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$α$ particles and the pasta phase

The effects of the $α$ particles in nuclear matter at low densities are investigated within three different parametrizations of relativistic models at finite temperature. Both homogeneous and inhomogeneous matter (pasta phase) are described for neutral nuclear matter with fixed proton fractions and stellar matter subject to $β$-equilibrium and trapped neutrinos. In homogeneous matter, $α$ particles are only present at densities below 0.02 fm$^{-3}$ and their presence decreases with the increase of the temperature and, for a fixed temperature, the $α$ particle fraction decreases for smaller proton fractions. A repulsive interaction is important to mimic the dissolution of the clusters in homogeneous matter. The effects of the $α$ particles on the pasta structure is very small except close to the critical temperatures and / or proton fractions when it may still predict a pasta phase while no pasta phase would occur in the absence of light clusters. It is shown that for densities above 0.01 fm$^{-3}$ the $α$ particle fraction in the pasta phase is much larger than the $α$ particle fraction in homogeneous matter.