Stars as Laboratories for Fundamental Physics - MPP Theory Group

Processes in a Nuclear Medium 133
it is based on the same matrix element with the neutrinos “crossed”
into the initial state. Then one finds in analogy to Sect. 4.3.1
λ −1 =
(
CA G F
√
2
) 2
n B
2ω 1
∫
= 3C2 A G 2 F n B
2π 2
∫ ∞
0
d 3 k 2
2ω 2 (2π) 3 f 2 S µν N µν
dω 2 ω 2 2 f 2 S σ (ω 1 + ω 2 ), (4.31)
where 1 refers to the ν for which the mfp is being determined while
2 refers to a ν from the thermal environment (occupation number
f 2 ). With the detailed-balance relationship S σ (ω) = S σ (−ω) e ω/T (see
Eq. 4.43) and writing S σ (|ω|) = (Γ σ /ω 2 ) s(ω/T ) as before one finds
λ −1 = 3C 2 A G 2 F n B Γ σ T
∫ ∞
0
dx 2 f 2
x 2 2 s(x 1 + x 2 )
2π 2 (x 1 + x 2 ) 2 , (4.32)
where x i = ω i /T . Then one may use the previously determined Γ σ and
s(x) to find the mfp for given conditions (degenerate or nondegenerate).
The ratio between the inverse mfp’s from elastic scattering and pair
absorption is, ignoring the contribution from C 2 V ,
λ −1
pair
λ −1
scat
= Γ σ
T
∫ ∞
0
dx 2 f 2
x 2 2 s(x 1 + x 2 )
2π x 2 1(x 1 + x 2 ) 2 . (4.33)
An average with regard to a thermal x 1 distribution yields about 0.02
for the dimensionless integral where Fermi-Dirac distributions with
chemical potentials µ ν = 0 were used. The main figure of merit, however,
is Γ σ /T , the ratio of a typical spin-fluctuation rate and the ambient
temperature. In the nondegenerate limit one finds with Eq. (4.7)
γ σ ≡ Γ σ
T = 4π1/2 α 2 π
m 5/2
N
n B
with the nuclear density ρ 0 = 3×10 14 g/cm 3 .
4.5.3 Inelastic Scattering
T ≈ 16 ρ ( ) 30 MeV 1/2
, (4.34)
1/2 ρ 0 T
It appears that for typical conditions of a young SN core, pair absorption
is almost as important as elastic scattering. However, even though
the quantity γ σ is larger than unity, the pair-absorption rate has an
unfavorable phase-space factor from the initial-state ν so that the dimensionless
integral in Eq. (4.33) is a small number. This would not
be the case for the inelastic scattering process νNN → NNν shown in