Quartic Anharmonicity Research Articles

Kinks in systems with cubic and quartic anharmonicity

For a classical system of interacting particles with on-site cubic or quartic anharmonicity explicit analytic solutions of the d'Alembert equation are obtained in the form of kinks in the presence of dissipation (viscous or Rayleigh) and a constant force. These kinks will be asymptotically stable in the case of quartic anharmonicity and unstable in the case cubic anharmonicity.

Intrinsic Localized Vibrational Modes in Anharmonic Crystals

An intrinsic localized vibrational mode is shown to appear in a pure anharmonic crystal lattice with quartic anharmonicity. This is a discrete-space-version of a spatially localized configuration in nonlinear field theory. Two types of quartic anharmonic potential, optical-mode-type and acoustical-mode-type, are considered to illustrate the existence of an s-like localized mode in a simple cubic lattice with nearest neighbour interactions. The calculation is done by using the lattice Green's function formalism in close analogy with the case of a localized mode due to an impurity atom in the harmonic lattice and by solving a nonlinear eigenvalue problem with widely spread defect space in a successive manner. It is shown that such an intrinsic localized mode can appear at any lattice site if the anharmonicity is sufficiently strong. The linear stability analysis of the localized mode is made to show that a localized mode lying above the top of the phonon frequency band is always stable. A brief study is also made on the movability of such an intrinsic anharmonic localized mode

Lifetimes of degenerate benzene 1B2u levels split by vibrational angular momentum

This work examines, both experimentally and theoretically, the lifetimes of nominally degenerate benzene 1B2u levels split by differential coupling of the vibrational angular momentum components. We report fluorescence lifetimes for the 61162, 61102, and 61172 levels of the S1 electronic state in cold C6D6 in a supersonic jet. The higher energy vibrational angular momentum component has the longer lifetime. Calculations of the nonradiative decay rates for these levels are presented. Calculations based on conventional radiationless transition theory, assuming the prepared state is a single vibronic level, are not able to explain the experimental results, and we conclude that another mechanism must be responsible for the observed lifetime differences. We present a qualitative explanation of the data that takes into account perturbations of zeroth-order harmonic oscillator states by cubic and quartic vibrational anharmonicities and by Coriolis coupling. These perturbations affect the split vibrational angular momentum components differently and are capable of producing differences in lifetimes.

Broadening of biphonon lines caused by interaction with acoustic phonons

A Green function technique is used to study a two-phonon spectrum. For the first time was calculated beyond the RPA the damping containing a term due to the quartic anharmonicity and a term connected with the acoustic phonon interaction. The last term causes larger line broadening, especially at room temperature.

The orientation—inversion motions of ammonia trapped in matrix an inertial model

Good doorway stationary states for the inversion and orientation of ammonia trapped in rare gas and nitrogen matrices are determined in order to interpret the narrowing of the v 2 inversion doublet and the hindering of the molecular rotation. A pseudomotional crystal model (MC) is developed to describe the coupling between the optical (inversion + orientation) modes and the lattice vibrations. With respect to the rigid crystal model already studied, the MC model introduces additional statical, dynamical and kinetical contributions in the optical mode hamiltonian. Among these additional terms, the cubic and quartic anharmonicities of the motion of the molecule center of mass, the effective mass corrections and the damping due to the matrix surrounding are shown to be very different in the three matrices Ar, Xe and N 2. Numerical results on the inversion doublet splitting and on the rotational level spacing of ammonia agree with the observed data.

Temperature modulation and damping of an anharmonic two-phonon spectrum

In the present paper a two-phonon spectrum in the frequency region of the sum or difference of the frequencies of two optical phonons is studied. A quartic anharmonicity is investigated as a factor for damping of the one- and two-phonon states as well as a mechanism binding two phonons into a biphonon. The numerical calculations of the dynamical structure factor give the complete picture of the two-phonon spectrum, e.g. the temperature modulation of the frequencies, width and the intensity of the spectral lines for different values of the anharmonicity and at various widths of the phonon bands. The possibility for a temperature modulation of the biphonon levels in the sum of frequencies region and for the existence of bound states in the region of the difference of the phonon frequencies is shown.

Quartic anharmonicity and angular momentum effects in even-even spherical nuclei

Arguments are given that the main nonlinear features of low-lying quadrupole excitations of even-even soft spherical nuclei are due to the quartic phonon interaction and to the coherent rotational response of noncollective degrees of freedom. The corresponding hamiltonian is solved by group theory methods; the results give a good description of energy spectra as well as of transition probabilities, the number of fitted parameters being less than that of the IBA. Some regularities are found going through the periodical table.

Anharmonicity of Low Frequency Lattice Vibrations in Red-HgI2

The Raman scattering spectra in the layer compound red-HgI 2 have been measured in detail under hydrostatic pressure. Interatomic force constants are deduced from the data on the basis of a simple Born-von Karman model to estimate the harmonic frequency of the infrared active E u 1 (TO) mode. The frequency is found to be considerably smaller than the experimental value at low pressures. This discrepancy is attributed to the shift due to the quartic anharmonicity associated with the shear potential of a Hg–I bond. Its harmonic force constant increases rapidly with increasing pressure, so that the E u 1 (TO) mode becomes rather harmonic at high pressures. The anharmonicity due to the change in crystal volume is also discussed in relation to the pressure dependence of the interlayer interaction between iodines. Our experimental results show that the effective size of iodine ions decreases with increasing pressure.

Pade approximant method for the statistical thermodynamics of a quantum system. II. Quartic anharmonic oscillator

For pt.I see ibid., vol.17, p.1877-90 (1984). A method of successive approximation involving two-point Pade approximants developed in a previous paper is applied to the quartic oscillator and to the anharmonic oscillator with quartic anharmonicity. The input data are the first three energy levels and the coefficients in a high-temperature series derived from the first three terms of the Wigner-Kirkwood series. The method gives an approximate expression for the partition function which is then differentiated to find the internal energy and the heat capacity. Good results are obtained at all temperatures and for a wide range of anharmonicities.

Solitary waves in a 1D anharmonic lattice with two-component order parameter

A 1D crystal with two-component order parameter, quartic anharmonicity and anisotropy in the on-site harmonic terms is considered in classical continuum approximation. It is shown that the propagation of travelling waves in the crystal is governed by completely integrable dynamics. The possible solitary excitations of the ground states and those connecting different ground states are discussed on this background.

Calculation of the phonon width and phonon shifts in naphthalene-d8

Abstract The width Γ and the shift Δ of the q = (0, 0, 0) phonon at 2.52 THz was calculated with three different sets for the forces of a 6-exp-potential. The molecules were assumed to be rigid. We took into account cubic and quartic anharmonicity terms. The three models differ in magnitude for Γ and Δ for about 25%. A comparison is made with inelastic neutron scattering datas.

Bound states of two surface phonons at a crystal surface with an adsorbed monolayer

We obtain the Bethe-Salpeter equation for the Green's function describing the propagation of two surface phonons interacting via cubic and quartic anharmonicity. We solve that equation by keeping only the latter contribution to its kernel. The two-dimensional nature of the problem is reflected in the existence of a bound state of two high-frequency surface phonons associated with the presence of an overlayer of light atoms, for all values of the coupling strength. This is illustrated with the detailed solution of a simple model.

Phase-integral calculation of the energy levels of a quantal anharmonic oscillator

The phase-integral method developed by N. Froman and P. O. Froman is used for solving the quantal eigenvalue problem of an anharmonic oscillator with quartic anharmonicity. The generalized Bohr-Sommerfeld quantization condition up to the seventh-order phase-integral approximation is expressed explicitly in terms of complete elliptic integrals. Solving this quantization condition numerically, and comparing the results with recent very accurate numerical results obtained by Banerjee, we present curves exhibiting in a general way the accuracies of various orders of the phase-integral approximations. These curves clearly illustrate the utility of higher-order phase-integral approximations for the treatment of anharmonic oscillators.

Study of the elastic properties of K2SeO4at the phase transition to the incommensurate and the commensurate phase by ultrasonic techniques and Brillouin spectroscopy

Using the ultrasonic pulse-overlap technique and multipass Brillouin spectroscopy the complete elastic stiffness tensor was determined in the temperature range from 300 to 30K. The transition to the incommensurate phase (at T2=127K) is reflected in the elastic stiffness functions c11, c22, c33, c44, c55, c13 and c22. The lock-in transition into the commensurate phase (at T3=93K) has no significant effect on all the measured stiffness functions apart from c55, which experiences large variations around T3. The observed effects are interpreted in terms of coupling to energy-density fluctuations in the case of c33, T>T2. In the incommensurate phase the stiffnesses c22, c33 and c44 are augmented by a contribution from quartic anharmonic interaction with the onset of incommensurate ordering. Another type of quartic anharmonicity is responsible for the decrease in c55 in the commensurate phase. The gradual transition at the lower end of the incommensurate phase is an indication of soliton-like discommensurations.

Nonergodicity and dynamical approach to the anharmonic oscillator

General lower bounds for the time average C AA(∞)≡lim T→∞( 1 T )∝ T 0 C AA(t) dt of the correlation C AA ( t)≡〈 A( t) A(0)〉−〈 A〉 2 of an arbitrary variable A are derived, which depend only on the temperature derivatives of the canonical averages of A and the Hamiltonian of the system. The bounds may be used to give good estimations for C AA (∞) which is different from zero when A is nonergodic. It is important to take care of these terms when dynamical theories made for interacting systems are applied to isolated systems. We show explicitely that our recently developed dynamical approach to phonon systems with quartic anharmonicity yields excellent results for the corresponding isolated system, the anharmonic single well oscillator, when nonvanishing time averages are taken into account.

Thermal behavior of the Debye-Waller factor and the specific heat of anharmonic crystals

We study, within the framework of the variational method in statistical mechanics, the influence of the cubic and quartic crystalline anharmonicity on the classical and quantum thermal behavior of the specific heat, Debye temperature $\ensuremath{\Theta}$, Debye-Waller factor $W$, crystalline expansion, and phonon spectrum. The systems we mainly focalize are the single oscillator, the monoatomic linear chain, and the simple cubic crystal. The trial Hamiltonian is a harmonic one, therefore the various anharmonic influences are mainly absorbed into the renormalization of $\ensuremath{\Theta}(T)$. Several differences between the classical and quantum results are exhibited. Satisfactory qualitative agreement with experience was obtained in the low-temperature regime, in particular on what concerns the existence of a minimum in $\ensuremath{\Theta}(T)$ which has been observed in Cu, Al, Ag, Au, and Pb. For the intermediate-temperature regime the customary linear behavior of $W(T)$ [hence $\ensuremath{\Theta}(T)$ almost constant] is reobtained. Finally, in the high-temperature regime, the present treatment leads to a $\sqrt{T}$ dependence for the $W$ factor, which implies the wrong curvature with respect to experimental data. A possible explanation of this disagreement might be related to the melting phenomenon, which is not covered by the present theory.

Far-infrared dispersive-reflection measurements on NaCl, compared with calculations based on cubic and quartic anharmonicity. II. Low temperature

Accurate far-infrared dispersive-reflection measurements have been performed on a large single crystal of NaCl to yield the reflectance amplitude and phase between 15 and 500 cm−1 at 48 K. These have been augmented with power-transmission measurements in those regions where the phase is small. Some of the difficulties encountered while extending this technique to low temperatures have been discussed, including the unfortunate tendency of the evaporated aluminum reference surface to break away from the crystal. The measurements have been compared with calculations of the damping of the transverse-optic resonance in NaCl, resulting from one-phonon, two-phonon, and three-phonon relaxation processes, which result from isotopic impurities (35,37Cl), cubic anharmonicity, and quartic anharmonicity, respectively. The overall agreement is good but the experimental accuracy is less than that obtained at 290 K. The low temperature results show that a significant discrepancy which was observed at 290 K and which persisted down to 48 K is not the result of neglecting higher-order anharmonicity, as originally supposed, but rather a probable error in the two-phonon spectrum resulting from incorrect values of low derivatives of the crystal potential.

Exact solutions for a doubly anharmonic oscillator

Two classes of exact solutions and eigenvalues for an oscillator with both quartic and sextic anharmonicity as well as the conditions under which these solutions can occur are given.

Importance of quartic anharmonicity for bending partition functions in transition-state theory

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTImportance of quartic anharmonicity for bending partition functions in transition-state theoryBruce C. Garrett and Donald G. TruhlarCite this: J. Phys. Chem. 1979, 83, 14, 1915–1924Publication Date (Print):July 1, 1979Publication History Published online1 May 2002Published inissue 1 July 1979https://doi.org/10.1021/j100477a025RIGHTS & PERMISSIONSArticle Views164Altmetric-Citations77LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (1 MB) Get e-Alerts

Asymptotic estimates and borel resummation for a doubly anharmonic oscillator

The large-order behavior of perturbation theory for an oscillator with both quartic and sextic anharmonicity is examined. Two approaches are considered: an explicit double expansion in the two coupling constants and a single expansion in the number of loops. A two-variable extension of the Borel resummation technique is developed and applied to the double series, and the standard one-variable resummation technique is applied to the loop expansion. Numerical values for the ground-state energy are calculated in each case and the two procedures are compared.