6.5 Appendix E: Formal Derivation of the Bohr-Wheeler Spontaneous Fission Limit
199
6.5 Appendix E: Formal Derivation of the Bohr-Wheeler
Spontaneous Fission Limit
6.5.1 Introduction
In Sects. 1.7 and 1.10, we used a simplified model of a fissioning nucleus to get a
sense of how the limit against spontaneous fission (SF), (Z
2 /A) = 2(a S /a C ) ~ 48,
arises, a result first derived by Bohr and Wheeler (1939). Given the significance of
this result, a formal derivation of it is presented here. This approach is somewhat
unusual in comparison to most texts, which do not present detailed derivations of this
work. Some do offer partial treatments based on starting from expressions for the
surface area and self-energy of an ellipsoid of variable eccentricity (see, for example,
Bernstein and Pollock 1979 or Cottingham and Greenwood 2001), but the ellipsoidal
model does not reflect the approach taken by B&W, who used a sum of Legendre
polynomials to describe the shape of the surface of a distorted nucleus. While it
is true that it should not matter how the distortion is modeled if the SF limit is a
matter of instability against slight distortions, it seems unfortunate that pedagogical
tendency has shifted away from historical accuracy.
The popularity of the ellipsoidal model is due to the fact that the mathematics
of the B&W analysis is tricky, even if one is facile with multivariable calculus and
Legendre polynomials. B&W published virtually none of the details of their work,
which they referred to as a “straightforward calculation.” Soon after B&W’s paper
appeared, Present and Knipp (1940a, b) pointed out that it contained an internal
inconsistency, and that B&W had changed the definition of some of the surfacedistortion parameters part-way through the derivation. In a paper that now seems
largely forgotten, Plesset (1941) reconstructed the details of the B&W derivation,
but his work is difficult to follow in view of some tangled notation and the fact that
he carried through his algebra to higher orders of perturbation than are necessary to
understand the SF limit.
In reconstructing the B&W derivation, one faces the question of what level of detail
to present. To lay out every step of the algebra would be far too lengthy for sensible
publication. Conversely, the danger of brevity is that subtle but important points can
get overlooked. Here I try to tread a middle path by setting down benchmark steps in
the calculations between which readers should be able fill in the intervening details.
No treatment is given here of the much more complex question of the fission barrier,
which requires carrying the algebra to higher orders of perturbation.
This derivation, which is adopted from Reed (2009), is rather lengthy. In
Sect. 6.5.2, the Legendre-polynomial model of a distorted nucleus is described, and
the calculation of the volume of the nucleus is carried out. The surface area energy
is calculated in Sect. 6.5.3. Section 6.5.4 deals with the calculation of the Coulomb
self-energy of the nucleus, which, when combined with the results of the preceding
sub-sections, leads to understanding how the SF limit arises.
199
6.5 Appendix E: Formal Derivation of the Bohr-Wheeler
Spontaneous Fission Limit
6.5.1 Introduction
In Sects. 1.7 and 1.10, we used a simplified model of a fissioning nucleus to get a
sense of how the limit against spontaneous fission (SF), (Z
2 /A) = 2(a S /a C ) ~ 48,
arises, a result first derived by Bohr and Wheeler (1939). Given the significance of
this result, a formal derivation of it is presented here. This approach is somewhat
unusual in comparison to most texts, which do not present detailed derivations of this
work. Some do offer partial treatments based on starting from expressions for the
surface area and self-energy of an ellipsoid of variable eccentricity (see, for example,
Bernstein and Pollock 1979 or Cottingham and Greenwood 2001), but the ellipsoidal
model does not reflect the approach taken by B&W, who used a sum of Legendre
polynomials to describe the shape of the surface of a distorted nucleus. While it
is true that it should not matter how the distortion is modeled if the SF limit is a
matter of instability against slight distortions, it seems unfortunate that pedagogical
tendency has shifted away from historical accuracy.
The popularity of the ellipsoidal model is due to the fact that the mathematics
of the B&W analysis is tricky, even if one is facile with multivariable calculus and
Legendre polynomials. B&W published virtually none of the details of their work,
which they referred to as a “straightforward calculation.” Soon after B&W’s paper
appeared, Present and Knipp (1940a, b) pointed out that it contained an internal
inconsistency, and that B&W had changed the definition of some of the surfacedistortion parameters part-way through the derivation. In a paper that now seems
largely forgotten, Plesset (1941) reconstructed the details of the B&W derivation,
but his work is difficult to follow in view of some tangled notation and the fact that
he carried through his algebra to higher orders of perturbation than are necessary to
understand the SF limit.
In reconstructing the B&W derivation, one faces the question of what level of detail
to present. To lay out every step of the algebra would be far too lengthy for sensible
publication. Conversely, the danger of brevity is that subtle but important points can
get overlooked. Here I try to tread a middle path by setting down benchmark steps in
the calculations between which readers should be able fill in the intervening details.
No treatment is given here of the much more complex question of the fission barrier,
which requires carrying the algebra to higher orders of perturbation.
This derivation, which is adopted from Reed (2009), is rather lengthy. In
Sect. 6.5.2, the Legendre-polynomial model of a distorted nucleus is described, and
the calculation of the volume of the nucleus is carried out. The surface area energy
is calculated in Sect. 6.5.3. Section 6.5.4 deals with the calculation of the Coulomb
self-energy of the nucleus, which, when combined with the results of the preceding
sub-sections, leads to understanding how the SF limit arises.