Two-phonon States Research Articles

Electron-phonon interaction in a strongly correlated Hubbard system

The electron-phonon (local) interactions have been considered in a single-band Hubbard model with strong on-site correlation. It has been shown that when no holes are present (i.e., one electron per site) the ground state of the system corresponds to the conventional coherent state of the phonon subsystem and the polaron has high effective mass, wheras for non-zero hole concentration the two-phonon coherent state of the phonon subsystem corresponds to the ground state of the system and the effective mass of the resulting squeezed polaron is reduced. If the superconductivity is due to Bose condensation of bipolarons the effective mass of the bisqueeps (squeezed bipolarons) at appropriate hole concentration may be reduced by 100 times or more in comparison to the conventional bipolarons and the corresponding Bose condensation temperature would be high.

Pressure tuning of Fermi resonance in crystal CO2

A solid state theory on pressure tuning of Fermi resonance in crystals is presented and applied to the CO2 case. The theory relies on consideration of delocalized ω2+ω2 two-phonon states anharmonically coupled to ω1 and it is developed along the formalism of the Green function technique. This gives a straightforward method to split bound states (or biphonons) out of the continuum. Explicit expressions for their energy separation and intensity are obtained. Calculations are performed on the pressure dependence of the main parameters governing the Fermi resonance and discussed in terms of the crystal approach. In contrast with the more limited molecular treatment, W is seen to decrease with pressure. The effect is specifically related to the spread of the two-phonon continuum over a finite energy interval. Other quantities, for instance the intensity ratio between bound states, do not suffer such a big discrepancy between the two models. Limits of applicability of our approach are also sketched.

Particle-hole and vibrational states in doubly closed subshell 96Zr from in-beam inelastic neutron and proton scattering

Through in-beam (n, n′ γ) and (p, p′ γ) reaction studies, we have identified 33 levels in 96Zr and, from (n, n′) cross section and γ-ray angular distribution data, we have made spin-parity assignments for twenty of these levels. Recent DSAM nuclear level lifetimes were combined with electronic timing measurements in interpreting these states. The 1750 keV 2 + and 2439 keV 3 + states are suggested as the lowest neutron particle-hole multiplet, of ν(3 s 1 2 , 2 d 5 2 −) type, expected from the double closed subshell picture. On the other hand, the band built on the 1582 keV 0 + shape isomer is determined to have vibrational collectivity up to the two-phonon level. Candidates for two-phonon vibrational states of quadrupole-octupole and double octupole type, associated with the ground-state structure exhibiting octupole softness, are proposed.

Description of low-lying vibrationalK ??0+ states of deformed nuclei in the quasiparticle-phonon nuclear model

The QPNM equations are derived taking account of p-h and p-p interactions. The calculated quadrupole, octupole and hexadecapole vibrational states in168Er,172Yb and178Hf are found to be in reasonable agreement with experimental data. It is shown that distribution of theEλ strength in some deformed nuclei differs from the standard one. There are cases when for a givenK π theEλ strength is concentrated not on the first but on higher-lying states. The assertion made earlier about the absence of collective two-phonon states in deformed nuclei is confirmed.

Effect of quantum fluctuations on the Peierls dimerization in the one-dimensional molecular-crystal model.

The effect of quantum lattice fluctuations on the Peierls dimerization is investigated in a half-filled one-dimensional molecular-crystal model. The nonadiabatic effects due to finite phonon frequency \ensuremath{\omega} are treated through a variational polaron wave function: A part of the lattice distortion is carried over by the electrons and interferes with the static (frozen) lattice deformation. This leads to a weakening of the effective (adiabatic) potential stabilizing the dimerized state. Our results in the \ensuremath{\omega}=0 (adiabatic) and \ensuremath{\omega}=\ensuremath{\infty} (nonadiabatic) limits are in agreement with previous results. Moreover, in the finite-\ensuremath{\omega} case we show that the lattice fluctuations are ``squeezed'' by the electron band motion, and the phonon subsystem is in the so-called two-phonon coherent state. The finite-\ensuremath{\omega} results are in good agreement with both Monte Carlo simulations and renormalization-group analysis: For the spinless case an order-disorder transition is found on increasing the phonon frequency. On the contrary, the system for spin-1/2 electrons is always ordered, but the order parameter is slightly reduced when \ensuremath{\omega} increases. The competition with singlet superconductive ordering is also examined. For \ensuremath{\omega}=\ensuremath{\infty} the system is found to be on the borderline between charge-density wave (CDW) and superconducting, while for finite-\ensuremath{\omega} CDW ordering prevails. Further extensions of this work are indicated.

Variational treatment of the strong electron-phonon interaction

Via discussions of the simple model of a strong electron-phonon interacting system, the authors develop a new variational treatment for dealing with the superconducting properties of such a system. The key idea of their treatment is to introduce a two-phonon coherent state for the phonon subsystem so that the ground-state energy of the interacting system can arrive at a stable minimum. They have shown that the superconducting condensation energy in their variational state is of a reasonable magnitude and that the effective mass of the polarons is much smaller than that considered by previous researchers. The gap equation and the expression for critical temperature Tc in the theory are also different from those in ordinary polaronic state as well as from those in Bardeen-Cooper-Schrieffer theory. The authors have pointed out the possibility of high-Tc superconductivity in such an interacting system.

Temperature-dependent (10-240 K) relaxation of vibrational excitons in KClO 4 crystal

The relaxation times of vibrational excitons in KClO 4 have been measured between 10 and 240 K by picosecond time-resolved CARS spectroscopy. The low temperature data are consistent with energy decay mechanisms based on three-phonon scattering processes. At higher temperature the linewidth of the symmetric stretching ν 1 has a well defined linear dependence on T, showing that three-phonon scattering is still the main relaxation mechanism. In contrast, the ν 2 bending mode has a marked T 2 dependence, showing that four-phonon and higher-order depopulation and dephasing mechanisms control the overall relaxation rate. Finally, the strong temperature dependence of the linewidth of the 2ν 2 overtone is interpreted in terms of the relaxation of an undispersed, quasi-free two-phonon state.

Variational ground state of a system with strong electron-phonon interaction.

A new variational ground state of a system with strong electron-phonon interaction has been proposed in which the electron subsystem is in a superconducting pairing state and the phonon subsystem is in a two-phonon coherent state. We have shown that such a variational state is necessary for obtaining reasonable magnitudes for the superconducting condensation energy and for the effective mass of the charge carriers.

Three-state mixing among 0+ states in even Cd isotopes.

Ratios of (t,p) transfer intensities to ${0}^{+}$ states in the cadmium isotopes have been analyzed using a three-state mixing model. The data allows determination of the degree of mixing of the rotational (intruder) state into the zero-phonon and two-phonon ${0}^{+}$ states. The three lowest ${0}^{+}$ states are found to be relatively pure and the intruder state lies above the two-phonon state in the heavier isotopes.

Vibrational circular dichroism and vibrational optical rotatory dispersion in molecular crystals. IV. Overtones of degenerate vibrations

Two-quanta anharmonic spectra, namely overtones and combination bands with degenerate vibrations, in uniaxial gyrotropic crystals from point group of symmetry C3, D3, C4, D4, are investigated, with due account taken of the time inversion. The symmetry of two-phonon states is studied. A general model of the anharmonicity and the influence of the crystal chiral structure on intermolecular exchange of the circularly polarized degenerate vibrations is elaborated. The contribution of overtones and of the combination bands to dielectric permittivity tensor, and to the gyration tensor is studied too. The paper also deals with the IR and vibrational circular dichroism (VCD) spectra near the frequencies of the unbound and of the bound two-phonon states as well as the general features of two-phonon circularly polarized states appearing in the linear and in nonlinear optical processes in gyrotropic crystals.

Squeezed polarons in one dimension

Starting from a simple one-dimensional model for many-electron and phonon interacting systems we have shown that in order to arrive at a stable minimum of the total energy the ordinary polaronic state must be replaced by the squeezed polaronic state in which the phonon subsystem is in the two-phonon coherent state.

New type of Cooper pairing in systems with strong electron-phonon interaction.

We have investigated the ground-state properties of a model system with a strong electron-phonon interaction. It is found that as long as the renormalized intersite electron correlation is attractive, a new type of Cooper pairing will occur whether the renormalized on-site electron correlation is repulsive or attractive. Furthermore, we have shown that a new phonon coherent state, named a two-phonon coherent state, must be introduced to serve as the ground state of the phonon subsystem, so that the ground-state energy of the interacting system can be made a stable minimum and an appropriate condensation energy of the superconducting ground state may be obtained. We have performed numerical calculations for the n=1 case.

Phonon-induced hybridisation and its effects on valence transition phenomena

The electron-phonon interactions in valence fluctuation systems have been reconsidered and a new procedure has been developed to transform the author's model Hamiltonian, which is a periodic Anderson model with indirect hybridisation, by a unitary transformation of the displacement operator type. Then, a variational ground state of the phonons subsystem has been constructed; thus the electron and the phonon subsystems could be separated approximately and an effective Hamiltonian for the electron subsystem could be obtained. Finally, a functional integration approach has been used to investigate the ground-state properties of the effective Hamiltonian, and the ground-state phase diagrams have been given. By requiring that the total ground-state energy of the model system should be a minimum, he has found that the ground state of the phonon subsystem must be a two-phonon coherent state which is different from any eigenstate of phonon number operators. Also, the author has shown that through the electron-phonon interaction the f-f Hubbard U-term could contribute to the d-f hybridisation and he considers that this contribution might be used to explain why most of valence fluctuation systems have a strongly correlated f band but are not in any magnetic state.

Variational ground state for the periodic Anderson model with an indirect hybridization.

As the direct on-site hybridization is forbidden by inversion symmetry in most of the mixed-valence compounds, an indirect on-site hybridization mediated by phonons has been introduced into the periodic Anderson model to constitute our model system. Then we try to construct a variational ground state for the model Hamiltonian by the following steps. First, we develop a new procedure to transform the model Hamiltonian by a unitary transformation of the displacement-operator type. Second, a two-phonon coherent state is taken as the trial-state vector for the ground state of the phonon subsystem and the parameters of the two-phonon coherent state, which are regarded as the adjustable parameters of the variational treatment, remain to be determined. Third, a Bogoliubov transformation is introduced to deal with the electron subsystem; the ground state and low-lying excited states are constructed directly. Finally, the parameters of the two-phonon coherent state are adjusted to ensure that the energy functional of our variational ground state is a stable minimum. Numerical calculations have been done and a nonzero energy gap and fluctuating valence have been obtained in various cases. Our results could be used to explain the small energy gap and valence-fluctuation phenomena observed in some Sm-based compounds.

High-pressure Raman study of the vibrational modes in AlPO4 and SiO2 ( alpha -quartz).

The pressure dependence of the optical-phonon frequencies in berlinite (${\mathrm{AlPO}}_{4}$) and \ensuremath{\alpha}-quartz has been investigated by high-pressure Raman scattering experiments, using a diamond cell; berlinite was studied up to 12 GPa and \ensuremath{\alpha}-quartz to 20 GPa. The ${A}_{1}$ soft phonon mode at 216 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ in ${\mathrm{AlPO}}_{4}$ and the corresponding mode in \ensuremath{\alpha}-quartz at 206 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ exhibit similar but anomalously large and nonlinear pressure dependences. Further, these two modes exhibit striking decreases in their widths with increasing pressure. These changes appear to be connected with the decoupling of the strong anharmonic interactions of the ${A}_{1}$ soft mode phonon with the two-phonon (acoustic) states in \ensuremath{\alpha}-quartz and with one-phonon (optical) and two-phonon (acoustic) states in ${\mathrm{AlPO}}_{4}$. Pressure experiments help in distinguishing the relative strength of these interactions in the two materials. The mode-Gr\"uneisen parameters have been obtained and, again, the ${A}_{1}$ soft mode has a large ${\ensuremath{\gamma}}_{i}$, namely, 3.45 in ${\mathrm{AlPO}}_{4}$ and 4.32 in \ensuremath{\alpha}-quartz. Using refractive-index matching with \ensuremath{\alpha}-quartz and ${\mathrm{AlPO}}_{4}$, the n(P) of a 4:1 methanol-ethanol mixture has been evaluated.

Scattering of 25.6 MeV protons on 94Mo, 96Mo and 100Mo

Elastic and inelastic scattering cross sections of 25.6 MeV protons on the isotopes 94,96,100Mo were measured. Level energies and J π values were deduced using DWBA and coupled channels analysis. CC calculations on vibrational quadrupole one-phonon and two-phonon states prove to fit the experimental cross sections well in the anharmonic approach to first and second order. Quadrupole-octupole two-phonon states could be identified by harmonic vibrational model CC calculations. A semi-microscopic model was applied to some of the collective states, resulting in a satisfactory description of the cross sections.

Study of the shift of two-phonon poles in deformed nuclei by the method of boson expansions

The situation with regard to two-phonon collective states in even-even deformed nuclei is obscure. According to the Bohr-Mottelson model deformed nuclei must have two-phonon collective states; this was recently pointed out once more in [i]. According to the interacting boson model the low-lying states must include states with large two-boson components [2]. In the framework of the quasiparticle-phonon nuclear model [3] the study of the twophonon states showed [4] that allowance for the Pauli principle, i.e., allowance for the exact commutation relations between the phonons, leads to a shift of the two-phonon poles to higher energies. In [5], the corresponding secular equations were solved. Allowance for the Pauli principle leads to a shift of the centroids of the energy of the collective two-phonon states by 1-2 MeV. At excitation energies of 3-4 MeV the strength of the collective two-phonon states must be distributed over many nuclear levels. It was concluded on this basis in [5] that in even-even deformed nuclei there are no collective twophonon states. In the description of two-phonon collective states in deformed nuclei there is a contradiction between the quasiparticle-phonon model and the Bohr-Mottelson and interacting boson models. As is shown in [6], there is a contradiction between the quasiparticle-phonon model and the interacting boson model in the description of a number of other nonrotational states in deformed nuclei. We note that according to [7] allowance for the Pauli principle in even-even spherical nuclei does not lead to a large shift in the energies of the two-phonon states, in agreement with the experimental data on the twophonon collective states in spherical nuclei. An analysis of experimental data led to the conclusion in [8] that there are no twophonon collective states in deformed nuclei. New experimental data on 1~SEr in [9,10] showed that the levels previously treated as two-phonon levels have a dominant one-phonon component. The absence of two-phonon quadrupole states with energy less than 2 MeV in 168Er is explained in [i,Ii] by the large anharmonicity of the $ and ~ vibrations due to the shape of the nucleus in the form of a triaxial ellipsoid. The absence of two-phonon octupole states in a number of Ra, Th, and U isotopes [12] is explained in many studies by the existence of a stable octupole deformation. We note that the new experimental data on 168Er [9,13] agree well with calculations in the quasiparticle-phonon model and with the calculations in [14] of the structure of the single-phonon states and do not agree with the calculations in the interacting boson model. In view of the importance of clarifying the situation with regard to two-phonon collective states of even-even deformed nuclei, it is necessary to study the behavior of the two-phonon states by a different mathematical formalism. Since allowance for the Pauli principle in the two-phonon components of the wave functions plays a decisive role, it is expedient to make the calculations with phonons constructed from the operators of "true" bosons. It is necessary to establish whether there is a large shift of the twophonon poles in the problem when formulated in this way. The present paper is devoted to

Electromagnetic properties of low-spin states in102, 104Pd

Electromagnetic transitions from low-lying 0+, 2+, 4+ and 3− states in102, 104Pd have been studied using (p, 2n) and (p, p′) reactions and Coulomb excitation. TheE2 transition probabilities from the 0 +3 state in102Pd (13 W.u.) and from the 02/+ state in104Pd (14 W.u.) are somewhat low for two-phonon states. Generally, theE2 transition rates are reasonably well reproduced by the IBA-2 and by the boson-expansion description. The intruding 0 +2 state (t1/2=14.3 ns) in102Pd is connected to the 22/+ and 2 +3 states via strong E2 transitions: B(E2; 0 +2 → 2 +2 )=96±40 W.u.; B(E2; 2 +3 → 0 +2 =17±11 W.u.

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.

Calculation of anharmonic phonon couplings in C, Si, and Ge.

Frozen-phonon total-energy calculations are used to extract anharmonic phonon couplings for the tetrahedral elements C, Si, and Ge. The local-density approximation is employed, with a localized-orbital basis used for C and a plane-wave expansion used for Si and Ge. The bare interactions between optical phonons are completely determined through fourth order at the Brillouin-zone center. These are used to compute renormalized couplings, in which a vertex is screened by virtual phonons. The renormalized couplings are found to have the wrong sign to allow formation of a proposed two-phonon bound state in diamond.