Single-Particle Electrodynamics - Assassination Science

We shall, at this point, perform a small feat of mathematical clairvoyance,
and predict that the only even integrals that we shall ultimately have need
to perform in r s -space, in the remainder of this thesis, are of two types: those
independent of r s , namely,
∫
V s
d 3 r s 1; (6.20)
and those with two factors of r s , namely,
∫
V s
d 3 r s r i sr j s , (6.21)
with no extra factors of the magnitude r s in either case. For the case (6.20),
we find
hence,
∫
V s
d 3 r s 1 =
=
=
∫ 2ε(1−α)
dz
−2ε(1−α)
∫ 2ε(1−α)
dz
−2ε(1−α)
∫ 2ε(1−α)
−2ε(1−α)
∫ ρ(z,α)
dy
−ρ(z,α)
∫ ρ(z,α)
−ρ(z,α)
√ ∫ ρ 2 (z,α)−y 2
dx
√
− ρ 2 (z,α)−y 2
√
dy 2 ρ 2 (z, α) − y 2
dz π { 4ε 2 − (|z| + 2εα) 2}
= 4 {
3 π(2ε)3 1 − 3 2 α + 1 }
2 α3
∫V s
d 3 r s 1 = 4 3 π(2ε)3 {
1 − 3 2
;
( ) rd
+ 1 ( ) }
rd
3
, (6.22)
2ε 2 2ε
where we have used (6.18) to replace α by r d /2ε. Although unfamiliar, the
result (6.22) can be verified to have intuitively correct properties. For r d = 0,
the region of integration in r s -space is just a three-sphere of radius 2ε, and
(6.22) gives the expected volume 4π(2ε) 3 /3. For r d = 2ε, the two offset threespheres
cease to intersect, and the volume (6.22) vanishes as expected. If we
integrate (6.22) in turn over all V d , using spherical coördinates, we find that
d
∫V 3 r d d
d
∫V 3 r s 1 = 4 ∫ {
2ε
s
3 π(2ε)3 dr d 4πrd
2 1 − 3 ( ) rd
+ 1 ( ) }
rd
3
0
2 2ε 2 2ε
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