Mean-field Potential Approach Research Articles

Studying hcp-bcc transition of Be via anisotropy of modulus and sound velocity

Abstract Based on ab initio calculations, we utilize the mean-field potential approach with the quantum modification in conjunction with stress-strain relation to investigate the elastic anisotropies and sound velocities of hcp and bcc Be at high-temperature (0-6000 K) and high-pressure (0-500 GPa) conditions. We propose the general defintion of anisotropy for elastic moduli and sond velocities. Results suggest that the elastic anisotropy of Be is more significantly influenced by pressure than by temperature. The pressure induced increase of c/a ratio makes the anisotropy of hcp Be significantly strengthen. Whereas the hcp Be still exhibits smaller anisotropy than bcc Be in terms of elastic moduli and sound velocities. We suggest that measuring the anisotropy in shear sound velocity may be an approach to distinguishing the hcp-bcc phase transition under extreme conditions.

Temperature effects on thermodynamic properties of rare-earth ytterbium

Thermodynamic quantities (including Debye temperature and frequency, atomic mean-square displacement, specific heats at constant volume and constant pressure, linear thermal expansion coefficient, and total entropy) of face-centered cubic ytterbium metal have been studied by means of the statistical moment method in statistical mechanics. This theoretical approach allows us to calculate these thermodynamic quantities taking into account the anharmonicity of lattice vibrations. Numerical calculations for ytterbium have been conducted in temperature range from 0 K to 1100 K using Lennard-Jones potential to describe the interaction between ytterbium atoms. Our derived results are compared with those of mean-field potential approach as well as previous experimental data showing the good agreement. We have shown that the atomic mean-square displacement contains the zero-point vibration (quantum effect) at low temperature. At high temperature it changes linearly with respect to temperature. Furthermore, our theoretical calculations have also pointed out the anharmonic contribution to specific heat at constant volume of ytterbium metal is negative with temperature. This research presents an effective statistical approach to study temperature effects on thermodynamic quantities of materials.

Structural and Thermodynamic Properties of the High-Entropy Alloy AlCoCrFeNi Based on First-Principles Calculations

During the past two decades, the high-entropy alloy AlCoCrFeNi has attracted much attention due to its outstanding thermal and mechanical properties under ambient conditions. However, the exploration on the thermodynamic properties of this alloy under high temperatures and high pressures is relatively insufficient. Combining structural modeling with the similar atomic environment (SAE) method and first-principles simulations with the modified mean-field potential (MMFP) approach, we studied the lattice and magnetic structure as well as the thermodynamic properties of the body-centered-cubic AlCoCrFeNi, through supercell simulations. AlCoCrFeNi was found to display a strong local lattice distortion compared with typical 3d high-entropy alloys; the ferromagnetic structure stable at 0 K was predicted to transform to the paramagnetic structure at the Curie temperature TC = 279.75 K, in good agreement with previous calculations; the calculated equilibrium volumes, bulk modulus, and shock Hugoniot all agree well with available experimental data and other theoretical values. These results demonstrate the validity and reliability of our methods used to study the dynamic properties of AlCoCrFeNi, providing a promising scheme for accessing the dynamic properties of sophisticated high-entropy alloys.

Mean-field potential approach for thermodynamic properties of lanthanide: Europium as a prototype

In the present paper, a simple conjunction scheme [mean-field potential (MFP) + local pseudopotential] is used to study the thermodynamic properties of divalent lanthanide europium (Eu) at extreme environment. Present study has been carried out due to the fact that divalent nature of Eu arises because of stable half-filled 4f-shell at ambient condition, which has great influence on the thermodynamic properties at extreme environment. Due to such electronic structure, it is different from remaining lanthanides having incomplete 4f-shell. The presently computed results of thermodynamic properties of Eu are in good agreement with the experimental results. Looking to such success, it seems that the concept of MFP approach is successful to account contribution due to nuclear motion to the total Helmholtz free energy at finite temperatures and pressure-induced inter-band transfer of electrons for condensed state of matter. The local pseudopotential is used to evaluate cold energy and hence MFP accounts the s–p–d–f hybridization properly. Looking to the reliability and transferability along with its computational and conceptual simplicity, we would like to extend the present scheme for the study of thermodynamic properties of remaining lanthanides and actinides at extreme environment.

Thermophysical Properties of Cerium and Ytterbium via Local Pseudopotential with Mean Field Potential Approach at Extreme Environment

In the present communication, the ion motional contribution (\(\hbox {F}_{\mathrm{ion}}\)) to the total Helmholtz free energy has been accounted for by using mean field potential (MFP) approximation. The MFP is constructed using the local pseudopotential for divalent ytterbium and trivalent cerium. Further, MFP is used to evaluate static as well as temperature-dependent thermodynamic properties of these metals up to their melting temperature. Computed results are compared with experimental findings as well as results obtained by applying other theoretical methods. Present conjunction scheme with its computational simplicity, physical transparency and transferability of local pseudopotential explains the role of pressure-induced interband transfer of electrons which is crucial in the determination of thermodynamic properties of complex metals like lanthanides.

Thermodynamic properties of rhodium at high temperature and pressure by using mean field potential approach

The thermophysical properties of rhodium are studied up to melting temperature by incorporating anharmonic effects due to lattice ions and thermally excited electrons. In order to account anharmonic effects due to lattice vibrations, we have employed mean field potential (MFP) approach and for thermally excited electrons Mermin functional. The local form of the pseudopotential with only one effective adjustable parameter r c is used to construct MFP and hence vibrational free energy due to ions – F ion . We have studied equation of state at 300 K and further, to access the applicability of present conjunction scheme, we have also estimated shock-Hugoniot and temperature along principle Hugoniot. We have carried out the study of temperature variation of several thermophysical properties like thermal expansion (β P ), enthalpy (E H ), specific heats at constant pressure and volume (C P and C V ), specific heats due to lattice ions and thermally excited electrons (C V ion and C V el, isothermal and adiabatic bulk moduli (B T and B s ) and thermodynamic Gruneisen parameter (γ th ) in order to examine the inclusion of anharmonic effects in the present study. The computed results are compared with available experimental results measured by using different methods and previously obtained theoretical results using different theoretical philosophy. Our computed results are in good agreement with experimental findings and for some physical quantities better or comparable with other theoretical results. We conclude that local form of the pseudopotential used accounts s-p-d hybridization properly and found to be transferable at extreme environment without changing the values of the parameter. Thus, even the behavior of transition metals having complexity in electronic structure can be well understood with local pseudopotential without any modification in the potential at extreme environment. Looking to the success of present scheme (MFP + pseudopotential) we would like to extend it further for the study of liquid state properties as well as thermophysical properties of d and f block metals.

Thermophysical properties of iridium at finite temperature**Project supported by the Department of Science and Technology–Fund for Improvement of Science and Technology Infrastructure Project (DST-FIST) (Level 1) of Department of Sciences and Technology (DST), New Delhi (Grant No. SR/FST/PST-001/2006).

The bulk properties of materials in an extreme environment such as high temperature and high pressure can be understood by studying anharmonic effects due to the vibration of lattice ions and thermally excited electrons. In this spirit, in the present paper, anharmonic effects are studied by using the recently proposed mean-field potential (MFP) approach and Mermin functional which arise due to the vibration of lattice ions and thermally excited electrons, respectively. The MFP experienced by a wanderer atom in the presence of surrounding atoms is constructed in terms of cold energy using the local form of the pseudopotential. We have calculated the temperature variation of several thermophysical properties in an extreme environment up to melting temperature. The results of our calculations are in excellent agreement with the experimental findings as well as the theoretical results obtained by using first principle methods. We conclude that presently used conjunction scheme (MFP+pseudopotential) is simple computationally, transparent physically, and accurate in the sense that the results generated are comparable and sometimes better than the results obtained by first principle methods. Local pseudopotential used is transferable to extreme environment without adjusting its parameters.

The Theoretical Study of High Pressure Equation of State and Relative Volume Thermal Expansion of fcc Transition Metals Ni, Pd and Pt Using Mean Field Potential Approach

In the theoretical study of thermophysical properties of materials at high temperature and pressure anharmonic effects are important. In this regard, in the present communication, anharmonic effects are taken into account by coupling recently proposed mean field potential (MFP) in conjunction with local pseudopotential for fcc transition metals Ni, Pd and Pt. Using present approach, we have computed volume variation of binding energy at 0K, vibrational free energy of lattice ions (Fion), Helmholtz free energy and relative volume thermal expansion up to melting temperature of Ni, Pd and Pt. Further, applicability of present coupling scheme has been tested by investigating equation of state (EOS) at 300K which explains the microscopic internal structure of material. Our results are in good agreement with experimental and other theoretical findings which demonstrate the capability of present coupling scheme of MFP with local pseudopotential alongwith its computational simplicity.

Wave-packet propagation based calculation of above-threshold ionization in the x-ray regime

We investigate the multi-photon process of above-threshold ionization for the light elements hydrogen, carbon, nitrogen, and oxygen in the hard x-ray regime. Numerical challenges are discussed and by comparing Hartree–Fock–Slater calculations with configuration–interaction–singles results we justify the mean-field potential approach in this regime. We present a theoretical prediction of two-photon above-threshold-ionization cross sections for the mentioned elements. Moreover, we study how the importance of above-threshold ionization varies with intensity. We find that for carbon, at x-ray intensities around , two-photon above-threshold ionization of the K-shell electrons is as probable as one-photon ionization of the L-shell electrons.

Thermodynamics of theα−γtransition in cerium studied by an LDA + Gutzwiller method

Utilizing the local-density approximation (LDA) + Gutzwiller method, we have studied the $\ensuremath{\alpha}\text{\ensuremath{-}}\ensuremath{\gamma}$ transition in cerium. Our results indicate that the volume collapse transition between $\ensuremath{\alpha}$ and $\ensuremath{\gamma}$ phases is present at zero temperature with negative pressure. By further providing a newly finite temperature generalization of the LDA + Gutzwiller method (using the mean-field potential approach), the entropy contributed by both electronic quasiparticles and lattice vibration included, we obtain the Gibbs free energy at a given volume and temperature, from which we get the $\ensuremath{\alpha}\text{\ensuremath{-}}\ensuremath{\gamma}$ transition at finite temperature and pressure. Our results indicate that the electronic entropy and lattice vibrational entropy both play important roles in the $\ensuremath{\alpha}\text{\ensuremath{-}}\ensuremath{\gamma}$ transition. We also calculated the equation of state and phase diagram of cerium, finding good agreement with the experiments.

Theoretical Study on Equation of State of Porous Mo and Sn

We present a first-principles scheme to investigate the equation of state (EOS) of porous materials, based on our recently developed modified mean-field potential approach. By taking the effect of the structural parameters on the free energy into account, we calculate the total energy of materials with initial different densities and then study the EOS of porous Mo and Sn as a prototype. The calculated results are in good agreement with the experimental data available, which demonstrates that our scheme is suitable for investigating EOS of porous materials over a wide range of porosities and pressures.

Thermal Properties of Divalent Metal Oxides: CaO as a Prototype

Commonly employed quasiharmonic approximation (QHA) is inadequate to account for intrinsic anharmonism such as phonon-phonon interaction, vacancy contribution, etc. Though anharmonic contributions are important at high temperatures and low pressure, complete ab initio calculations are scanty due largely to laborious computational requirements. Nevertheless, some simple semi-empirical schemes can be used effectively to incorporate the anharmonism. In this regards, in the present study we have proposed a simple computational scheme to include the effect of vacancy directly into the description within the mean-field potential approach, which calculates vibrational free energy of ions. Validity of the scheme is verified by taking calcium oxide as a test case. Equilibrium properties at (T,P) = (0,0) condition is obtained within the tight-binding second-moment approximation (TB-SMA), whose parameters were determined through first principles density functional theory. Kohn-Sham equations for core electrons were solved using ultrasoft plane-wave pseudopotential employing the generalized gradient approximation for exchange and correlation. Present findings for thermal expansion and high-T EOS clearly show perceptible improvement over the case when vacancy contribution was not included. Some related thermodynamic properties are also calculated and compared with the available experimental and theoretical data.

Analytic equation of state and thermodynamic properties for solid FCC C60 over a wide range of pressures and temperatures

As an analytic approximation to the free volume theory (FVT), the analytic mean field potential (AMFP) approach is applied to the multi-exponential model solid. The analytic expressions for the equation of state (EOS) and thermodynamic quantities are derived. The formalism for the case of the double-exponential (DE) model is applied to the solid fcc C60. The four potential parameters are determined by fitting the experimental data of cohesive energy, lattice constant and compression curve of solid C60. The numerical results calculated show that the Girifalco potential is too hard and gives compression curve prominently deviated from experiments at high pressure, whereas the DE potential can well describe the thermo-physical properties of C60 system within wide pressure range. The calculated variation of lattice constant versus temperature relationship and compression curve for solid C60 by using the DE potential agree well with experimental data available. Especially the AMFP with DE potential predicted that the fcc C60 solid is stable up to 2440 K, the spinodal temperature, in good agreement with the result from molecular dynamic (MD) simulation. The AMFP is a useful approach to consider the anharmonic effects at high temperature for solid C60.

General construction of mean-field potential and its application to the multiphase equations of state of tin

The mean-field potential (MFP) approach is an efficient way to evaluate the free energy contribution of ion motions for both solid and liquid states. In this paper the MFP is generally constructed with a volume-dependent term and a shape function. The former is derived in accordance with quasi-harmonic approximation. The latter is given semi-empirically. Application to multiphase equations of state for β-, γ- and liquid-tin has been examined. The theoretical phase diagram and thermodynamic properties of isotherm, thermal expansion, heat capacity, Hugoniot states as well as phase transitions are all in excellent agreement with experiments.

High-Temperature Vibrational Properties and Melting Curve of Aluminum

The mean-field theory due to Wang and Li (Phys Rev B 63:196, 2000) to calculate the effective mean potential experienced by vibrating ions in a crystal is used to compute the ion-motional free energy. An improvement is sought by treating the parameter λ, entering an expression of the mean-field potential (MFP), as a free parameter for the case of aluminum. Although a corresponding expression for the Gruneisen parameter (γ) is significantly different then the known cases, namely, those due to (i) Slater, (ii) Dugdale and MacDonald, (iii) free volume theory, and (iv) Barton and Stacey, its value is very close to the experimental result. Significant improvement is observed for high-temperature thermodynamics of aluminum with the new choice of λ, or equivalently γ. Also, the present improved scheme is extended to measure the vibrational response of the crystal. Recently, Bhatt et al. (Philos Mag 90:1599, 2010) have demonstrated that the mean frequency (ω′) calculated by the MFP approach in conjunction with the density-dependent local pseudopotential suffices to characterize the crystal at finite temperatures. Relating ω′ to the Debye frequency, vibrational properties like the Debye temperature, the mean-square displacement, and entropy are obtained as a function of temperature. Further, a generalized melting law is derived by combining the MFP approach to Lindemann’s law, where the effect of different choices of the parameter λ is now explicitly included into the description. Results so obtained for different physical properties are analyzed and discussed in the light of recent first principles and experimental findings.

High-temperature vibrational response and high-pressure melting curve of lead

The concept of mean-field potential (MFP) is an adequate description to evaluate vibrational contribution to the Helmholtz free energy at finite temperatures and pressures for condensed-state of matter. Once the total free energy is available, as an explicit function of temperature and volume (≡pressure), several thermodynamic properties can be calculated numerically using the text-book equations. While the MFP approach is robust, its applicability is extended to estimate atomic and vibrational properties of materials within the Debye model. In the present scheme, we propose energy-calculation based formula to calculate the Debye temperature. We discuss temperature dependence of Debye temperature, entropy and atomic mean-square displacements for elemental fcc-Pb. It is demonstrated that the present extended MFP (EMFP) scheme, allows one to include the effect of three choices for thermodynamic Grüneisen parameter, namely, due to Slater, due to Dugdale and MacDonald and the one due to free volume theory, on different physical properties. A much debatable dynamical phenomenon, a high pressure melting curve is also obtained, where melting temperature is now explicitly dependent on these three assumptions for the Grüneisen parameter. Results so obtained are discussed and compared with the recent first principles theoretical and experimental findings.

High temperature and pressure thermodynamics of strontium: A macroscopic approach

Close proximity of d-bands (above) to the Fermi level (E F) makes the heavy alkaline earth metals (Ca, Sr and Ba) fairly sensitive to external influences like temperature and pressure. Softening of some of the phonon modes at high temperatures and/or pressures implies that anharmonic effects can play an important role in determining lattice dynamics and related properties. In the conventional approach, phonon density of states (p-dos) have to be calculated at each volume to compute free energy and thereby the other thermodynamic properties, which is computationally quite demanding. Using an alternative technique, the mean-field potential (MFP) approach was combined with the relatively soft local pseudopotential to obtain the free energy at different temperatures and pressures. The results for phonon frequency shifts at finite temperatures using the MFP approach and those calculated from p-dos within the quasiharmonic approximation are very similar. This validates the use of the MFP approach coupled with the local pseudopotential to estimate vibrational response of the system at high-temperature and high-pressure environments. The present scheme was used to study various thermophysical properties for elemental strontium at elevated temperatures and pressures, including the high-pressure melting curve and temperature along the shock Hugoniot. Computed results are affirmatively compared and analyzed with other reported data. The present scheme completely bypasses traditional cumbersome calculations, and it is computationally convenient yet accurate.

Thermodynamic properties of solid FCC C84 based on an analytic mean field approach

The analytic mean field potential (AMFP) approach is applied to the poly-exponential model solid. The analytic expressions for the Helmholtz free energy, internal energy and equation of state (EOS) are derived. The formalism for the case of the double-exponential (DE) model is applied to fcc C84. One set of potential parameters are determined by fitting the experimental compression data of C84 up to 9.24 GPa at ambient temperature (297 K). The equilibrium distance and well depth for C60, C70 and C84 molecules are plotted. The thermophysical properties including the isothermals, thermal expansion, isochoric heat capacity, Helmholtz free energy and internal energy are calculated and analyzed. The theoretical results agree well with the experimental data available of C84. Basing the results of our calculations, we may also predict the behaviors of C84 at extreme conditions.

Analytic equation of state for solids with multi-exponential potential based on analytic mean field potential approach and applied to the epsilon phase of solid oxygen

The thermodynamic properties of the ε phase of solid oxygen are studied by using the analytic mean field approach (AMFP). Analytic expressions for the Helmholtz free energy, internal energy and equation of state of solid oxygen have been derived based on the multi-exponential potential. The formulism for the case of double-exponential (DE) model is applied to the ε phase of solid oxygen. Its four potential parameters are determined through fitting the experimental compression data of the ε phase of solid oxygen. Numerical results of the pressure dependence of the volume calculated by using the AMFP are in good agreement with the original experimental data. This suggests that the AMFP is a useful approach to study the thermodynamic properties of the ε phase of solid oxygen. Furthermore, we predict the variation of the volume, lattice parameters and intermolecular distances with pressure, and some thermodynamic quantities versus volume, at several higher temperatures.

An analytic equation of state based on analytic mean-field potential approach and application to three phases of metallic Zirconium

Abstract The analytic mean-field approach (AMFP) was applied to study the thermodynamic properties of Zirconium (Zr). The analytic expressions for the Helmholtz free energy, internal energy and equation of state have been derived. The formalism for the case of the Morse potential is used in this work. The four potential parameters are determined by fitting the molar volume of the three phases of Zr. The calculated molar volume of α, β and ω Zr are in fairly good agreement with the available experimental data. The results presented in this paper verify that the AMFP is a useful approach to study the thermodynamic properties of Zr. Furthermore, we predict the variation of the relationship of free energy and internal energy versus the molar volume at various temperatures and the dependence of the bulk modulus, the thermal expansion coefficient and the heat capacity on temperature at zero pressure of α, β and ω Zr.