1.10 A Semi-empirical Look at the Fission Barrier
37
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.00
0.20
0.40
0.60
0.80
1.00
1.20
f(x)
x
Fig. 1.12 Straight line: fission barrier function f linear (x) of (1.87). The curved line is the
interpolating function of (1.89) and (1.90), configured to give f smooth (x) → 0 as x →~ 1
Now consider, as did Bohr and Wheeler, fission into equal-mass product nuclei:
f = 1. In this case we have α = 1.25992, β = 0.62996, γ = 0.26248, and hence
a S A 2/3 = f linear (x) = 0.25992 − 0.21511 x.
(1.87)
Equation (1.87) predicts that
a S A
2/3 will decline linearly with x until it
reaches zero at x = (0.25992/0.21511) = 1.208; this behavior is shown as the straight
line in Fig. 1.12.
That this result predicts a fission barrier of zero for x > 1 indicates that a simple
“two-sphere” model of fission cannot be an accurate representation of the real shape
of a fissioning nucleus; we should have f (x) → 0 as x → 1. Presumably, f (x) should
have some shape more akin to the curve shown in Fig. 1.12. The precise recipe for
the curve shown is elucidated in what follows.
Following Bohr and Wheeler, we develop a plausible interpolating function for
f (x). Presuming (as did they) that f linear (x) accurately models the fission barrier for
nuclei with small values of x, we seek an interpolating function that satisfies four
criteria:
(i) f (x) = (a − 1) at x = 0
(ii) d f/dx = 2(β + γ − 1) at x = 0
(iii) f (x) = 0 at x = 1
(iv) d f/dx = 0 at x = 1
(1.88)
Conditions (i) and (ii) demand that f (x) behave as (1.87) for small values of x.
Condition (iii) is the Bohr and Wheeler limiting condition of (1.86), and condition
37
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.00
0.20
0.40
0.60
0.80
1.00
1.20
f(x)
x
Fig. 1.12 Straight line: fission barrier function f linear (x) of (1.87). The curved line is the
interpolating function of (1.89) and (1.90), configured to give f smooth (x) → 0 as x →~ 1
Now consider, as did Bohr and Wheeler, fission into equal-mass product nuclei:
f = 1. In this case we have α = 1.25992, β = 0.62996, γ = 0.26248, and hence
a S A 2/3 = f linear (x) = 0.25992 − 0.21511 x.
(1.87)
Equation (1.87) predicts that
a S A
2/3 will decline linearly with x until it
reaches zero at x = (0.25992/0.21511) = 1.208; this behavior is shown as the straight
line in Fig. 1.12.
That this result predicts a fission barrier of zero for x > 1 indicates that a simple
“two-sphere” model of fission cannot be an accurate representation of the real shape
of a fissioning nucleus; we should have f (x) → 0 as x → 1. Presumably, f (x) should
have some shape more akin to the curve shown in Fig. 1.12. The precise recipe for
the curve shown is elucidated in what follows.
Following Bohr and Wheeler, we develop a plausible interpolating function for
f (x). Presuming (as did they) that f linear (x) accurately models the fission barrier for
nuclei with small values of x, we seek an interpolating function that satisfies four
criteria:
(i) f (x) = (a − 1) at x = 0
(ii) d f/dx = 2(β + γ − 1) at x = 0
(iii) f (x) = 0 at x = 1
(iv) d f/dx = 0 at x = 1
(1.88)
Conditions (i) and (ii) demand that f (x) behave as (1.87) for small values of x.
Condition (iii) is the Bohr and Wheeler limiting condition of (1.86), and condition