1.7 The Bohr-Wheeler Theory of Fission …
25
Table 1.2 Fission reactions, derived surface energy parameter, and derived spontaneous fission
limit
Fission products of 1 n + 235 U
Q (MeV)
f
a S (MeV)
(Z 2 /A) lim
141
56 Ba + 92
36 Kr + 3
1
0 n
173.2
1.53
18.3
61.4
139
54 Xe + 95
38 Sr + 2
1
0 n
183.6
1.46
17.5
58.6
116
46 Pd + 116
46 Pd + 4
1
0 n
177.0
1.00
19.0
63.7
208
82 Pb + 26
10 Ne + 2
1
0 n
54.2
8.00
16.3
54.7
The values of a S derived here are consistent with those quoted in numerical fits of
the semi-empirical mass formula, ~ 18 MeV.
With a S in hand, we face the question of what value of f to use in (1.59) to establish
an estimate of the stability limit for Z
2 /A against spontaneous fission.
If the limiting value of Z
2 /A is to be a matter of fundamental physics, it should
in principle be independent of any choice for f , although evaluating (1.59) does
depend on the choice; in particular, it would make no sense to use the mass ratio for
an induced reaction used to calibrate a S to determine a limit against spontaneous
fission! To resolve this dilemma, recall that f could also have been defined as the
inverse of what was adopted in (1.45). The only value of f that is in any sense “unique”
is therefore f = 1. Indeed, plots of the right side of (1.59) versus f for fixed values
of a S reveals that a minimum always occurs at f = 1, symmetric in the sense of f →
1/f about f = 1. To establish a lower limit to the spontaneous-fission condition, let
us consequently take (1.59) evaluated at f = 1:
Z
2
/A
lim
∼ 3.356a S .
(1.62)
Limiting values of Z
2 /A so calculated are given in the last column of Table 1.2;
in each case these are based on the a S values in the preceding column of the Table.
While these are somewhat high compared to Bohr and Wheeler value of 48, the
agreement is respectable given the simplicity of the model.
Spreadsheet TwoSphereFission.xls allows a user to enter mass numbers and -
values for reactions like those in Table 1.2; the spreadsheet calculates values for Q,
f , α, β, γ , a S , and the limiting value of Z
2 /A.
To close this section, we use this analysis to estimate the value of Z beyond which
nuclei will be unstable against spontaneous fission. From data given in the online
version of the Nuclear Wallet Cards, one finds that there are 352 isotopes that are
either permanently stable or have half-lives > 100 years. A plot of A vs. Z for these
isotopes can be approximately fit by a power law, as shown in Fig. 1.5; the result is
A ∼ 1.6864Z
1.0870
r
2
= 0.9965
.
(1.63)
This fit slightly underestimates A(Z) for heavy nuclei, giving A ~ 230 for Z =
92, but is sufficiently accurate for our purposes. For a limiting Z
2 /A of 60, (1.63)
predicts a maximum stable Z of about 157; the Bohr and Wheeler value of 48 gives
a maximum Z of about 123.
25
Table 1.2 Fission reactions, derived surface energy parameter, and derived spontaneous fission
limit
Fission products of 1 n + 235 U
Q (MeV)
f
a S (MeV)
(Z 2 /A) lim
141
56 Ba + 92
36 Kr + 3
1
0 n
173.2
1.53
18.3
61.4
139
54 Xe + 95
38 Sr + 2
1
0 n
183.6
1.46
17.5
58.6
116
46 Pd + 116
46 Pd + 4
1
0 n
177.0
1.00
19.0
63.7
208
82 Pb + 26
10 Ne + 2
1
0 n
54.2
8.00
16.3
54.7
The values of a S derived here are consistent with those quoted in numerical fits of
the semi-empirical mass formula, ~ 18 MeV.
With a S in hand, we face the question of what value of f to use in (1.59) to establish
an estimate of the stability limit for Z
2 /A against spontaneous fission.
If the limiting value of Z
2 /A is to be a matter of fundamental physics, it should
in principle be independent of any choice for f , although evaluating (1.59) does
depend on the choice; in particular, it would make no sense to use the mass ratio for
an induced reaction used to calibrate a S to determine a limit against spontaneous
fission! To resolve this dilemma, recall that f could also have been defined as the
inverse of what was adopted in (1.45). The only value of f that is in any sense “unique”
is therefore f = 1. Indeed, plots of the right side of (1.59) versus f for fixed values
of a S reveals that a minimum always occurs at f = 1, symmetric in the sense of f →
1/f about f = 1. To establish a lower limit to the spontaneous-fission condition, let
us consequently take (1.59) evaluated at f = 1:
Z
2
/A
lim
∼ 3.356a S .
(1.62)
Limiting values of Z
2 /A so calculated are given in the last column of Table 1.2;
in each case these are based on the a S values in the preceding column of the Table.
While these are somewhat high compared to Bohr and Wheeler value of 48, the
agreement is respectable given the simplicity of the model.
Spreadsheet TwoSphereFission.xls allows a user to enter mass numbers and -
values for reactions like those in Table 1.2; the spreadsheet calculates values for Q,
f , α, β, γ , a S , and the limiting value of Z
2 /A.
To close this section, we use this analysis to estimate the value of Z beyond which
nuclei will be unstable against spontaneous fission. From data given in the online
version of the Nuclear Wallet Cards, one finds that there are 352 isotopes that are
either permanently stable or have half-lives > 100 years. A plot of A vs. Z for these
isotopes can be approximately fit by a power law, as shown in Fig. 1.5; the result is
A ∼ 1.6864Z
1.0870
r
2
= 0.9965
.
(1.63)
This fit slightly underestimates A(Z) for heavy nuclei, giving A ~ 230 for Z =
92, but is sufficiently accurate for our purposes. For a limiting Z
2 /A of 60, (1.63)
predicts a maximum stable Z of about 157; the Bohr and Wheeler value of 48 gives
a maximum Z of about 123.