Stars as Laboratories for Fundamental Physics - MPP Theory Group

Processes in a Nuclear Medium 145
second term takes account of interference effects between the radiation
emitted in different collisions.
The interference term yields a suppression of the incoherent summation
because the ∆σ in subsequent collisions are anticorrelated. The
spin is constrained to move on the unit sphere whence a kick in one
direction is more likely than average followed by one in the opposite
direction. A similar argument pertains to the velocity; a ∆v in one direction
is more likely than average to be followed by one in the opposite
direction. The radiation spectrum with ω ∼ > Γ coll will remain unaffected
while for ω ∼ < Γ coll it is suppressed. The low-ω suppression of bremsstrahlung
is known as the Landau-Pomeranchuk-Migdal effect (Landau
and Pomeranchuk 1953a,b; Migdal 1956; Knoll and Voskresensky 1995
and references therein).
The summation in Eq. (4.66) can be viewed as an integration over
the relative time coordinate ∆t = t i − t j with a certain distribution
function f(∆t). For a random sequence of “kicks” one expects an exponential
distribution of the normalized form f(∆t) = 1 4 Γ σe −Γ σ∆t/2
where Γ σ is some inverse time-scale. This implies the Lorentzian shape
(e.g. Knoll and Voskresensky 1995)
F (ω) =
ω 2
ω 2 + Γ 2 σ/4 . (4.67)
A Lorentzian model is familiar, for example, from the collisional broadening
of spectral lines. A comparison with Eq. (4.60) indicates that
Γ coll ⟨(∆σ) 2 ∫
⟩ +∞
ω 2 + Γ 2 σ/4 = dt e ⟨ iωt σ(t) · σ(0) ⟩ . (4.68)
−∞
An integral over dω/2π reveals a normalization ⟨σ 2 ⟩ so that
Γ σ = ⟨(∆σ)2 ⟩
⟨σ 2 ⟩
Γ coll . (4.69)
Therefore, Γ σ is identified with a collisional spin-fluctuation rate.
For nucleons interacting by an OPE potential one may estimate Γ σ
without much effort. The NN cross section is dimensionally απ/m 2 2 N.
A typical thermal nucleon velocity is v = (3T/m N ) 1/2 yielding for a
typical collision rate Γ coll = ⟨vσ NN ⟩n B ≈ απT 2 1/2 m −5/2
N n B . Because
|∆σ| ≈ 1 in a collision, Γ σ ≈ Γ coll . This estimate agrees with the
detailed result of Eq. (4.7) apart from a numerical factor 4 √ π.