeV, only 0.04% high.

Figure 3 shows how the results of the present model

compare to those Myers and Swiatecki’s seven-parame-

ter model. For the present model, this plot was formed

by computing ∆E(

) for values of A = 10, 15, 20, ···,

300 and searching for the maximum value of ∆E for

each A; the curve is interpolated. In computing these

maximum ∆E values it was assumed that the atomic

number Z for a given value of A could be modeled as Z ~

0.627A0.917 (1 < Z < 98), a purely empirical result based

on the (Z, A) trend of 352 nuclides with half-lives > 100

years [8]. The Myers and Swiatecki curve was generated

by fitting the empirical formula

n

o

AA

p

ECAe

(16)

to Figure 18 of their paper by a least-squares calculation.

The values of the parameters are (C, p, Ao, n) = (5.115

17, 0.592 21, 156.630 3, 2.537 96). This fit predicts a

slightly high peak value for the curve (~ 58 MeV vs. a

true value of ~ 55 MeV), but is convenient for compar-

ing the models. The present model predicts somewhat

high values of

E at both low and high values of A and

somewhat low ones for intermediate values, but it does