  A hybrid ab initio theoretical approach for examining thermal properties in
magnetic systems of unknown entropy is presented. Commonly used theoretical
approaches interrogate thermal properties from Gibbs/Helmholtz free energies,
which require an accurate model of magnetic interactions. The present approach
avoids this requirement by instead calculating system pressure from thermally
disordered microstates that properly incorporate vibrational and spin
subsystems at each temperature as well as the coupling between these
subsystems. In place of a specific model for magnetic interactions, the
approach integrates measurements of temperature dependent magnetization of the
studied material. We apply the approach to calculate phonon modes and to
investigate the anomalously low thermal expansion of the classical Invar alloy,
Fe_0.65Ni_0.35. The calculated phonon dispersions for Invar are in excellent
agreement with measured data. The Invar thermal expansion is shown to remain
small between 50 K and room temperature, consistent with the experimentally
observed low thermal expansion value in this same temperature range. This
anomalously small thermal expansion is directly connected to a small positive
contribution from lattice thermal disorder that is nearly canceled by a large
negative magnetic disorder contribution. By contrast, calculations for bcc Fe
show a much larger thermal expansion, consistent with experiment, which is
dominated by a large contribution from lattice thermal disorder that is reduced
only slightly by a small negative contribution from that of magnetism. These
findings give insights into the unusual nature of magnetism and spin-lattice
coupling in Invar and Fe, and they support the presented new methodology as a
complementary way to investigate thermal properties of magnetic materials.
