Quartic Anharmonic Contribution Research Articles

Structural Anharmonicity Explains Continuous Frequency Shifts of Intramolecular Ring Vibrations in a Hydrogen-Bonded Antiferroelectric Crystal

Rational principles from which one can design the energy storage capability of antiferroelectric crystalline materials composed of hydrogen-bonded molecular constituents remain deficient. In particular, there remains gaps in our understanding of the fundamental physical mechanisms by which microscopic electrostatics in this class of materials control macroscopic properties central to potential energy storage technologies. While molecular vibrations have been applied as probes of microscopic electrostatics in other areas of materials physical chemistry, their ability to report the electrostatics in antiferroelectrics remains an open question. In this study, we use temperature-dependent Raman spectroscopy to investigate the fundamental physical mechanism by which aromatic ring distortional vibrational peaks shift in the antiferroelectric phase of crystalline 2-trifluoromethylbenzimidazole (TFMBI). With density functional theory (DFT) calculations and a theoretical model of anharmonic contributions to vibrational frequencies, we propose the quartic anharmonic contribution to the interatomic potential energy of TFMBI explains the continuous vibrational peak shifts we observe in the temperature-dependent Raman spectra of this material. Specifically, we find the shifts likely result from thermally driven changes to the average occupation of other, lower frequency intramolecular vibrations interacting with the Raman-active ring distortion vibrations of crystalline TFMBI. To support this conclusion, we use explicit fits of our measured peak shifts to an equation for the fundamental vibrational transition we develop from a theoretical model of material anharmonicity and the harmonic frequencies found from DFT calculations. In addition to highlighting the importance of anharmonic contributions to the structural dynamics of molecular crystals, this analysis also allows us to assess the ability of DFT calculations to predict the harmonic frequency of the aromatic ring vibrations of TFMBI correctly. Furthermore, we characterize the correlation between the structure of a low-temperature monoclinic phase of TFMBI unreported previously and discontinuous shifts in the positions of three peaks in the Raman spectrum of TFMBI. This characterization indicates the quartic anharmonic coupling does not depend on the hydrogen-bonding structure of the material. Our results illustrate the care researchers must take to appropriately interpret the physical mechanism explaining vibrational peak shifts in hydrogen-bonded antiferroelectric materials.

Lattice dynamics of Sr2TiO4

Lattice dynamics of Sr2TiO4 is investigated by means of inelastic neutron scattering and Raman scattering experiments. The phonon density of states is experimentally determined by inelastic neutron scattering experiment and all the peaks except the one at 88 cm-1 can be attributed to some optical modes, whereas the peak at 88 cm-1 is probably due to the acoustic phonons at the zone boundary. Raman scattering experiments done at low temperature reveal that the explicit contribution to the anharmonicity from the thermal population of the vibrational levels should be stronger for the high energy A1g mode at about 570 cm-1 than for the other Raman active modes. This contribution increases anomalously with the temperature due to a quartic anharmonic contribution.

Observation of the broadening and shift of the frustrated translation vibrational mode of CO on Cu(001) by high resolution helium atom scattering

The lifetime broadening of the frustrated translation (T) mode parallel to the surface of isolated CO molecules [≲6% of a c(2×2) coverage] chemisorbed on Cu(001) has been studied with high resolution inelastic helium atom scattering as a function of the surface temperature and the parallel momentum transfer. The T excitation peak shows a temperature dependent peak shift and a broadening. The latter is attributed to a quartic anharmonic potential contribution described by a spectroscopic constant χe=−0.0085±0.0008. The extrapolated TS=0 K peak broadening is then γ0=85±5 μeV, corresponding to a vibrational lifetime τ=8±1 ps. We attribute this lifetime to substrate phonon mediated damping and electronic damping.

Atomic mean-square displacement in fcc metals: Repulsive potentials

We present a method for the calculation of the atomic mean-square displacement (u2) of an anharmonic crystal from potentials involving repulsive interactions. The quasiharmonic and the lowest-order cubic and quartic anharmonic contributions to (u2) are evaluated from the knowledge of seven Brillouin zone (BZ) sums which are tabulated in the interval -0.1 a1 0.0. The parameter a1 characterizes the volume dependence of the BZ sums and is negative for repulsive potentials. All the BZ sums are evaluated in the limit L, where L is the step length from the origin to the boundary of the BZ. The method is applicable to a nearest-neighbour central force model of the fcc lattice, and it can be extended to a bcc lattice. We present two applications of the method. One is the calculation of (u2) from the r-1 2 repulsive potential, and our results are in good agreement with those obtained by Monte Carlo methods. The other is to the nearest-neighbour Born-Mayer potential with a volume-dependent e ective coe cient alpha. In the case of Cu, the agreement with other theoretical calculations as well as the experimental values from X-ray data at 300 and 400 K is excellent.

Erratum: Low-order anharmonic contributions to the internal energy of the one-component plasma

We present precise calculations of the second-order cubic and first-order quartic anharmonic contributions to the internal energy of the one-component plasma (OCP), which are then used with available computer-simulation data to estimate the size of higher-order anharmonic effects. We show that in both the zero-temperature and high-temperature limits the OCP is very harmonic. At zero temperature, we find that the two lowest-order anharmonic contributions are the largest anharmonic contributions, and of the two the second-order cubic contribution is nearly three times the first-order quartic. At high temperatures, we find that the second-order cubic contribution is about twice that of the first-order quartic, but their combined contribution to the internal energy is small compared with our estimate of the next-order anharmonic terms.

Temperature dependence of optical modes of vibration in MgO thin film

The authors observe the thermal emission spectra of the metal-backed thin film of MgO at the emission angle of 25 degrees . The film deposition on a platinum plate is performed by means of an electron beam technique in a high-vacuum system. The observed spectra measured at elevated temperatures are analysed by using the response function based on the virtual-mode theory of Kliewer and Fuchs for the ionic slab to estimate the temperature dependence of the lattice dynamical quantities such as the phonon frequency and the damping. The dielectric function derived from the theory based on the thermodynamic Green function technique is considered in the spectrum analysis. No significant difference is found in the values of the parameters between a film and a bulk single crystal. The damping around the TO frequency shows the linear temperature dependence, while the damping around the LO frequency varies nonlinearly with temperature. The quartic anharmonic contribution to the phonon self-energy is significant as well as the cubic contribution around the TO frequency.

Convergence of the quartic anharmonic contribution to the Helmholtz free energy in metals

The convergence of the quartic term of the Helmholtz free energy (${F}_{4}$) in metals is investigated by comparing the numerical results of ${F}_{4}$ by two methods. In the first method, ${F}_{4}$ is calculated by summing over the direct lattice vectors of the product of the fourth-rank tensor derivatives of a two-body potential and correlation functions involving Brillouin-zone and branch-index summations. In the second method, ${F}_{4}$ is calculated from an alternative expression which requires beforehand the knowledge of wave-vector-dependent fourth-rank tensors. The alternative expression for ${F}_{4}$, which is derived in this paper, is more appropriate for the calculation of ${F}_{4}$ in metals where the forces are long range and oscillatory. Calculations are carried out by the two methods, for different volumes, from almost 0 K to the melting temperature range for Na and K using the same first-principle potential as used by Shukla and Taylor in their previous calculations of ${F}_{4}$. Representing the ${F}_{4}$ results of the two methods by ${F}_{4}(s)$, where $s$ is the shell index at which the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{l}}$ summation is truncated in the first method and ${F}_{4}$ (Ewald), we find the two methods giving the same answer for ${F}_{4}$ only in some cases. For example, in Na, ${F}_{4}$ (Ewald) agrees exactly with ${F}_{4}(s)$, where $s=18$, at the 90-K volume. The agreement between the two is almost exact at 160-K and 361-K volumes with $s=17 \mathrm{and} 10$, respectively. At the other two volumes referred to $T=5 \mathrm{and} 293$ K, ${F}_{4}$ ({Ewald) and ${F}_{4}(s)$ differ by about 1% where $s=22 \mathrm{and} 21$, respectively, for these volumes. In K, ${F}_{4}$ (Ewald) and ${F}_{4}(s)$ differ by less than 1% in each case for the three volumes corresponding to temperatures 9, 215, and 299 K where for each of these volumes $s=15, 7, \mathrm{and} 10$, respectively. The largest disagreement between ${F}_{4}$ (Ewald) and ${F}_{4}(s)$ ($s=7$) arises for the volume at 99 K. In this case they differ by 3%. It is clear from the above discussion that ${F}_{4}$ (Ewald) and ${F}_{4}(s)$ are in agreement with each other for "some" shell index $s$, but $s$ changes with volume. Thus, without a priori knowledge of an exact value of ${F}_{4}$, such as the one obtained by the Ewald procedure, one cannot decide the shell index ($s$) at which the summation in the first method should be truncated to obtain a correct answer for ${F}_{4}$.

High-temperature specific heat of crystals

The Monte Carlo method is used to estimate the specific heat of a model of rubidium. Both the specific heat at constant volume, ${C}_{\ensuremath{\nu}}$, and the specific heat at constant pressure, ${C}_{p}$, are obtained for a range of temperatures up to the instability point of this lattice. These results for the fully anharmonic perfect crystal are compared with those obtained by perturbation theory to lowest order in the anharmonicity, (i.e., only cubic and quartic anharmonic contributions to the Helmholtz free energy are considered). It is shown that the fully anharmonic Monte Carlo calculation yields a more rapidly increasing specific heat than the linear temperature dependence given by lowest-order perturbation theory in the high-temperature limit. The Monte Carlo calculations also indicate that, at temperatures much higher than the Debye temperature, large-scale atomic displacements can occur without disrupting the lattice. When this happens, there is a further increase in the specific heat.

Lattice anharmonicity of alkali metals

Frequencies and eigenvectors are obtained for alkali metals by using the force constants calculated by Shyu and Gaspari. These are used to calculate the cubic and quartic anharmonic contributions to the specific heat in the high-temperature limit with nearest-neighbor central-force interaction, in which the effect of thermal expansion is included. Calculated values of the anharmonic coefficient $A$, defined by $\frac{{C}_{v}}{3Nk}=1+AT$, are in good agreement with the experimental results. The values of $A$ are positive for alkali metals. Calculated dispersion curves lie higher than the experimental ones. The sum $\ensuremath{\Sigma}{i=1}^{3}{{\ensuremath{\omega}}_{i}}^{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}})$ is independent of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ for sodium, potassium, and rubidium, but not for lithium. It is concluded that a potential which may be adequate in estimating the effects of anharmonicity may not be essentially suitable to describe the harmonic properties.

Lattice anharmonicity of diamond-structure crystals

Cubic and quartic anharmonic contributions to the Helmholtz free energy of a diatomic crystal are derived as functions of temperature for a general force-constant model of the crystal. The corresponding cubic and quartic anharmonic contributions to the specific heat at constant volume for diamond, germanium, silicon, and grey-tin are evaluated in the high-temperature limit with central force interactions between the atoms up to second-nearest neighbours. The effect of thermal expansion of the lattice is taken into consideration. There is excellent agreement between the experimental and calculated values of anharmonic coefficient A, defined by Cv/3Nk=1+AT, for germanium and silicon. Values of A for diamond and grey-tin are predicted for which the experimental results are not available. It is concluded that the second-nearest neighbours' contribution is much smaller than that of the nearest neighbours in the calculations of anharmonic coefficients. A for diamond-structure crystals.

Lattice dynamics and anharmonic self-energies for the lithium halides

Lattice-dynamical calculations, using both the modified deformation-dipole model and the breathing-shell model, of the lithium halides are presented. Using the phonon frequencies and eigenvectors generated from these calculations, the frequency and temperature dependence of the Hermitean and anti-Hermitean parts of the $q\ensuremath{\approx}0$ transverse-optic-mode self-energy are calculated. The results of the anharmonic calculations reveal that the zero-frequency self-energy shift is dominated at low temperatures for all four halides by the first-order quartic anharmonicity, but that, as the temperature is increased, the second-order cubic and quartic anharmonic contributions become increasingly more dominant, causing a changeover in sign for the zero-frequency self-energy shift, a result observed experimentally for other alkali halides crystalizing in the NaCl structure. This same phenomenon is also found for the far-infrared self-energy shift for LiBr and LiI, but for LiF and LiCl it appears that the second-order cubic and quartic anharmonicity provides the dominant contribution to the far-infrared self-energy shift at all temperatures. For all four materials the second-order cubic anharmonicity provides the dominant contribution to the inverse lifetime at low temperatures, but as the temperature is increased the second-order quartic contribution becomes increasingly important and finally provides the dominant contribution. At low temperatures the calculations yield lifetimes which are much longer than are actually observed, but for LiF, where experimental data exist at temperatures up to 1100 K, the calculations are in reasonable agreement for the range 200-1100 K.

Anharmonicity in silicate crystals: Temperature dependence of A u type vibrational modes in ZrSiO 4 and LiAlSi 2O 6

Infrared reflection spectra for the extraordinary ray of zircon and spondumene have been obtained in the frequency range from 270 to 1500 cm −1 and from room temperature up to 1300°K. Using a combination of Kramers-Kronig analysis and a classical dispersion theory, the oscillator parameters of the A 2u modes in zircon and A u modes in spodumene were determined. The temperature dependence of observed mode damping is analysed on the basis of quantum theoreticall results involving both cubic and quartic anharmonic contributions. The comparison of the present results with other data reported in previously studied silicate crystals, shows that the observed anharmonic behavior of internal and external modes is consistent with simple considerations based on the site vicinity of each ion.

Evaluation of the Free Energy of an Anharmonic Crystal

The anharmonic contributions of order η 4 to the free energy of a crystal have been evaluated for a central-force nearest-neighbor model in the Ludwig approximation. The accuracy of the Ludwig approximation in evaluating the Brillouin-zone sums occurring in the expression for the anharmonic contributions to the free energy has been investigated in some detail. It is found that the Ludwig approximation gives exact results for some simple sums. While most of the remaining sums are underestimated by about 18%, the sums occurring in the expression for the second-order quartic anharmonic contribution to the free energy are overestimated by about 20%. The coefficient of the T 2 term in the specific heat of lead at high temperatures is also estimated for a Morse potential and is found to be of the same order of magnitude as the experimental value obtained by Leadbetter.

Anharmonicity in the Silver and Thallium Halides: Far-Infrared Dielectric Response

The isobaric temperature dependence of the $q\ensuremath{\approx}0$ transverse- and longitudinal-optic modes of AgCl, AgBr, TlCl, and TlBr is reported in the range 4-450 K and the isothermal pressure dependence of the transverse-optic modes for pressures up to 10 kbar. These measurements are used to determine the thermal-strain and anharmonic contributions to the self-energy for the $q\ensuremath{\approx}0$ transverse-optic mode for these materials. The results reveal a near cancellation of the cubic and quartic anharmonic contributions to the self-energy for AgCl and AgBr with the cubic contribution dominating at all temperatures. In the thallium halides the quartic contributions are dominant at all temperatures and are sufficiently large as to dominate the thermal-strain contribution to the self-energy, thus causing the mode to be weakly soft.

Manifestations of Anharmonicity in the Specific Heat of Crystalline 63Cu

Accurate measurements of the specific heat of a high purity single crystal of 63Cu have been carried out for T<50°K. The data are analyzed in terms of deviations from the specific heat of a physically realistic phonon model for 63Cu. The sixth neighbor general-force model determined for normal Cu at 49°K by neutron diffraction measurements has been adapted to 63Cu by a scaling method. Precision calculations of phonon spectra were done for both general-force and simple Leighton models. A statistical analysis shows deviation curve features are statistically significant with a minimum close to that of the theoretical quartic anharmonic contribution. Smoothed experimental Cp and Cv data are given for crystalline 63Cu.

Second order quartic anharmonic contribution to the free energy of simple solids

The second order quartic anharmonic contribution of the free energy of simple solids is estimated using a central force model with nearest neighbour interactions. The estimate is compared with the recent experimental result of Leadbetter on anharmonic effects in lead.

Lattice anharmonicity of rare-gas solids, copper and sodium

An expression for the cubic and quartic anharmonic contributions to the specific heat of the monatomic face-centered and body-centered cubic crystals with nearest neighbours, central force interactions, is given in the high temperature limit, in which the effect of the thermal expansion is included. It is applied to the rare-gas solids, copper and sodium. The results are in excellent agreement with the experimental values.

Thermodynamics of anharmonic crystals

A general expression for the free energy of anharmonic crystals has been obtained using the equation of motion method of the double time Green's function. It is applied to obtain the second order quartic anharmonic contribution to the free energy.

Lattice-Dynamical Properties ofFe57Impurity Atoms in Pt, Pd, and Cu from Precision Measurements of Mössbauer Fractions

Precision measurements of the M\"ossbauer fractions of ${\mathrm{Fe}}^{57}$ in single crystals of Cu, Pd, and Pt were obtained over a range from liquid-helium temperatures to around 750\ifmmode^\circ\else\textdegree\fi{}K in about 50\ifmmode^\circ\else\textdegree\fi{}K intervals, using the "black wide absorber" technique. The over-all accuracy is estimated to be better than 0.7%, with a somewhat larger error for Pt. Our data are consistent with a harmonic lattice with small cubic and quartic anharmonic contributions. Properly weighted Debye temperatures ${\ensuremath{\Theta}}_{D}(\ensuremath{-}1)$ and ${\ensuremath{\Theta}}_{D}(\ensuremath{-}2)$ were derived from the low- and high-temperature limits of our $f$ measurements, which were compared with equivalent data for the pure hosts. We conclude that Fe in Cu is about 20% more strongly bound than Cu in Cu, while Fe in Pd and Pt shows about a 20% weaker average force constant than the host atoms. Fe in Cu also has a much larger degree of anharmonicity than pure Cu. Analysis of earlier data on Fe in Ni shows a similar behavior. Our data were also examined for evidence of localized impurity modes. The low-temperature data for Fe in Pt are consistent with the existence of a predicted localized mode, but they do not provide unambiguous evidence.

Temperature Dependence of the Width of the Fundamental Lattice-Vibration Absorption Peak in Ionic Crystals. II. Approximate Numerical Results

The results of numerical calculations of the temperature dependence of the width and position of the fundamental lattice-vibration absorption peak in NaCl and LiF are presented. The calculations are carried out in the high-temperature limit, on the basis of the Hardy-Karo deformation dipole model of these crystals. Cubic and quartic anharmonic terms are retained in the crystal Hamiltonian, but the approximation of neglecting the anharmonicity of the Coulomb forces has been made. The expressions for the Fourier-transformed anharmonic force constants have been approximated and simplified by a method suggested by Peierls. The results of the calculations show that the quartic anharmonic terms in the crystal potential energy make a contribution to the width of the fundamental absorption peak which is comparable in magnitude with the contribution from the cubic anharmonic terms, in agreement with the theoretical arguments of Gurevich and Ipatova. Since the quartic anharmonic contribution to the width is proportional to the square of the absolute temperature at high temperatures, these results provide an explanation for the experimental observations that the width increases with a power of the absolute temperature which is intermediate between the first and the second. Quantitatively, the theoretical results are in quite good agreement with the experimental data of Heilmann for the variation with temperature of the width of the fundamental absorption peak in LiF. In the case of NaCl, the agreement between theory and experiment is somewhat poorer, but the theoretical values are still within a factor of about 2 of the experimental values of Hass. The frequency dependence of the imaginary part of the dielectric constant of these two crystals has also been calculated, and is compared with experimental data.