Extended Thomas-Fermi theory at finite temperature

We present a generalization of the extended Thomas-Fermi (ETF) theory to finite temperatures T. Starting from the Wigner-Kirkwood expansion of the Bloch density in powers of (h/2π), we derive the gradient expansion of the free energy and entropy density functionals F[rho] and sigma[rho] up to fourth order with their correct temperature-dependent coefficients. (Effective mass and spin-orbit contributions are taken into account up to second order.) For a harmonic-oscillator potential we show that both the (h/2π)-expansion of the free energy and the entropy and the gradient expansion of the functionals F[rho] and sigma[rho] converge very fast and yield the exact quantum-mechanical results for kT > or approx. 3 MeV, where the shell effects are washed out. Finally we discuss the Euler variational equation obtained with the new functionals and use its numerical solutions for semi-infinite symmetric nuclear matter to test the quality of parametrized trial densities. As an application, we present liquid-drop model parameters, calculated with a realistic Skyrme interaction, as functions of the temperature. (orig.)