Single-Particle Electrodynamics - Assassination Science

Substituting (6.151) into (6.152), we thus find the evolution equation of motion
for P t (t):
P˙
t (t) = − 2 µ 5 B 3 {
1 − P
2
3 4πs 4 t (t) } . (6.153)
This differential equation is nonlinear, but fortunately its general solution is
simple:
where
P t (t) = − tanh t − t 0
τ cl.
, (6.154)
{ 2
τ cl. ≡
3
µ 5 B 3 } −1
, (6.155)
4πs 4
and where the arbitrary constant t 0 in (6.154) specifies the initial polarisation:
P t (0) ≡ tanh t 0
τ cl.
. (6.156)
Before we compare these classical results with the spontaneous radiation
slip-flip results of the previous sections, we shall first briefly indicate how the
classical calculations generalise for the Sokolov–Ternov effect. Because of the
acceleration of the electron, an evaluation of the effect in the rest frame is not
feasible (see the detailed discussion of this problem in Jackson’s review [114]);
to analyse it in the lab frame, we first write the torque equation of motion
(6.146) in covariant terms, using the expressions listed in Section G.2.2:
(Ṡ) = 2
3 µ2 η 0 U ×Σ × { ( ...
Σ ) + ˙U 2 ( ˙ Σ ) } . (6.157)
After some algebra, one can then show that
{ 1
˙σ = σ×
γ C − 1 }
γ + 1 C0 v + ˙σ T
≡ σ×Ω RR + ˙σ T , (6.158)
where ˙σ T is the Thomas precession contribution to ˙σ, and
C ≡ 2 3
µ 2
s η { ...
0 ( Σ ) + ˙U 2 ( Σ ˙ ) } .
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