Quartic Anharmonic Terms Research Articles

Higher Genus Solutions for the N × N Matrix Model

For the N × N Hermitian matrix model with a potential of quartic anharmonic term, a general expression of vacuum energy EH for arbitrary genus H is presented, and higher genus solutions E3, E4, and E5, are evaluated explicitly.

A simplified transformation of the rotational–vibrational Hamiltonian to eliminate a Coriolis term and perturbation theory for higher order terms

A quantum mechanical canonical transformation, applied previously to eliminate a Coriolis term from the rotational–vibrational Hamiltonian of a polyatomic molecule [J. Chem. Phys. 94, 461 (1991)], is simplified to permit the computation of rotational energy levels for highly excited vibrational states. An approximate matrix representation of the transformed Hamiltonian is presented which is shown to be very accurate for a two-mode model of formaldehyde. The quantum dynamics of the two nearly degenerate vibrational modes, strongly coupled by a Coriolis term, is studied using the approximate representation. It is shown that the second order Coriolis term and quartic anharmonic terms in the potential can be treated effectively as perturbations using the transformed Hamiltonians as the zeroth-order model.

Picosecond measurements of relaxation of internal modes in crystalline benzene as a function of temperature

The relaxation rates of four internal vibrational levels of benzene crystal have been measured by time-resolved coherent anti-Stokes Raman scattering (CARS) at different temperatures. The important information on the dephasing mechanisms provided by the experiment has been supported by anharmonic calculations, which include the full contribution of the density of phonon states, by utilizing average coupling coefficients. Different behaviors have been observed for the different modes considered. Three of the four vibrations (v1,v6, and v10) show linewidths that increase linearly in the classical regime with temperature; the experimental evidence for the important role played by three-phonon processes (driven by cubic anharmonicity) is confirmed by the calculations, which give a quantitative agreement with the observed linewidths. For the above-mentioned vibrations the role of pure dephasing results in a minor contribution, while the effect of isotopic impurities is important in determining the low temperature relaxation rate. On the other hand, the linewidth of v9 increases quadratically with T: Both decay processes of high order and pure dephasing may be responsible for such a behavior. This ambiguity cannot be overcome by our calculations, since these do not include the effect of high order (mainly quartic) anharmonic terms. Finally, the analysis of the decay mechanisms as predicted by the calculation shows that the anharmonic coefficients may differ from mode to mode: the relaxation mechanism is highly mode selective, and its efficiency depends greatly on the nature of the molecular normal coordinates involved in the process.

Calculation of dielectric susceptibility for complex ionic systems: Application to a predicted superlattice.

The Gordon-Kim theory of interionic interactions between closed-shell ions, which has had considerable success for existing alkali halide systems, has been applied to the determination of the structure, phonon dispersion, and room-temperature dielectric properties of a proposed superlattice, RbF(NaCl${)}_{2}$, of alkali halides. The dielectric properties of such a hypothetical material can have an unusually strong low-frequency dependence because of the presence of Brillouin-zone-folded ``infrared''-active modes. In evaluating the standard expressions for the dielectric properties, in the rigid-ion approximation, we make use of standard cubic and quartic anharmonic terms in addition to the instantaneous-phonon cubic anharmonic term, which is needed because of the lack of a center of inversion. Our procedure for numerically calculating the absorption for complex materials is described in detail. Effects of off-diagonal terms in the Green's function and of the long-range ionic Coulomb interactions are examined. In particular, an absorption peak at 30 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ is found to be especially sensitive to these contributions. This paper demonstrates (a) the theoretical existence of metastable phases of an alkali halide superlattice and (b) the application of anharmonic perturbation theory to determine the dielectric response in the infrared and millimeter-wave region for such complex materials. The particular structure studied is just one of many possibilities in this class of materials, which offers the opportunity for joint theoretical and experimental research to fabricate structures with properties tailored for specific applications.

Overtone and combination stretching vibrational bands of hydrogen peroxide, ethene, and propadiene

Symmetry allowed Coriolis and H22 operators have been constructed for the OH stretching vibrational states of hydrogen peroxide and for the CH (CD) stretching vibrational states of ethene and propadiene. The behavior of the coefficients of these terms has been investigated at a special local mode limit with no kinetic or potential energy coupling between the bond oscillators and with special constraints on the molecular geometry and the atomic masses. It is found that the Coriolis coefficients disappear but that the H22 coefficients remain important at this limit. For the stretching vibrational overtones and combinations the effects of quartic anharmonic resonance terms, Darling–Dennison resonance terms, on effective Coriolis and H22 constants are investigated. Finally the effects of the tunneling motion of H2O2 on the rotational energy level structure are discussed.

Rotational energy level structure of stretching vibrational states in some small symmetrical molecules

The vibrational dependence of various vibration–rotation parameters in XY2, XY3, and XY4 molecules with C2v, C3v, and Td point groups has been investigated between normal mode and local mode limits. The results at these two limiting cases differ drastically from each other. The rotational constants of the two lowest vibrational states in each overtone manifold (v=v1+v3=constant) are equal for v≥2 in the local mode limit. The effective Coriolis constants between these vibrational pairs disappear but the vibrationally off-diagonal H22 resonance terms remain important as the local mode limit is approached. The vibrational dependence of the rotational constants of the stretching vibrational states in the symmetrically substituted acetylenes and in normal silane has been analyzed. It is shown that it is necessary to couple levels by quartic anharmonic resonance terms in the normal coordinate space in order to explain the vibrational behavior of the rotational constants.

Thermodynamic properties of bcc crystals at high temperatures: The alkali metals

We have calculated the thermodynamic properties of monatomic bcc crystals at high temperatures from the Helmholtz free energy $F(V,T)$ for a second-neighbor, central-force model of the bcc lattice. $F(V,T)$ includes cubic and quartic anharmonic terms in perturbation theory evaluated in the high-temperature limit. The 25 Brillouin-zone sums that enter the calculation of $F(V,T)$ are expressed as functions of three parameters, ${\ensuremath{\kappa}}_{1},{\ensuremath{\kappa}}_{2}$, and ${\ensuremath{\kappa}}_{3}$, that depend on first and second derivatives of the pair potential $\ensuremath{\varphi}(r)$, thus, knowledge of $\ensuremath{\varphi}(r)$ and the Brillouin-zone sums completely determines the outcome of the calculation. Numerical results have been obtained for the alkali metals Li, Na, K, Rb, and Cs. The linear thermal expansion $\ensuremath{\epsilon}$ is well represented by this theory. The specific heat at constant volume, ${C}_{V}$, agrees with experiment to within 3% when allowance is made for the vacancy contribution. The results for the specific heat at constant pressure, ${C}_{P}$, are less satisfactory than for ${C}_{V}$ (they are within 6% of the experimental values) since they depend on the bulk moduli which, in turn, depend very sensitively on the method of treating the electrons. The free-electron theory is quite inadequate for the alkali metals. We have taken electron correlation into account and obtain results for the bulk moduli that are reasonable (within 10% of the experimental values in most cases, and within 25% for the worst case, Rb), considering the wide range in the experimental values. We discuss the limitations of this approach in dealing with systems where the electrons play a major role in stabilizing the structure of the crystal.

Thermal conductivity of an imperfect anharmonic crystal

The thermal conductivity of an anharmonic crystal containing randomly distributed substitutional defects due to impurity-phonon scattering is theoretically investigated with the use of the method of double-time thermal Green's functions and the Kubo formalism considering all the terms, i.e., diagonal, nondiagonal, cubic anharmonic, and imperfection terms in the energy-flux operator as propounded by Hardy. The study uses cubic, quartic anharmonic, and defect terms in the Hamiltonian. Mass changes as well as force-constant changes between impurity and host-lattice atoms are taken into account explicitly. It is shown that the total conductivity can be written as a sum of contributions, namely diagonal, nondiagonal, anharmonic, and imperfection contributions. For phonons of small halfwidth, the diagonal contribution has precisely the same form which is obtained from Boltzmann's transport equation for impurity scattering in the relaxation-time approximation. The present study shows that there is a finite contribution of the nondiagonal term, cubic anharmonic term, and the term due to lattice imperfections in the energy-flux operator to the thermal conductivity although the contribution is small compared with that from the diagonal part. We have also discussed the feasibility of numerical evaluation of the various contributions to the thermal conductivity.

Thermodynamic properties of fcc metals at high temperatures

We have carried out an exact and consistent calculation of the thermodynamic properties of monatomic fcc crystals at high temperatures. These properties are obtained from the Helmholtz free energy of the crystal $F(V,T)$ by means of the appropriate thermodynamic relations. It is crucial to the success of the calculation that we have been able to obtain the volume dependence of the free energy. $F(V,T)$ includes the static lattice energy and the vibrational contributions from the harmonic and lowest-order (cubic and quartic) anharmonic terms in perturbation theory evaluated in the high-temperature limit. The atoms interact via an effective nearest-neighbor central-force potential $\ensuremath{\varphi}(r)$. We have calculated the specific heat at constant volume and at constant pressure, the thermal expansion, the coefficient of linear expansion, the isothermal and adiabatic bulk moduli, and the Gr\uneisen parameter for the following fcc metals: Cu, Ag, Ca, Sr, Al, Pb, and Ni. Good agreement with experiment is obtained in all cases. We discuss the implications of these results for further studies of the properties of metals.

Single vibronic level fluorescence and relaxation spectra of s-tetrazine- d0 and d2 vapors

The third strongest cold band in the 5500Å absorption spectrum of s-tetrazine vapor is assigned to 6b 0 2 by analysis of single-vibronic-level (SVL) fluorescence spectra obtained with a tunable cw dye laser. Fermi resonance model calculations for the doublet 6a 0 1…6b 0 2 simulate the perturbations of relative intensities in ν 6a ″ fluorescence progressions which arise fron interference between the two terms of the Franck-Condon factors. Addition of a quartic anharmonic term to the quadratic ν 6b ′ potential accounts for the observed spacing of the observed 6b 1 and 6b 2 levels and suggests a likely candidate for the heretofore missing 6b 0 4 member of the Fermi resonance triad. Weak bands with in-plane polarization are revealed in fluorescence and contribute to a list of newly measured frequencies of the ground state. Experiments with added n-pentane show that rotational relaxation proceeds at the hard-sphere rate while vibrational relaxation is about five times slower. Vibrational relaxation exhibits non-statistical distribution in the early stages; it is rather distinctive when starting from the zero-level of the excited state, but is reminiscent of earlier work (on benzene) when starting from the 16b 1 level. Identification of the level 6b 1 is supported by these relaxation experiments.

Shear viscosity of isotopically disordered insulating solids

An expression is derived for the shear viscosity of an imperfect anharmonic Bravais crystal, containing randomly distributed impurities, using the correlation function formalism and double-time thermal Green's function technique. The effects of mass change and force constant changes between impurity atom and host lattice atoms as well as cubic- and quartic-anharmonic terms in the expansion of potential energy are taken into account explicitly. In the first approximation, it is shown that the viscosity tensor reduces to the relaxation time expression obtained by Rice.

Amplitudes of thermal vibrations in molecular solids

The mean-square nuclear displacements 〈 U 2〉 T have been evaluated for the molecular (rare-gas) solids by a lattice dynamical rigid-atom Model. The model derives the inter-molecular interaction from a “two-piece four parameter” pair-potential, includes the contribution of zero-point vibration through potential parameters by a self-consistent method, and accounts for the cubic and quartic anharmonic potential terms as perturbations to the harmonic Hamiltonian. The effect of many body interactions has been included on the basis of Axilrod-Teller approximation. The effect of including three-body forces as well as anharmonicity is found to decrease the values of 〈 U 2〉 T at all temperatures for all the solids. However, the ratio of the root mean-square nuclear displacements at melting to the nearest-neighbour distance, i.e. the Lindemann parameter (δ) is approximately the same for all the solids under study, which shows that the Lindemann parameter is structure as well as interaction dependent. The results are consistent with the other conventional calculations.

H.c.p.-f.c.c. phase transition in solid parahydrogen

The difference of free energies for the h.c.p. and f.c.c. parahydrogen is calculated as a function of molar volume. It is shown that the observed transition in solid parahydrogen 1 can be explained by differences in the structure of the cubic and quartic anharmonic terms.

Theory of surface force-constant changes in body-centered cubic lattices

A calculation has been made of force-constant changes at the (100) and (110) surfaces of body-centered cubic crystals. A model consisting of first, second and third-neighbor Lennard-Jones interactions together with harmonic angle-bending interactions is employed. The parameters characterizing the interactions are chosen to fit the equilibrium lattice spacing, the elastic constants, and certain phonon frequencies at high symmetry points in the Brillouin zone. The procedure consists of first calculating the static displacements produced by the creation of the surface and then calculating the change in force constants produced by the cubic and quartic anharmonic terms in the potential energy. Specific results have been obtained for chromium, iron, tungsten, and molybdenum.

Thermal Conductivity of an Anharmonic Crystal

An expression is obtained for the lattice thermal conductivity of an anharmonic crystal by the method of double-time Green's functions using the energy-flux operator propounded by Hardy. The study uses cubic and quartic anharmonic terms in the crystal Hamiltonian. It is found that the nondiagonal part of the energy-flux operator gives a finite contribution to the transport of energy, though its contribution is small compared to that from the diagonal part.

Far-Infrared Properties of Lattice Resonant Modes. V. Second-Order Stark Effect

Small electric-field-induced frequency shifts have been observed for resonant modes associated with three defect systems. For NaI: ${\mathrm{Cl}}^{\ensuremath{-}}$, the shifts have been used to measure the quartic anharmonic terms of the interionic potential of the impurity ion. For KBr: ${\mathrm{Li}}^{+}$, the quartic anharmonic terms are found to be very small, and an harmonic potential which includes a central barrier with the barrier height less than the zero-point energy of the oscillator is required to explain the experimental results. For NaCl: ${\mathrm{Cu}}^{+}$, only an on-center resonant-mode configuration is consistent with the experimental results.

Evidence for a central instability in KBr: Li +

The observed frequency shifts of the lattice resonant mode in KBr:Li + are not consistent with a model potential which includes quartic anharmonic terms, but these results can be explained by a potential with a central barrier with the barrier height less than the zero point energy of the oscillator.

Optical Excitation Spectrum of Interacting Polariton Waves in Molecular Crystals

In the optical region of frequencies, the excitation spectrum of interacting polariton waves in molecular crystals has been studied by means of the polariton Hamiltonian, which includes both cubic and quartic anharmonic terms with respect to the polariton operators. A general expression for Dyson's equation is developed which describes the system of interacting polariton fields and the polarization operator is a function of two-, three-, and mixed-polariton Green's functions. The polarization operator and, therefore, the polariton Green's function is then evaluated in successive approximations. In the zero approximation, the quartic anharmonic terms in the polariton Hamiltonian renormalize the frequencies of each polariton mode. The polarization operator is calculated in the lowest order via a zeroth-order renormalized Hamiltonian; the derived expression contains terms arising from the cubic and quartic anharmonicity and is a function of the renormalized frequencies of the polariton modes. In this approximation, the spectral function for the polariton spectrum is found to have a shape of an asymmetric Lorentzian line when the frequency dependence of the energy shift and spectral width is neglected. Resonances occur (i) when either two renormalized polaritons are created or one is created and the other is absorbed (two-polariton process), and (ii) when three renormalized polaritons are created or two are created and one is absorbed (three-polariton process) by a single incident renormalized polariton and vice versa. Formulas are derived for the probability amplitudes corresponding to the third- and fourth-order nonlinear scattering processes in the transparent range of frequencies of the crystal; an expression for the energy of excitation is developed where the frequency of the polariton mode is dressed by the field of all the others in the first approximation and propagates in the crystal without damping. A discussion is given on how the derived results can be improved by calculating the polarization operator in the next order in such a self-consistent manner as to become a function of the frequencies of the polariton modes, which are correct in the first approximation.

Thermodynamic Properties of Solid Ar, Kr, and Xe Based Upon a Short-Range Central Force and the Conventional Perturbation Expansion of the Partition Function

Thermodynamic properties of the rare-gas solids Ar, Kr, and Xe are calculated using phenomenological two-body central potentials. The cubic and quartic anharmonic terms in the crystal Hamiltonian have been included, using conventional perturbation theory. Although some definite discrepancies with experiment exist for $T<\frac{1}{3}{T}_{\mathrm{melting}}$ these simple models give a fair over-all account of both the volume and temperature dependence of the Helmholtz energy. However, for $T>\frac{1}{3}{T}_{\mathrm{melting}}$ (rms deviations greater than \ensuremath{\sim}6% of the nearest-neighbor distance), anharmonic contributions to the thermodynamic properties become unrealistically large, indicating, we believe, the breakdown of perturbation theory to the order considered in this work.

Isotopic differences in the 0 K volume and sublimation energies of solid neon and argon

Calculations are presented of the relative volume differences and sublimation energy differences of solid neon and argon isotopic lattices, assuming isotopic invariance of the intermolecular forces, which are chosen to be of the Lennard-Jones type. The calculations, performed at 0 °K on 20Ne and 22Ne, and 36Ar and 40Ar, make use of the anharmonic potential parameters presented by Brown in 1966. It is found that the relative volume and sublimation energy differences in solid neon and argon are well accounted for by the inclusion of cubic and quartic anharmonic terms at 0 °K. Comparison of our calculated sublimation energy differences with experiment is not possible owing to the lack of any measurements, and the only measured volume difference which is available is that of solid neon. It is suggested that the accuracy of the measurements has advanced to a point where non-adiabatic contributions may contribute slightly.