24
1 Energy Release in Nuclear Reactions, Neutrons, Fission, and Characteristics …
We can now begin to consider the question of the limiting value of Z
2 /A. If the
total energy of the two nuclei in the fission circumstance shown in Fig. 1.4b is less
than that for the original nucleus of Fig. 1.4a, then the system will proceed to fission.
That is, spontaneous fission will occur if
U
orig
S
+ U
orig
C
> U
f iss
S
+ U
f iss
C
.
(1.58)
Substituting (1.48)–(1.50), (1.52), and (1.54)–(1.57) into (1.58) shows that
spontaneous fission will occur for
Z
2
A
>
a S (α − 1)
0.72 (1 − β − γ )
.
(1.59)
Estimating the Z
2 /A stability limit apparently demands selecting an appropriate
mass ratio and knowing the value of a S . For the latter, we could adopt a value from the
semi-empirical mass formula, but it is more satisfying to derive one based on some
direct physical grounds. We take up this issue now; the question of an appropriate
mass ratio will be addressed shortly.
To calibrate the value of a S , we appeal to the fact that fission can be induced by
slow neutrons with Q ~ 170 MeV of energy being liberated. In the present notation
this appears as
U
orig
S
+ U
orig
C
−
U
∞
S
+ U
∞
C
= Q,
(1.60)
where U
∞
S and U
∞
C respectively designate the areal and Coulombic energies of the
system when the product nuclei are infinitely far apart. Since the areas of the product
nuclei do not change following fission,U
∞
S = U
f iss
S
; see (1.49) and (1.50). U
∞
C is
given by (1.53) without the point-charge interaction term, that is, (1.54) without the
γ term. From these we find
a S =
Q/A
2/3
− 0.72
Z
2
/A
(1 − β)
(1 − α)
,
(1.61)
where A and Z refer to the parent nucleus in the fission reaction, not the general Z
2 /A
spontaneous-fission limit we seek.
Values of a S derived in this way from a number of fission reactions involving
235 U
are shown in Table 1.2. The first reaction is representative of the Hahn and Strassmann
fission-discovery reaction. The second one is concocted to have the masses of the
fission products as 139 and 95, values claimed by Weinberg and Wigner (1958, p. 30)
to be the most probable mass yields in slow-neutron fission of
235 U. The last two
reactions are less probable ones chosen to give a sense of how sensitive a S is to
the choice of calibrating reaction. The mass ratios are those of the fission products,
neglecting any neutrons emitted. As one might hope if a S reflects some fundamental
underlying physics, its value is fairly insensitive to the choice of calibrating reaction.
1 Energy Release in Nuclear Reactions, Neutrons, Fission, and Characteristics …
We can now begin to consider the question of the limiting value of Z
2 /A. If the
total energy of the two nuclei in the fission circumstance shown in Fig. 1.4b is less
than that for the original nucleus of Fig. 1.4a, then the system will proceed to fission.
That is, spontaneous fission will occur if
U
orig
S
+ U
orig
C
> U
f iss
S
+ U
f iss
C
.
(1.58)
Substituting (1.48)–(1.50), (1.52), and (1.54)–(1.57) into (1.58) shows that
spontaneous fission will occur for
Z
2
A
>
a S (α − 1)
0.72 (1 − β − γ )
.
(1.59)
Estimating the Z
2 /A stability limit apparently demands selecting an appropriate
mass ratio and knowing the value of a S . For the latter, we could adopt a value from the
semi-empirical mass formula, but it is more satisfying to derive one based on some
direct physical grounds. We take up this issue now; the question of an appropriate
mass ratio will be addressed shortly.
To calibrate the value of a S , we appeal to the fact that fission can be induced by
slow neutrons with Q ~ 170 MeV of energy being liberated. In the present notation
this appears as
U
orig
S
+ U
orig
C
−
U
∞
S
+ U
∞
C
= Q,
(1.60)
where U
∞
S and U
∞
C respectively designate the areal and Coulombic energies of the
system when the product nuclei are infinitely far apart. Since the areas of the product
nuclei do not change following fission,U
∞
S = U
f iss
S
; see (1.49) and (1.50). U
∞
C is
given by (1.53) without the point-charge interaction term, that is, (1.54) without the
γ term. From these we find
a S =
Q/A
2/3
− 0.72
Z
2
/A
(1 − β)
(1 − α)
,
(1.61)
where A and Z refer to the parent nucleus in the fission reaction, not the general Z
2 /A
spontaneous-fission limit we seek.
Values of a S derived in this way from a number of fission reactions involving
235 U
are shown in Table 1.2. The first reaction is representative of the Hahn and Strassmann
fission-discovery reaction. The second one is concocted to have the masses of the
fission products as 139 and 95, values claimed by Weinberg and Wigner (1958, p. 30)
to be the most probable mass yields in slow-neutron fission of
235 U. The last two
reactions are less probable ones chosen to give a sense of how sensitive a S is to
the choice of calibrating reaction. The mass ratios are those of the fission products,
neglecting any neutrons emitted. As one might hope if a S reflects some fundamental
underlying physics, its value is fairly insensitive to the choice of calibrating reaction.