Stars as Laboratories for Fundamental Physics - MPP Theory Group

300 Chapter 8
the other (Fig. 8.8, lowest panel) when it moves across the resonant
density region.
A linear density profile near the resonance region, which is always
a first approximation, yields the “Landau-Zener probability”
p = e −πγ/2 (8.41)
which was first derived in 1932 for atomic level crossings. The adiabaticity
parameter γ was defined in Eq. (8.39); in the adiabatic limit
γ ≫ 1 one recovers p = 0.
For a variety of other density profiles and without the assumption
of a small mixing angle one finds a result of the form
p = e−(πγ/2) F − e −(πγ/2) F ′
1 − e −(πγ/2) F ′ , (8.42)
where F ′ = F/ sin 2 θ 0 and F is an expression characteristic for a given
density profile. For a linear profile n e ∝ r one has F = 1 so that for
a small mixing angle one recovers the Landau-Zener probability. The
profile n e ∝ r −1 leads to F = cos 2 2θ 0 / cos 2 θ 0 . Of particular interest is
the exponential n e ∝ e −r/R 0
which yields
F = 1 − tan 2 θ 0 . (8.43)
All of these results are quoted after the review by Kuo and Pantaleone
(1989) where other special cases and references to the original literature
can be found.
Going beyond the Landau-Zener approximation requires assuming
one of the above specific forms for n e (r) for which analytic results exist.
Recently, Guzzo, Bellandi, and Aquino (1994) used a somewhat
different approach which is free of this limitation. They derived an
approximate solution to the equivalent of Eq. (8.36) by the method of
stationary phases for the space-ordered exponential. They found
p =
( 1 − γ
′
1 + γ ′ ) 2
sin 2 (θ 0 − θ) + 2γ′
(1 + γ ′ ) 2 [
cos 2 (θ 0 − θ) + cos 2 (θ 0 + θ) ] ,
(8.44)
where γ ′ ≡ πγ/16. It was assumed that a resonance occurs between
the production point (mixing angle θ) and the detection point which is
taken to lie in vacuum (mixing angle θ 0 ). Eq. (8.44) can be used only
in the nonadiabatic regime as it works only for γ < 16/π.