Explicit Anharmonic Contributions Research Articles

Structural stability and anharmonicity of phonon modes of metastable Zn4V2O9: In-situ Raman spectroscopic investigation

The structural stability of metastable Zn4V2O9 has been investigated using in-situ Raman spectroscopy at high pressures up to 25 GPa at ambient temperature and high temperature, upto 1073 K at ambient pressure. Pressure dependent Raman studies show pressure induced structural phase transitions around 5 GPa and 14 GPa, followed by an onset of structural disorder above17 GPa. The transition observed at 5 GPa is reversible while the transition to disordered phase is irreversible. The changes in Raman spectra above 14 GPa point out towards an increase in coordination of vanadium atoms. The disordered high pressure phase could be retrieved upon release of pressure. Temperature dependent Raman studies indicated many anomalous phonon modes but none of them show softening behaviour with pressure. The anharmonicities of Raman modes have been estimated for all the observed Raman modes in the ambient monoclinic phase and interestingly most of the modes show positive explicit anharmonic contribution.

Stability and Lattice Dynamics of Ruddlesden–Popper Tetragonal Sr2TiO4

We report a combined experimental and theoretical lattice dynamics study of the Ruddlesden–Popper layered compound Sr2TiO4. From inelastic neutron scattering experiments, we derive the generalized phonon density of states of Sr2TiO4. We also report its heat capacity, thermal expansion, and thermodynamic Gruneisen parameters using the calculated bulk modulus and find a large value of about 2. Using Raman scattering experiments under pressure, we discuss a potential structural distortion of the tetragonal structure above 11 GPa, which could be due to nonhydrostatic compression. The mode Gruneisen parameters of the four Raman-active modes are determined and shown to be in reasonable agreement with those obtained by density functional perturbation theory (DFPT) calculations. The temperature behavior of the Raman-active modes was studied, allowing us to determine the implicit volume and explicit anharmonic contributions. Above 400 K, the implicit volume contribution dominates the temperature-induced variation of the four Raman-active modes, whereas, below this temperature, the explicit anharmonic contribution is the dominant contributor to the highest energy mode. Our results underline the importance of anharmonicity in vibration-related properties of Sr2TiO4.

Temperature dependence of the Gibbs energy of vacancy formation of fcc Ni

Quantum-mechanical calculations are used to determine the temperature dependence of the Gibbs energy of vacancy formation in nickel. Existing data reveal a discrepancy between the high-temperature estimates from experiments and low-temperature approximations from density functional theory. Our finite-temperature calculations - which include the effects of magnetism and fully interacting phonon vibrations - demonstrate that this discrepancy is mostly caused by the previously neglected explicit anharmonic contribution.

Phonons and colossal thermal expansion behavior of Ag3Co(CN)6 and Ag3Fe(CN)6

Recently colossal volume thermal expansion has been observed in the framework compounds Ag3Co(CN)6 and Ag3Fe(CN)6. We have measured phonon spectra using neutron time-of-flight spectroscopy as a function of temperature and pressure. Ab initio calculations were carried out for the sake of analysis and interpretation. Bonding is found to be very similar in the two compounds. At ambient pressure, modes in the intermediate frequency part of the vibrational spectra in the Co compound are shifted slightly to higher energies as compared to the Fe compound. The temperature dependence of the phonon spectra gives evidence for a large explicit anharmonic contribution to the total anharmonicity for low-energy modes below 5 meV. We have found that modes are mainly affected by the change in size of the unit cell, which in turn changes the bond lengths and vibrational frequencies. Thermal expansion has been calculated via the volume dependence of phonon spectra. Our analysis indicates that Ag phonon modes within the energy range 2–5 meV are strongly anharmonic and major contributors to thermal expansion in both systems. The application of pressure hardens the low-energy part of the phonon spectra involving Ag vibrations and confirms the highly anharmonic nature of these modes.

Quasi-harmonic and leading order anharmonic parameters for some minerals estimated from available thermodynamic data

The thermodynamics of the Earth's interior is usually approached through the quasi-harmonic theory of solids, relying on the tacit assumption that explicit anharmonic contributions, which are difficult to calculate even for the most simple solids, are not important. The basic qualitative idea assumes that, as anharmonicity increases with temperature but decreases with pressure, in the Earth's interior the two effects are self-cancelling. A safer approach is to consider not the absolute but the homologous temperature, i.e. to take the melting temperature as a scaling factor. Since the geotherm lies rather close to melting, the relative importance of anharmonic terms in the thermodynamics of the Earth's interior is presumably high. At room pressure, the leading order anharmonic coefficients and their volume derivatives, as well as the quasi-harmonic terms from available thermodynamic data, can be estimated by fitting to experiment the theoretical expressions to the fourth order in the Hamiltonian of three functions of compressibility, heat capacity and thermal expansion, tailored to minimize the effect of experimental errors. The capability of this procedure to produce stable and accurate results has been previously tested through extensive applications to NaCl, and to several metals, chosen because of the large number of available independent data, which allowed the study in detail of the effect of external as well as internal errors. Comparatively accurate estimates were found to be possible. The procedure is here applied to forsterite, periclase, rutile and pyrope, allowing the estimation of the relative magnitude of the quasi-harmonic and leading order anharmonic terms. At temperatures of half the melting temperature the difference between the two is of the order of 20% on the Grüneisen parameter and just a few per cent on the heat capacity, which is nevertheless typically 10% from the classical 3 R limit.

Atomic mean-square displacements in f.c.c. metals

Abstract In two recent letters, one by Zoli in 1991, and the other by Schober in 1992, the evaluation of the quasiharmonic and anharmonic contributions to the atomic mean-square displacement (MSD) for f.c.c. metals has been discussed. In Zoli's work, the difference in the two contributions is found to be 91%. Schober, on the other hand, has not evaluated the explicit anharmonic contribution to MSD. The huge difference in Zoli's work is shown here to be due to an inaccurate evaluation of the explicit anharmonic contribution to MSD. A proper self-contained method as presented here, which employs the same model in the quasiharmonic and anharmonic calculations of MSD or Debye-Waller factor, indeed shows that the two contributions differ from each other by 10–15% depending on the temperature. Larger differences exist at higher temperatures. Some numerical results are given for a model of the f.c.c. lattice, namely a nearest-neighbour central force model employing a Lennard-Jones potential (applicable to rare-g...

A calculation of the anharmonic phonon frequencies in solid deuterated naphthalene-d8

A detailed procedure leading to the evaluation of the explicit anharmonic contributions to the phonon frequency shift and the linewidth to the lowest order is described for a molecular crystal. An application is then made to obtain numerical results for phonon frequency shifts and the linewidths in deuterated naphthalene. The assumptions are made that the molecules behave as rigid units and the intermolecular potential is a pairwise sum of the atom-atom potentials. Using a 6-exp potential, and the potential parameters provided by Kitaigorodski (1973), the results for all the external modes at the Gamma points (q=0) are presented and compared with the available experimental data. Keeping in view the inadequacy of many of the measured results, and the fact that there is only a fair agreement between the measured and calculated harmonic phonon frequencies with the chosen potential, the results from the calculation presented here for phonon shifts and linewidths are very encouraging. An attempt made to compare the results from the three sets of potential parameters-those of Kitaigorodski, Mackenzie et al. (1977) and Williams (1966, 1967)-reveals the superiority of Williams's parameters.

Temperature dependence of the phonon frequencies in deuterated anthracene

Neutron inelastic scattering experiments have been carried out to measure the phonon energies of deuterated anthracene C14D10 for the lowest 16 dispersion branches along (0, 0, xi ) direction at various temperatures ranging from 14 to 250K. The results for the temperature dependence of the phonon energy shift and width are presented. The phonon energies show a negative shift with increasing temperatures. A quasiharmonic treatment using Kitaigorodsky potential parameters is found to explain quite well the observed shift for some of the phonons. For the remaining phonons the measurements indicate a positive shift due to the explicit anharmonic contribution. For these phonons the magnitude of the observed widths is also large.

Numerical estimate of the anharmonic contribution to thermal expansion and free energy to orderλ4

Thermal-expansion contributions to the free energy and heat capacity of an anharmonic crystal have been estimated. It has been found that the perturbation expansion for the free energy converges faster by taking into consideration the thermal-expansion contribution. Due to thermal expansion, the ${T}^{2}$ coefficient of the heat capacity at constant volume is reduced to about 50% of its explicit anharmonic contributions. The expressions for thermal expansion have also been obtained to $O({\ensuremath{\lambda}}^{3})$. The contribution of $O({\ensuremath{\lambda}}^{3})$ has been found to enhance the lower-order results by about (4-7)% near room temperature.

Thermal Properties of Alcaline-Earth-Oxides / III. Analysis of Anharmonic Effects

Abstract Experimental heat capacity data for the alcaline-earth-oxides have been used to analyse by a new method the high temperature thermodynamic properties of these crystals in terms of quasi-harmonic and explicit anharmonic contributions. The explicit anharmonic contribution ΔC is consistent with theoretical predictions; ΔC is proportional to T in a first order approximation. The factor of anharmonicity A is negative for MgO, A = (-0.4 + 0.4) · 10-5 K-1 , for SrO A= (-2.8±1) · 10-5 K-1 and for BaO A=(-3.9±1) ·10-5 K-1 , but positive for CaO A = (2.3 ± 1.2) · 10-5 K-1. Comparison with the results of analysis of the anharmonic effects in the alcaline-halides shows that the alcaline-earth-oxides may be treated by the same models valuable for the alcaline-halides. The results suggest that the type of anharmonic forces is determined in the alcaline-halides as well as in the alcaline-earth-oxides primarily by the cation of the salt.

Anharmonic contribution to the entropy of solids. Analysis of KF

A method is described for analysing the explicit anharmonic contribution to the entropy which is obtained from experimental Cp data. It is known that there exists a temperature interval in which the experimental entropy at fixed volume, Sexp(V0, T), fulfils Salter's expansion; it is called the apparent quasiharmonic region. According to Barron's frequency shift, the anharmonic contribution to the entropy is found to be Delta Sanh(V0, T)=(A(V0)/3NK)EvCv where A(V0) is a constant characteristic of the crystal, EvCv is fitted in the apparent quasiharmonic region to an expansion, Sigma n=0infinity anT-2n, up to a temperature T. For T>T, the difference between the experimental values and the extrapolated values, obtained with the Salter expansion, is called Delta Sexpanh(V0, T) and therefore is equal to the difference between the anharmonic contribution to the experimental entropy and the anharmonic contribution to the extrapolated entropy. The method is applied to KF, and A=(-2.1+or-0.5)10-5K-1 is found. In the potassium halides the absolute value of A theta (2) is found to decrease when the radius of halide ion increases.

Anharmonic effects in the thermodynamic properties of solids VI. Germanium: heat capacity between 30 and 500 $deg$C and analysis of data

The heat capacity of germanium has been measured between 30 and 500 °C with an accuracy of ± 03% for T less, similar 400 °C and ± 05% at the highest temperatures. The average deviation of the experimental points from a smoothed curve is less than 01%. The entropy, the constant volume heat capacity and the Gruneisen function have been analysed to yield quasi-harmonic and explicit anharmonic contributions. The explicit anharmonic contributions to the entropy and heat capacity are consistent with the predictions of anharmonic perturbation theory, and give A(V0) = 31 ± 10 × 10-5 degk-1 and B(V0) = 14 ± 1 × 10-8 degk-2 for the first- and second-order anharmonic coefficients respectively. The analysis of γ(T, V) gives γinfinityqh(V0) = 070 ± 004 and γanh = 45 ± 15. The explicit anharmonic contributions are much larger than those arising through thermal expansion of the crystal, in contrast to the crystals previously studied. The temperature dependence of the geometric mean frequency calculated from the results of the anharmonic analysis is somewhat lower than that derived from measurements on nine normal modes by inelastic neutron scattering. The value of the first-order anharmonic coefficient calculated using a phenomenological elastic continuum model is in good agreement with experiment.

Anharmomic effects in the thermodynamic properties of solids V. Analysis of data for NaCl, KCl and KBr

Experimental data for the entropy, the constant volume heat capacity and the Gruneisen function for NaCl, KCl and KBr have been analysed to yield quasi-harmonic and explicit anharmonic contributions. Values have been obtained for γinfinityqh(V0), γanh = (1/ΔCvanh)(partial differential ΔSanh/partial differential ln V) and for ΔSanh and ΔCvanh as functions of temperature and volume. The explicit anharmonic contributions to the entropy and heat capacity are consistent to within experimental uncertainty with the relations and ΔSanh(T, V0)/6R = AT + BT2 ΔCvanh(T, V0)/6R = AT + 2BT2 which are the expected forms for anharmonic effects up to second order. For NaCl, A = -15±09 × 10-5 degk-1 and B is probably positive; for KCl, A = 28±07 × 10-5 degk-1 and B similar 1 × 10-8 degk-2; and for KBr, A = 35±10 × 10-5 degk-1 and B similar 14 × 10-8 degk-2. The following values for γinfinityqh(V0) and γanh(V0) were obtained: NaCl, 158±005 and 7±3; KCl, 148±005 and -5±2; KBr, 152±005 and -5±2. The above data may be used to calculate the volume dependence and explicit temperature dependence of the geometric mean frequency. For KBr the total temperature dependence determined in this way agrees very well with that derived from inelastic neutron scattering experiments; such experiments have not been reported for NaCl or KCl. Calculations of explicit anharmonic effects for specific models of these alkali halides give results too high by factors of 3 or 4 even using a fairly sophisticated shell model. These effects provide a very stringent test for a model, however, because they are the resultant of two terms of similar magnitude but opposite sign arising from the cubic and quartic terms in the crystal potential energy. It is the different balancing of these terms which accounts for the different signs of the explicit anharmonic effects in sodium and potassium salts. This suggests that the type of interatomic force is determined primarily by the alkali ion.

Calculation of 0cdifferences for the face-centred cubic and close-packed hexagonal lattices in the ideal inert gas solids

Θ0c(cph) and the ratio K ≡ 100{Θ0c(cph) - Θ0c(fcc)}/Θ0c(fcc) for the ideal inert gas solids was calculated, using the quasi-harmonic approximation and an (m - 6) Lennard-Jones all-neighbour force model; K was found to be about 2%. The neglect of explicit anharmonic contributions to K is discussed. A table of the relevant all-neighbour sums is given. It was found that the use of the ideal axial ratio γI = Θ(8/3) to characterize the close-packed hexagonal lattice limits the accuracy to which K can be calculated to about one decimal place, and Θ0c(cph) to about two decimal places.

Thermal Expansion and Other Anharmonic Properties of Crystals

The temperature dependences of the thermodynamic functions, as derived from lattice dynamics, are examined for the limit of low temperature and also for high temperatures (those above a high characteristic temperature). Particular attention is given to the effects of anharmonic terms in the lattice potential energy. Detailed calculations are reported for central-potential models for fcc, bcc, and hcp lattices. In particular, the normal-mode frequencies and Gr\"uneisen parameters were calculated for a large number of points in the Brillouin zone as a function of volume; and the specific heat, compressibility, thermal-expansion coefficient, and macroscopic Gr\"uneisen parameter were calculated as functions of temperature and volume. At fixed volume the isothermal compressibility shows little temperature dependence and the explicit anharmonic contribution is small; at zero pressure the compressibility increases with increasing temperature and the explicit anharmonic contribution is again small. The thermal-expansion coefficient exhibits similar behavior at high temperatures. The anharmonic specific heat is proportional to temperature at high temperatures, and also depends strongly on the volume. The effective Debye temperatures and the macroscopic Gr\"uneisen parameters exhibit a wide variety of temperature and volume dependences. Approximations are developed for quantities which determine the behavior of thermodynamic functions at low and high temperatures, and approximate relations between several anharmonic properties are found. These approximations are tested by comparison with accurate calculations for the central-potential models.