Theory of thermal expansion: Quasi-harmonic approximation and corrections from quasi-particle renormalization

"Quasi-harmonic" (QH) theory should not be considered a low-order theory of anharmonic effects in crystals, but should be recognized as an important effect separate from "true" anharmonicity. The original and widely used meaning of QH theory is to put T = 0 volume-dependent harmonic phonon energies omega(Q) (V) into the non-interacting phonon free energy. This paper uses that meaning, but extends it to include the use of T = 0 V-dependent single-particle electron energies epsilon(K)(V). It is demonstrated that the "bare" quasi-particle (QP) energies omega(Q)(V) and epsilon(K)(V) correctly give the first-order term in the V-dependence of the Helmholtz free energy F(V, T). Therefore, they give the leading order result for thermal expansion alpha(T) and for the temperature-dependence of the bulk modulus B(T) - B-0. However, neglected interactions which shift and broaden omega(Q) with T, also shift the free energy. In metals, the low T electron-phonon mass enhancement of states near the Fermi level causes a shift in free energy that is similar in size to the electronic QH term. Before T reaches the Debye temperature Theta(D), the mass renormalization essentially disappears, and remaining electron-phonon shifts of free energy contribute only higher-order terms to thermal expansion. Similarly, anharmonic phonon-phonon interactions shift the free energy, but contribute to thermal expansion only in higher order. Explicit next order formulas are given for thermal expansion, which relate "true" anharmonic and similar free energy corrections to QP self-energy shifts.