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##
Screening effects

Nuclear reactions between two nuclei occur at a distance of the order of
10^{-13} cm after tunnelling the Coulomb barrier. Screening is
related to the presence of target/projectile electrons in laboratory experiments
or to the presence of other charged particles in a stellar plasma, which
change the Coulomb barriers in comparison to bare nuclei. Thus, the screening
problem for astrophysics has two aspects. (i) **Laboratory screening**
has to be corrected in order to obtain the cross sections for bare nuclei,
(ii) the stellar environment gives rise to screening effects in **astrophysical
plasmas** as well, but under quite different conditions. In the laboratory,
the targets and projectiles are usually neutral atoms/molecules and ions,
causing a larger cross section, ,
with an enhancement factor over bare nuclei ,
where
is the Sommerfeld parameter and *U*_{e} is the electron-screening
potential energy. For *E*/*U*_{e}<100, shielding effects
become important for understanding and extrapolating low energy data. The
observed enhancements in recent low-energy studies of several fusion reactions
might be larger for some reactions than could be accounted for from available
atomic-physics models [24]. As the **stopping
power corrections** at these low energies enter strongly into the determination
of the bombarding energy for ,
and they are quite uncertain, additional efforts must also provide energy-loss
data far below the Bragg peak, since such data (not available in the literature)
influence sensitively the analysis of the fusion cross sections. In astrophysical
environments which permit nuclear reactions, nuclei are fully ionized,
but polarisation of the electron cloud around nuclei or the Coulomb lattice
effects of electrons and ions have to be accounted for. For instance, in
the case of a pure carbon plasma with a density
g/cm^{3} and a temperature *T*=10^{9} K, the rate
of the reaction ^{12}C + ^{12}C increases by a factor 10^{16}
due to the influence of the surrounding particles. Usually, the screening
factor is written as *f*=*r*/*r*_{b}= ,
where *r* and *r*_{b} are the rates with and without
(bare nuclei) the inclusion of screening, *U* corresponds to the Coulomb
corrections to the purely nuclear Q-value, and
is due to the different radial behaviour of the potential. The weak and
strong screening regimes are described by the conditions |*f*-1|<<1
and |*f*-1|>>1 (see [25] for detailed
discussions). *U*/*kT* is always dominant, but
becomes non-negligible for the strong screening regime. If densities are
high enough, the barrier penetration is driven by the energy of ground
state lattice oscillations of the nuclei rather than the thermal energy,
defining the so called pycnonuclear regime. The theory of the **weak screening**
regime has been considered satisfactory, since corrections are usually
smaller than the current uncertainties in many rates. For the **strong
screening** regime the results are a bit controversial but seem to converge.
Question are related to the (non-uniform?) density distribution of the
degenerate electrons. The evaluation of the reaction rate in the **pycnonuclear
regime** is quite complicated since the thermal distribution of nuclei
is no longer given by a Boltzmann distribution. Two limiting cases have
been traditionally used: (i) the "static" case, that assumes that all the
nuclei and the center of mass of the reacting particles are frozen in their
equilibrium positions and (ii) the "relaxed" case, that assumes that the
position of the centre of mass is fixed and the remaining lattice points
polarise into the positions determined by the separation of the two reacting
nuclei. Monte-Carlo simulations of the interparticle potential seem to
indicate that the relaxed approximation is the correct one, but the proper
accounting of the polarisation energy gives a final fusion rate very near
to the static one. Thus, the understanding of the dynamics of the crystal
needs further investigation as well as the transition from the strong screening
to the pycnonuclear regime.

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