Constraints on the phase diagram of molybdenum from first-principles free-energy calculations

We use first-principles techniques to re-examine the suggestion that transitions seen in high-P experiments on Mo are solid-solid transitions from the bcc structure to either the fcc or hcp structures. We confirm that in the harmonic approximation the free energies of fcc and hcp structures become lower than that of bcc at P > 325 GPa and T below the melting curve, as reported recently. However, we show that if anharmonic effects are fully included this is no longer true. We calculate fully anharmonic free energies of high-T crystal phases by integration of the thermal average stress with respect to strain as structures are deformed into each other, and also by thermodynamic integration from harmonic reference systems to the fully anharmonic system. Our finding that fcc is thermodynamically less stable than bcc in the relevant high-P/high-T region is supported by comparing the melting curves of the two structures calculated using the first-principles reference-coexistence technique. We present first-principles simulations based on the recently proposed Z method which also support the stability of bcc over fcc.Comment: 33 pages, 10 figure

The earth’s core: an approach from first principles

The Earth’s core is largely composed of iron (Fe), alloyed with less dense elements such as sulphur, silicon and/or oxygen. The phase relations and physical properties of both solid and liquid Fe-alloys are therefore of great geophysical importance. As a result, over the past fifty years the properties of Fe and its alloys have been extensively studied experimentally. However, achieving the extreme pressures (up to 360 GPa) and temperatures (~6000K) found in the core provide a major experimental challenge, and it is not surprising that there are still considerable discrepancies in the results obtained by using different experimental techniques. In the past fifteen years quantum mechanical techniques have been applied to predict the properties of Fe. Here we review the progress that has been made in the use of first principles methods to study Fe and its alloys, and as a result of these studies we conclude: (i) that pure Fe adopts an hexagonal close packed structure under core conditions and melts at ~6200 K at 360 GPa, (ii) that thermodynamic equilibrium and observed seismic data are satisfied by a liquid Fe alloy outer core with a composition of ~10 mole% S (or Si) and 8 mole% O crystallising at ~ 5500 K to give an Fe alloy inner core with ~8 mole% S (or Si) and 0.2 mole % O, and (iii) that with such concentrations of S (or Si), an Fe alloy might adopt a body centred cubic structure in all or part of the inner core. In the future the roles of Ni, C, H and K in the core need to be studied, and techniques to predict the transport and rheological properties of Fe alloys need to be developed

Thermodynamic properties and Debye-Waller factor of fee materials

We have calculated the equation of state and the various thermodynamic properties of monatomic fcc crystals by minimizing the Helmholtz free energy derived in the high temperature limit for the quasiharmonic theory, QH, and the lowest-order (cubic and quartic), 'A2, anharmonic terms of the perturbation theory, PT. The total energy in each case is obtained by adding the static energy. The calculation of the thermal properties was carried out for a nearest-neighbour central-force model of the fcc lattice by means of the appropriate thermodynamic relations. We have calculated the lattice constant, the thermal expansion, the coefficient of volume expansion, the specific heat at constant volume and at constant pressure, the isothermal and adiabatic bulk moduli, and the Griineisen parameter, for the rare-gas solids Kr and Xe, and gold. Morse potential and modified Morse potential were each used to represent the atomic interaction for the three fcc materials. For most of the calculated thermodynamic properties from the QH theory, the results for Kr and Xe with the modified Morse potential show an improvement over the results for the Morse potential when compared with the experimental data. However, the results of the 'A 2 equation of state with the modified Morse potential are in good agreement with experiment only in the case of the specific heat at constant volume and at constant pressure. For Au we have calculated the lattice contribution from the QH and 'A 2 PT and the electronic contribution to the thermal properties. The electronic contribution was taken into account by using the free electron model. The results of the thermodynamic properties calculated with the modified Morse potential were similar to those obtained with the Morse potential. U sing the minimized equation of state we also calculated the Mossbauer recoilless fraction for Kr and Xe and the Debye-Waller factor (DWF) for Pb, AI, eu, Ag, and Au. The Mossbauer recoilless fraction was obtained for the above two potentials and Lennard-Jones potential. The L-J potential gives the best agreement with experiment for Kr. No experimental data exists for Xe. At low temperature the calculated DWF results for Pb, AI, and eu show a good agreement with experimental values, but at high temperature the experimental DWF results increase very rapidly. For Ag the computed values were below the expected results at all temperatures. The DWF results of the modified Morse potential for Pb, AI, eu and Ag were slightly better than those of the Morse potential. In the case of Au the calculated values were in poor agreement with experimental results. We have calculated the quasiharmonic phonon dispersion curves for Kr, Xe, eu, Ag, and Au. The calculated and experimental results of the frequencies agree quite well for all the materials except for Au where the longitudinal modes show serious discrepancies with the experimental results. In addition, the two lowest-order anharmonic contributions to the phonon frequency were derived using the Green's function method. The A 2 phonon dispersion curves have been calculated only for eu, and the results were similar to those of the QH dispersion curves. Finally, an expression for the Griineisen parameter "( has been derived from the anharmonic frequencies, and calculated for these materials. The "( results are comparable with those obtained from the thermodynamic definition

Phonon dispersion curves and atomic mean square displacement for several fcc and bcc materials

The atomic mean square displacement (MSD) and the phonon dispersion curves (PDC's) of a number of face-centred cubic (fcc) and body-centred cubic (bcc) materials have been calclllated from the quasiharmonic (QH) theory, the lowest order (A2 ) perturbation theory (PT) and a recently proposed Green's function (GF) method by Shukla and Hiibschle. The latter method includes certain anharmonic effects to all orders of anharmonicity. In order to determine the effect of the range of the interatomic interaction upon the anharmonic contributions to the MSD we have carried out our calculations for a Lennard-Jones (L-J) solid in the nearest-neighbour (NN) and next-nearest neighbour (NNN) approximations. These results can be presented in dimensionless units but if the NN and NNN results are to be compared with each other they must be converted to that of a real solid. When this is done for Xe, the QH MSD for the NN and NNN approximations are found to differ from each other by about 2%. For the A2 and GF results this difference amounts to 8% and 7% respectively. For the NN case we have also compared our PT results, which have been calculated exactly, with PT results calculated using a frequency-shift approximation. We conclude that this frequency-shift approximation is a poor approximation. We have calculated the MSD of five alkali metals, five bcc transition metals and seven fcc transition metals. The model potentials we have used include the Morse, modified Morse, and Rydberg potentials. In general the results obtained from the Green's function method are in the best agreement with experiment. However, this improvement is mostly qualitative and the values of MSD calculated from the Green's function method are not in much better agreement with the experimental data than those calculated from the QH theory. We have calculated the phonon dispersion curves (PDC's) of Na and Cu, using the 4 parameter modified Morse potential. In the case of Na, our results for the PDC's are in poor agreement with experiment. In the case of eu, the agreement between the tlleory and experiment is much better and in addition the results for the PDC's calclliated from the GF method are in better agreement with experiment that those obtained from the QH theory

A new thermodynamic model for solid metals under elastic deformations

We present a theoretical model of the free Helmholtz energy (F) for solid metals that incorporates three contributions: the elastic part through a local strain description, the vibrational energy within a quasi-harmonic Einstein model with volume-dependent cohesive energy, and the electronic contribution in the free electron gas setting. To get F, we introduce discrete approximations of the Helmholtz energy defined in cubic lattices and show their convergence to F by finite element methods. For homogeneous deformations, the obtained model is applied to derive an equation of state (EOS) which shows a very good agreement with experimental data. Moreover, compared to other known theoretical EOSs, the present model is highly stable under different estimations of its parameters.Fil: Bertoldi, Dalía Surena. Universidad Nacional de Cuyo; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Ochoa, Pablo Daniel. Universidad Nacional de Cuyo; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

First-principles calculations of anharmonic phonons in diamond and silicon at high temperature and pressure

Many ab initio approaches for calculating anharmonic phonon dispersion relations have recently been developed, taking advantage of improvements in computational power. In this thesis, anharmonic phonons in the diamond-type semiconductors silicon and diamond are studied using two of these recently developed ab initio techniques to better understand the role of anharmonicity in these materials at elevated temperatures and pressures. The two techniques are the self-consistent phonon method as implemented in the alamode code and the temperature dependent effective potential approach implemented in the TDEP code. Both these approaches rely on density functional theory calculations to compute anharmonic phonon frequencies from first principles. The renormalisation of the zone-centre optical phonon of silicon is calculated using both methods. The TDEP approach accurately reproduces the experimentally observed temperature dependence of the zone-centre phonon, whereas alamode underestimates the renormalisation. This underestimation is determined to originate from the exclusion of certain phonon–phonon interaction processes in a series expansion central to the self-consistent phonon method. In particular, an interaction process involving three phonons is identified to contribute strongly to the anharmonic phonon renormalisation. An attempt was made to extend alamode to include this interaction, which was, regrettably, unsuccessful. The TDEP approach is then applied to diamond in the same manner as silicon. The zone-centre optical phonon is calculated and a comparison to available experimental data is made. The approach is again found to accurately reproduce the experimental data. Consequently, the TDEP approach is used to investigate the so-called quantum isotope effect in diamond. Deviations from the harmonic frequency ratio of the zone-centre phonons are used to investigate the anharmonic nature of the interatomic potential, as well as to search for an experimentally suggested “inversion” of the quantum isotope effect at high pressure. No such inversion of the quantum isotope effect is observed in the calculations made here. A detailed comparison of the effect of different exchange–correlation functionals and pseudopotentials on the density functional theory calculations is made, ultimately recommending local density approximation as the most accurate predictor of phonon frequencies in diamond. Finally, the Raman frequency of natural diamond is calculated at high temperature and pressure using the highly accurate TDEP method. Improvements are made to the stochastic sampling process, eliminating unwanted scatter from misaligned eigenvectors at degenerate points in the Brillouin zone and increasing the precision of the method. The calculated Raman frequency is used to suggest a calibration of the high-frequency edge of the Raman signal from a diamond anvil, which is used as a pressure marker in very-high-pressure experiments. The suggested calibration extends to pressures up to 1 TPa and temperatures up to 2000 K

Superconductive "sodalite"-like clathrate calcium hydride at high pressures

Hydrogen-rich compounds hold promise as high-temperature superconductors under high pressures. Recent theoretical hydride structures on achieving high-pressure superconductivity are composed mainly of H2 fragments. Through a systematic investigation of Ca hydrides with different hydrogen contents using particle-swam optimization structural search, we show that in the stoichiometry CaH6 a body-centred cubic structure with hydrogen that forms unusual "sodalite" cages containing enclathrated Ca stabilizes above pressure 150 GPa. The stability of this structure is derived from the acceptance by two H2 of electrons donated by Ca forming a "H4" unit as the building block in the construction of the 3-dimensional sodalite cage. This unique structure has a partial occupation of the degenerated orbitals at the zone centre. The resultant dynamic Jahn-Teller effect helps to enhance electron-phonon coupling and leads to superconductivity of CaH6. A superconducting critical temperature (Tc) of 220-235 K at 150 GPa obtained from the solution of the Eliashberg equations is the highest among all hydrides studied thus far.Comment: 19 pages, 4 figure