6.5 Appendix E: Formal Derivation of the Bohr-Wheeler Spontaneous Fission Limit
211
At this point, (6.61) goes back into (6.47) to give the overall Coulomb self-energy
of the nucleus as
U C =
ρ
2
8 π ε o
⎧
⎨
⎩
−
2 π
3
θ,φ
r 1( θ)
0
P 0(1) r
4
1 dr 1 dΩ 1 +
+ 2π R
2
O
(1 + α 0 )
2
+
1
5
α
2
2
θ,φ
r 1( θ)
0
P 0(1) r
2
1 dr 1 dΩ 1
+
4π
5
α 2
1 + α 0
−
πα
2
2
(1 + α 0 )
2
[2, 2, 2]
θ,φ
r 1( θ)
0
P 2(1) r
4
1 dr 1 dΩ 1
⎫
⎬
⎭
. (6.62)
The integrals in (6.62) proceed as did the U 2 integrals above. Integrating over r 1
in the first integral will yield r
5
1 (θ )
5, but it is not necessary to keep all of the terms
when writing out (6.25) raised to the fifth power; we need only keep terms up to
order α
2
2 . That integral then proceeds with use of (6.31) and (6.32). To order α
2
2 and
including a factor of 2π from integrating over φ, we have
−
2 π
3
θ,φ
r 1( θ)
0
P 0(1) r
4
1 dr 1 dΩ 1
= −
4 π
2 R
5
O
15
2(1 + α 0 )
5
+ 4(1 + α 0 )
3
α
2
2 + . . .
.
(6.63)
The second integral in (6.62) yields surviving terms that involve (1 + α 0 )
3 and
(1 + α 0 )α
2
2 . When multiplying the result by the square-bracketed prefactor, again
keep terms only up to order α
2
2 . The result in this case is
2π R
2
O
(1 + α 0 )
2
+
1
5
α
2
2
θ,φ
r 1( θ)
0
P 0(1) r
2
1 dr 1 dΩ 1
=
8π
2 R
5
O
3
(1 + α 0 )
5
+
4
5
(1 + α 0 )
3
α
2
2 + · · ·
.
(6.64)