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11.5 Neutrino Bremsstrahlung from electron-nucleus collisions

Proposed by Pontecorvo [335336] and Gandelman [154155], neutrino Bremsstrahlung from electron-nucleus collisions is one of the major neutrino emission mechanisms in the crust. The process can be written as
e− + (A, Z) −→ e− + (A, Z ) + νℓ + ¯νℓ, ℓ = e, μ,τ . (267 )
With obvious notations, the total momentum of an initial state is PPP = pppe + pppA, and that of a final state PPP ′ = ppp′e + ppp′A + pppνℓ + ppp ′¯νℓ. The corresponding total energies will be denoted by E and E′. We first consider the case of T > Tm, when nuclei form a Coulomb liquid, Tm being the melting temperature of the crust defined by Equation (15View Equation). Let us also neglect, for the sake of simplicity, the Coulomb correlations between ions. The neutrino emissivity from process (267View Equation) is then calculated by integrating the energy emission rate over initial and final momenta,
∫ QBrem ∝ d3p d3p′d3p d3p d3p d3p′ ν e e ν ¯ν A A ′ ′ ×WBrem δ(E − E)δ(PPP − PPP )fAfe [1 − fe′](εν + ε¯ν), (268 )
where fA is the Boltzmann distribution of ions (nuclei), fe is the Fermi–Dirac distribution for electrons,
1 fe(ppp) = -[(ε(p)−μe)∕kBT]----, (269 ) e + 1
and WBrem is the square of the transition amplitude corresponding to elementary process (267View Equation). The factor [1 − fe′] takes care of the Pauli exclusion principle for the electron in the final state. Let us consider the integration over the neutrino momenta. We can rewrite d3pν = d2ˆppp p2dp ν ν ν (ˆppp ≡ pppν∕pν ν) and replace pν with εν∕c. Relevant neutrino energies are εν ∼ kBT. Electrons are degenerate, and the momenta contributing to the integral are close to the electron Fermi surface. Consequently, the integration over energies is restricted to a thin shell around μe of a thickness ∼ kBT. Integration over neutrinos and antineutrinos yields a factor of T 3 each, neutrino energies give T, and the energy delta function removes one T factor. Moreover, W ∝ Z2 Brem and n ∝ ρ∕A N. To account for the nonideality of the plasma of nuclei, one introduces an additional dimensionless factor L [181Jump To The Next Citation Point]. All in all,
Brem 6 2 Q ν ∝ T Z ρL∕A . (270 )
For Γ ≪ 1 we have L ≈ 1, but for a strongly coupled plasma (Γ ≫ 1) L can be significantly smaller than one (see [181]).

Things become more complicated at low temperatures. Then the electron states are no longer described by plane waves. Instead, the Bloch functions consistent with crystal symmetry should be used (see the review of band theory in Section 3.2.4). The electron energy spectrum is no longer continuous, but is formed of bands. At high T, the thermal motion of electrons “smears out” this band structure. However, the gaps between energy bands strongly suppress Brem Q ν [231Jump To The Next Citation Point]. Detailed formulae valid for different domains of the density-temperature plane can be found in [428Jump To The Next Citation Point].

Strong magnetic fields affect the motion of electrons scattered off nuclei. However, the effect of BBB on Brem Q ν has not been calculated.


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