### 11.5 Neutrino Bremsstrahlung from electron-nucleus collisions

Proposed by Pontecorvo [335, 336] and Gandelman [154, 155], neutrino Bremsstrahlung from
electron-nucleus collisions is one of the major neutrino emission mechanisms in the crust. The process can
be written as
With obvious notations, the total momentum of an initial state is , and that of a
final state . The corresponding total energies will be denoted by
and . We first consider the case of , when nuclei form a Coulomb liquid,
being the melting temperature of the crust defined by Equation (15). Let us also neglect, for
the sake of simplicity, the Coulomb correlations between ions. The neutrino emissivity from
process (267) is then calculated by integrating the energy emission rate over initial and final momenta,
where is the Boltzmann distribution of ions (nuclei), is the Fermi–Dirac distribution for electrons,
and is the square of the transition amplitude corresponding to elementary process (267). The
factor 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 () and replace
with . Relevant neutrino energies are . 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 of a thickness . Integration over neutrinos and
antineutrinos yields a factor of each, neutrino energies give , and the energy delta function
removes one factor. Moreover, and . To account for the nonideality of
the plasma of nuclei, one introduces an additional dimensionless factor [181]. All in all,
For we have , but for a strongly coupled plasma () 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 , the thermal motion of electrons “smears out”
this band structure. However, the gaps between energy bands strongly suppress [231].
Detailed formulae valid for different domains of the density-temperature plane can be found
in [428].

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