Phonon Entropy and Thermal Expansion from Inelastic Neutron Scattering and Computation

Entropy comes from atomic-scale degrees of freedom. Most of the entropy in materials comes from the vibrations of atoms, which increase in amplitude with increasing temperature. This is "phonon entropy" (or "vibrational entropy"). The important question is usually how the entropy differs, sometimes slightly, between different states of a material (such as a crystal with chemical order or with disorder). Today both ab initio computation and inelastic neutron scattering can provide accurate entropies over a wide range of temperatures. High temperatures are where entropy is most thermodynamically significant, but is also where degrees of freedom become coupled. Normal modes of vibration are no longer independent, and interactions between vibrations and electrons become important. I will describe how such couplings are sorted out and how they alter thermal expansion and other thermophysical properties.<br/><br/>The phonon entropy can be obtained by inelastic neutron scattering in two ways. Most common for thermodynamics are measurements of phonon densities of states (DOS) using polycrystalline samples. Measurements of single crystal dispersions can also be performed. Some principles of the conversion of raw spectra into phonon DOS curves will be described for data from the chopper spectrometer, ARCS, at Oak Ridge National Laboratory. Calculations of phonon dispersions based on density functional theory (DFT) were also performed with both quasiharmonic and anharmonic methods, and compared to the measured phonons at different temperatures.<br/><br/>Three examples of thermodynamic measurements by inelastic neutron scattering will be presented. 1) The first is bcc Cr, which was measured from 6 to 1493 K. Here, the interest was in reconciling the entropy from phonons and electrons with the entropy assessed from calorimetry. This was successful to better than 2% at 1500 K, showing that magnetic contributions to the entropy are negligible at high temperatures. 2) The second is fcc Pd and Pt, which were also accurate to within 2% of calorimetric results. In both cases, this good agreement required accounting for the effects of the electron-phonon interaction on the electronic entropy. 3) The third is diamond-cubic silicon, where the phonon entropy agreed with the calorimetric entropy to about 1%.<br/><br/>For Cr and Si, the change in free energy with temperature and volume was used to calculate the thermal expansion by both quasiharmonic (QH) and anharmonic (AH) methods. The thermal expansion was calculated successfully by both methods. Curiously, the temperature dependence of the phonons was not calculated successfully by QH theory. The QH model predicted the wrong sign for the thermal shifts of many phonon energies. Some cancellations of errors in thermal shifts of different phonons evidently allow for the success of the QH method for calculating the thermal expansion of Cr and Si. This is not always the case, however. For NaBr, the thermal expansion from the quasi-harmonic method is too small by a factor of four [4], and AH calculations are necessary. It is possible to calculate bulk thermophysical properties with an unreliable microscopic theory, so more incisive measurements of individual phonons are important for our basic understanding of the underlying physics.<br/><br/>1. C.Bernal-Choban, et al., submitted. 2. Y.Shen, et al, PRB 93, 214303 (2016). 3. D.S. Kim, et al. PRB 91, 014307 (2015). 4. Y. Shen, et al., PRL 125, 085504 (2020).<br/><br/>This work is supported by DOE BES award No. DE-FG02-03ER46055.