Multicomponent Fermi systems at low densities

We calculate, to second order in the scattering length between two fermions, the Landau quasiparticle interaction for a low-density mixture of two fermion species with unequal densities at temperature zero. From the Landau parameters we evaluate the energy density and find agreement with the result of Kanno []. The calculations are then extended to the case of two fermion components with different total densities, each with two spin components, a situation of interest in nuclear physics and astrophysics, where the species are neutrons and protons. An interesting finding is that, for low proton concentrations, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>x</a:mi><a:mo>≪</a:mo><a:mn>1</a:mn></a:mrow></a:math>, the leading term in the energy density, beyond the <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:msup><b:mi>x</b:mi><b:mrow><b:mn>5</b:mn><b:mo>/</b:mo><b:mn>3</b:mn></b:mrow></b:msup></b:math> contribution from the kinetic energy and the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:msup><c:mi>x</c:mi><c:mn>2</c:mn></c:msup></c:math> one due to the two-body interaction in the mean-field approximation, varies as <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:msup><d:mi>x</d:mi><d:mrow><d:mn>7</d:mn><d:mo>/</d:mo><d:mn>3</d:mn></d:mrow></d:msup><d:mo form="prefix">ln</d:mo><d:mi>x</d:mi></d:mrow></d:math>. This is to be contrasted with the higher powers of <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mi>x</f:mi></f:math> implicit in many phenomenological energy-density functionals employed in nuclear physics, such as those of the Skyrme type.