From Elementary Excitations to Microstructures: The Thermodynamics of Metals and Alloys across Length Scales

An experimental investigation has been made into the components that determine the phase stability of metals and alloys. Contributions were found to be important across many length scales from electronic excitations to atomic vibrations and finally microstructural strains at the continuum level. The metals and alloy that have been studied are U, Ce, and Pd3V. Time-of-flight (TOF) inelastic neutron scattering spectra were measured on the three crystalline phases of uranium at temperatures from 50 K to 1213 K. Phonon density of states (DOS) curves were obtained from these spectra. For the α-phase, a large decrease in phonon energies with increasing temperature was observed over the entire temperature range. Analysis of the vibrational power spectrum showed that the phonon softening originates with continuous softening of a harmonic solid, as opposed to vibrations in anharmonic potentials. Without anharmonicty, it must be that thermal excitations of the electronic structure are changing the interatomic forces. State-of-the-art electronic band structure calculations are based on the assumption that temperature effects on the electronic structure can be neglected when compared to volume effects (where the volume effects are just a manifestation of anharmonicity). The present results turn that problem upside down by showing that temperature effects are actually more important than volume effects. Vibrational entropies of the phase transitions were (Sβ-Sα)vib = (0.15±0.1) kB/atom and (Sγ -Sβ)vib = (0.36±0.1) kB/atom.The former accounts for about 35% and the latter 65% of the total entropy of the phase transition. The remaining entropy must be electronic. TOF inelastic neutron scattering spectra were measured on cerium at temperatures near the fcc (γ) to bcc (δ) transition temperature. Phonon DOS curves were extracted from data acquired over a wide range of momentum transfers. A large softening of the phonon DOS was found in going from γ-cerium to δ-cerium, and this accounts for an increase in vibrational entropy of (0.71 ± 0.05) kB/atom. To be consistent with the latent heat of the γ-δ transition, this increase in vibrational entropy must be accompanied by a large decrease in electronic entropy. The results not only confirm the recent discovery of a significant electronic contribution to the γ-δ transition but also suggest that it may be twice as large as previously reported. TOF inelastic neutron scattering spectra were measured on β-cerium (dhcp) and γ-cerium (fcc) near the phase transition temperature. Phonon densities of states (DOS) were extracted from the TOF spectra. A softening of the phonon DOS occurs in the transition from β-cerium to γ-cerium, accounting for an increase in vibrational entropy of ΔSγ-βvib = (0.09 ±0.05) kB/atom. Crystal field levels were extracted from the magnetic scattering for both phases. The entropy calculated from the crystal field levels and a fit to calorimetry data from the literature was significantly larger in β-cerium than γ-cerium below room temperature. The difference was found to be negligible at the experimental phase transition temperature. There was a contribution to the specific heat from Kondo spin fluctuations that was consistent with the quasielastic magnetic scattering, but the difference between phases was negligible. To be consistent with the latent heat of the β-γ transition, the increase in vibrational entropy at the phase transition may be accompanied by a decrease in electronic entropy not associated with the crystal field splitting or spin fluctuations. At least three sources of entropy need to be considered for the β-γ transition in cerium. Differences in the heat capacity and thermal expansion of cubic (fcc-disordered) and tetragonal (DO22-ordered) Pd3V were measured from 40 K to 315 K. Below 100 K the heatcapacity difference was consistent with harmonic vibrations. At higher temperatures, however, the data show significant anharmonic effects. Measurements of elastic constants, densities, and thermal expansion showed that the anharmonic volume expansion contribution (Cp – Cv) could account for only about one-third of this anharmonic heat capacity difference. The remainder may originate with elastic and plastic deformation of the polycrystalline microstructure. Strain energy from anisotropic thermal contractions of grains in the tetragonal ordered phase contributes to the heat capacity, but some of this strain energy is eliminated by plastic deformation. The vibrational entropy difference of disordered and ordered Pd3V was estimated to be Sdis – Sord = (+0.035± 0.001) kB//atom at 300 K, with 70% of this coming from anharmonic effects. The microstructural contribution to the heat capacity of α-uranium was determined bymeasuring the heat capacity difference between polycrystalline and single crystal samples from 77 K to 320 K. When cooled to 77 K and then heated to about 280 K, the uranium microstructure released (3±2) J/mol of strain energy. On further heating to 300 K the microstructure absorbed energy as the microstructure began to redevelop microstrains. Neutron diffraction measurements on polycrystals predicted the total strain energy stored in the microstructure to be (3.7±0.5) J/mol at 77 K and (1±0.5) J/mol at room temperature in good agreement with the calorimetry.