Single-Particle Electrodynamics - Assassination Science

chrotron” radiation (i.e., the radiation computed from the Dirac equation—
not the Liénard–Wiechert radiation of a classical charge) in a highly asymmetrical
manner [197], the associated spin-flip transition rates, along the axis
of the magnetic field, are not equal. This was put on a more quantitative basis
by Sokolov and Ternov [198], who considered all of the relevant spin-flip
terms, and showed that this “transverse” polarisation
P t ≡ (σ· ˆB) (6.131)
(i.e., transverse to the velocity and acceleration; viz., in the direction of
the magnetic field) builds up, for zero initial polarisation, according to the
relation
P t (t) = P ST
(
1 − e
−t/τ ST
)
, (6.132)
where the asymptotic polarisation, P ST , is given by
P ST = 8
5 √ 3
and the characteristic time, τ ST , by
τ ST =
{ 5
√
3
8
e 2¯hγ 5
4πm 2 R 3 } −1
≡
∼ 92.4%, (6.133)
{ 5
√
3
8
¯λ C r c
R 3 } −1
, (6.134)
where R is the radius of curvature of the trajectory, ¯λ C ≡ ¯h/m is the reduced
Compton wavelength, r c ≡ e 2 /4πm is the classical electron radius, and we
are still using the notational conventions and units of Appendix A. That the
Sokolov–Ternov effect (as it has become known) is, relatively speaking, an
extremely small effect is recognised by the presence of the ratios ¯λ C /R and
r c /R: the Compton wavelength and classical radius of an electron are truly
tiny compared to the radii of typical terrestrial storage rings. The strong
energy dependence (γ 5 ) of the polarisation rate ameliorates the situation
somewhat; the resulting characteristic polarisation times of real rings are,
roughly speaking, on the order of minutes to hours: extremely long compared
300