Analytical study of relativisitic fermi gas and RKKY interaction

Abstract

We develope a new method to calculate the temperature dependent behavior of ideal relativistic Fermi gas using complex theory and Euler-Maclaurin formula. Our method does not require any restriction on the properties of integrand, namely integrand is nonsingular and not too rapidly varying about $ E = \mu $, as Sommerfeld expansion method does. Our method doesn``t need the exact value of chemical potential. In that reason, Fermi integral and every thermodynamic function can be calculated at every chemical potential value with ease. We investigate thermodynamic functions, especially density and pressure which are really important in studying compact objects in astrophysics. Their relationships are $ p \propto n^{\frac{5}{3}} $ for nonrelativistic limit and $ p \propto n^{\frac{4}{3}} $ for ultrarelativistic limit. For ultrarelativistic case, we can know that chemical potential is almost zero at any temperature. In that reason, we can fix chemical potenial as relativistic $E_f^r = mc^2(\sqrt{1+x_f^2}-1)$ at every temperature for ultrarelativistic case.