38
1 Energy Release in Nuclear Reactions, Neutrons, Fission, and Characteristics …
(iv) ensures that f (x) will approach this limiting condition “smoothly.” Apparently,
a virtual infinitude of interpolating functions could be conceived. Here we follow
Bohr and Wheeler, adopting
f smooth (x) = F(1 − x)
3
+ B(1 − x)
4
,
(1.89)
where F and B are constants to be calibrated.
It is not clear from B&W’s paper precisely how they were led to this expression.
Indeed, in papers published in the spring of 1940, R. D. Present and J. K. Knipp of
Purdue University pointed out some algebraic inconsistencies in Bohr and Wheeler’s
treatment. Purely empirically, however, f smooth (x) has the tremendous advantage that
it automatically satisfies conditions (iii) and (iv) above, leaving the coefficients F and
B to be determined by conditions (i) and (ii). Thus, to modify the two-sphere model,
I adopt (1.89), with the coefficients F and B adjusted so that f smooth (x) respects the
values of f linear (0) and (df linear /dx) 0 of (1.87). This gives
F = 4(α − 1) + 2(β + γ − 1)
B = −3 (α − 1) − 2(β + γ − 1)
.
(1.90)
For f = 1, these give (B, F) = (0.82457, –0.56465); this is the smooth curve in
Fig. 1.12. With this function that now respects the correct limiting value of Z
2 /A, we
can write the energy necessary to distort a nucleus to the point of fission as
E = a S A
2/3 f smooth (x).
(1.91)
To calibrate this model, it is necessary to know the value for ΔE for some nuclide
whose x-value is known, in order that we can pin down the limiting value of Z
2 /A in
(1.86). To do this, I again follow Bohr and Wheeler’s lead. They used an estimate of
the fission barrier for the compound nucleus
239 U (i.e, that formed by neutron capture
by
238 U) of E ~ 6 MeV. From the table of fission barriers given in Appendix A, the
value of this is now estimated to be E ~ 6.21 MeV, so their choice was reasonable.
Adopting this value and setting a S = 18 MeV for consistency with Sect. 1.1.7, (1.89)–
(1.91) can be solved numerically for x; the result is x ~ 0.7652. In (1.86), this gives
(Z
2 /A) lim ~ 46.3, close to Bohr and Wheeler’s 1939 value of 47.8. With this, the
model is completely calibrated and can be used to predict fission values across the
entire range of mass numbers, provided a relationship between Z and A is specified
in order that a value of x can be computed for each value of A. For this purpose, if
the fit of Eq. (1.63) is reversed, one finds that
Z ∼ 0.6274 A
0.9167
.
(1.92)
Figure 1.13 shows the run of E vs. A for f = 1 upon assuming (1.92), a S =
18 MeV, and (Z
2 /A) lim = 46.3 as above.
1 Energy Release in Nuclear Reactions, Neutrons, Fission, and Characteristics …
(iv) ensures that f (x) will approach this limiting condition “smoothly.” Apparently,
a virtual infinitude of interpolating functions could be conceived. Here we follow
Bohr and Wheeler, adopting
f smooth (x) = F(1 − x)
3
+ B(1 − x)
4
,
(1.89)
where F and B are constants to be calibrated.
It is not clear from B&W’s paper precisely how they were led to this expression.
Indeed, in papers published in the spring of 1940, R. D. Present and J. K. Knipp of
Purdue University pointed out some algebraic inconsistencies in Bohr and Wheeler’s
treatment. Purely empirically, however, f smooth (x) has the tremendous advantage that
it automatically satisfies conditions (iii) and (iv) above, leaving the coefficients F and
B to be determined by conditions (i) and (ii). Thus, to modify the two-sphere model,
I adopt (1.89), with the coefficients F and B adjusted so that f smooth (x) respects the
values of f linear (0) and (df linear /dx) 0 of (1.87). This gives
F = 4(α − 1) + 2(β + γ − 1)
B = −3 (α − 1) − 2(β + γ − 1)
.
(1.90)
For f = 1, these give (B, F) = (0.82457, –0.56465); this is the smooth curve in
Fig. 1.12. With this function that now respects the correct limiting value of Z
2 /A, we
can write the energy necessary to distort a nucleus to the point of fission as
E = a S A
2/3 f smooth (x).
(1.91)
To calibrate this model, it is necessary to know the value for ΔE for some nuclide
whose x-value is known, in order that we can pin down the limiting value of Z
2 /A in
(1.86). To do this, I again follow Bohr and Wheeler’s lead. They used an estimate of
the fission barrier for the compound nucleus
239 U (i.e, that formed by neutron capture
by
238 U) of E ~ 6 MeV. From the table of fission barriers given in Appendix A, the
value of this is now estimated to be E ~ 6.21 MeV, so their choice was reasonable.
Adopting this value and setting a S = 18 MeV for consistency with Sect. 1.1.7, (1.89)–
(1.91) can be solved numerically for x; the result is x ~ 0.7652. In (1.86), this gives
(Z
2 /A) lim ~ 46.3, close to Bohr and Wheeler’s 1939 value of 47.8. With this, the
model is completely calibrated and can be used to predict fission values across the
entire range of mass numbers, provided a relationship between Z and A is specified
in order that a value of x can be computed for each value of A. For this purpose, if
the fit of Eq. (1.63) is reversed, one finds that
Z ∼ 0.6274 A
0.9167
.
(1.92)
Figure 1.13 shows the run of E vs. A for f = 1 upon assuming (1.92), a S =
18 MeV, and (Z
2 /A) lim = 46.3 as above.