Phonon Operators Research Articles

New results for the fully renormalized proton–neutron quasiparticle random phase approximation

A many-body Hamiltonian describing a system of Z protons and N neutrons moving in spherical shell model mean field and interacting among themselves through proton–proton and neutron–neutron pairing and a dipole–dipole proton–neutron interaction of both particle–hole and particle–particle type, is treated within a fully renormalized (FR) pnQRPA approach. Two decoupling schemes are formulated. One of them decouples the equations of motion of particle total number conserving and non-conserving operators. One ends up with two very simple dispersion equations for phonon operators which are formally of Tamm–Dancoff types. For excitations in the ( N − 1 , Z + 1 ) system, Ikeda sum rule is fully satisfied provided the BCS equations are renormalized as well and therefore solved at a time with the FRpnQRPA equations. Next, one constructs two operators R 1 μ † , R 1 , − μ ( − ) 1 − μ which commutes with the particle total number conserving operators, A 1 μ † and A 1 , − μ ( − ) 1 − μ , and moreover could be renormalized so that they become bosons. Then, a phonon operator is built up as a linear combination of these four operators. The FRpnQRPA equations are written down for this complex phonon operator and the ISR is calculated analytically. This formalism allows for an unified description of the dipole excitations in four neighboring nuclei ( N − 1 , Z + 1 ) , ( N + 1 , Z − 1 ) , ( N − 1 , Z − 1 ) , ( N + 1 , Z + 1 ) . The phonon vacuum describes the ( N , Z ) system ground state.

Quasiparticle random phase approximation with a nonlinear phonon operator

We present model calculations of the quasiparticle random phase approximation (QRPA) with a new form of the phonon operator. This modification of the QRPA is applied to the proton-neutron Lipkin model and we shall review it briefly. The present calculations show that the inclusion of nonlinear terms in the phonon operator leads to a much better agreement with the exact results obtained by the diagonalization of the nuclear Hamiltonian. It is further found that if all relevant nonlinear terms of the phonon operator are taken into account, all odd excited states are also exactly reproduced.

RPA-like calculations within limited particle–hole spaces

Working in a boson formalism, we define a new class of phonon operators for RPA-like calculations. These operators remind the standard RPA ones, namely they have “upward” and “backward” amplitudes and an overall one-particle one-hole nature. However, their structure is more complex. As a basic feature, the vacuum of these phonon operators is characterized by only a limited (and variable at will) set of particle–hole excitations. In addition, multiphonon states constructed by repeated actions of these phonon operators on the vacuum state have a particle–hole structure never exceeding in complexity that of the vacuum itself. We discuss advantages and disadvantages deriving from the use of this new class of phonon operators and show an application of the formalism within the Lipkin model. Diagonalizations in multiphonon spaces built in terms of new and standard RPA phonon operators exhibit a faster convergence toward the exact results in the former case.

Testing a New Class of Phonon Operators for RPA-Like Calculations Within an Exactly Solvable Model

Working in a boson formalism, we introduce a phonon operator different from that commonly used in RPA calculations. As a major feature, this new phonon operator has a vacuum carrying only a limited number of particle-hole excitations. Therefore, the form of this vacuum is quite different from the exponential one of the RPA. In addition, multiphonon excitations have a particle-hole structure never exceeding in complexity that of this vacuum, no matter how large is the number of phonons involved. We present the results of some preliminary tests performed within the exactly solvable two-level Lipkin model.

Ground state of double beta decaying nuclei

Recent results concerning the ground state of proton neutron systems are reviewed. A schematic Hamiltonian is treated by a time dependent formalism. New harmonic motions and static properties are pointed out. Applications to single beta and 2νββ decays are given. Also the isospin symmetry restoration is discussed. It is shown that inclusion of scattering terms ill the expression of the pnQRPA phonon operator enables to account for the isospin wobbling motion. Predictions obtained in the new approaches are compared with the corresponding exact results.

Phonon damping model using random-phase-approximation operators

A fully microscopic interpretation of the phonon damping model ~PDM! is proposed. The phonon operator, which generates the collective excitation such as giant dipole resonance ~GDR!, is constructed from coherent particle-hole pairs. The phonon energy is determined as the solution of dispersion equation within the randomphase approximation ~RPA!. The phonon structure is found in term of RPA X and Y amplitudes. The formalism is illustrated by numerical calculations within a schematic model. The results of calculations demonstrate the same feature of GDR as a function of temperature obtained previously within PDM-1, which employs structureless phonons.

Limitations of the number self-consistent random phase approximation

The Quasiparticle Random Phase Approximation equations are solved taking into account the Pauli Principle at the expectation value level, and allowing changes in the mean field occupation numbers to minimize the energy while having the correct number of particles in the correlated vacuum. The study of Fermi pn excitations in $^{76}$Ge using a realistic Hilbert space shows that the pairing energy gaps in the modified mean field are diminished up to one half of the experimental value when strong proton-neutron correlations are present. Additionally, the Ikeda sum rule for Fermi transitions is violated due to the lack of scattering terms in the phonon operators. These results call for a critical revision of the double beta decay half-lives estimated using the QRPA extensions when standard QRPA calculations collapse.

Quantum theory of the vibronic solid-state laser

A quantum-mechanical description of a vibronic multimode standing-wave transition-metal ion laser is presented. The impurity ion levels are coupled both by the radiation field and by the phonons of the host lattice. A dynamic Heisenberg–Langevin set of equations for the material system and the photon and phonon operators, including radiative and nonradiative damping terms and quantum-stochastic forces, has been derived. The numerical solutions of those equations show the crucial role of the phonons in the laser dynamics. The competition between the photons and the phonons leads to regular stable self-pulsation for certain sets of parameters.

β- and double-β-decay transitions in a schematic model

Single- and double-$\ensuremath{\beta}$ decay transitions of Fermi type are discussed within a schematic model. The amplitudes associated with these transitions as well as some basic properties of the model like ground state and excitation energies are reproduced quite well in terms of a multistep variational procedure. The procedure, fully developed in a boson space, searches for the best ground state wave function through a series of minimizations. Excited states are constructed in terms of a phonon operator acting on the ground state so derived and their structure is also determined via a variational mechanism. Results obtained within the quasiparticle random phase approximation (QRPA) are shown for reference and also the comparison with other approaches like the renormalized QRPA is discussed. A considerable improvement is obtained within the present approach.

Self-consistent determination of the one-body density matrix and particle-hole excitations

We develop a procedure recently proposed for the treatment of correlations in finite Fermi systems in a more consistent way than the random phase approximation (RPA). The one-body density matrix in the correlated ground state is now no more assumed to be diagonal in the Hartree-Fock basis. An expression is rather provided where this quantity is expressed in terms of the X and Y amplitudes of the particle-hole phonon operators. A set of RPA-like equations depending only on the one-body density matrix is then constructed by means of the method of linearized equations of motion and solved via an iterative procedure. The procedure is employed to calculate spectra and strength distributions of several neutral and charged Na and Li metal clusters within the jellium approximation. Improvements are observed with respect to standard RPA calculations.

Extension of the renormalized quasiparticle random phase approximation

The quasiparticle random phase approximation is extended in order to restore of the Pauli principle beyond the renormalized approach by treating the so-called scattering terms in the QRPA phonon operators. It has been shown that this new framework can be described in a case of a single nuclear shell occupied by both protons and neutrons in terms of the QRPA(14,3) algebra. An application of the formalism to the double beta decay of calcium48Ca is discussed to some extent.

Sum-rule analysis of the renormalized quasiparticle random-phase approximation

The renormalized quasiparticle random-phase approximation (RQRPA) is analyzed by the use of sum-rule techniques in realistic model calculations. It is found that the RQRPA does not satisfy the Gamow-TellerS − −S +=3(N − Z) sum rule and that the violation mostly comes from theS − part of it. The violation also seems to be mass-dependent increasing towards lighter masses. At the same time also the double Gamow-Teller sum rule is seen to be violated. Possible restoration of the sum rule is seen in the inclusion of the scattering terms in the definition of the RQRPA phonon operators.

Excitonic polaron in the molecular crystal model: Nonlocal dynamic coherent-potential approximation

A nonlocal dynamic coherent-potential approximation is formulated as a further development of the dynamic coherent-potential method. The nonlocal dynamic coherent-potential approximation is an efficient method of determining the one-exciton Green’s function in a model with the Hamiltonian in the strong-coupling approximation, where a spectrum of optical phonons is assumed, and the exciton-phonon interaction operator is linear or quadratic in the phonon operators. A system of recursion equations is derived, from which the coherent potential is found as a function of the energy E and the wave vector k. An analytical expression is derived for the one-exciton Green’s function in the case of narrow (in comparison with the phonon energy) exciton bands and exciton-phonon interaction linear in the phonon operators. For broader exciton bands and more complex exciton-phonon interaction the system of equations determining the coherent potential represents a recursion algorithm, which can be effectively implemented by numerical means.

Relaxation jumps of strong vibration.

A nonperturbative theory of two-phonon relaxation of a strongly excited local mode is developed, whereby the mode is considered classically and phonons, quantum mechanically. The relaxation mechanism is analogous to that of black hole emission proposed by Hawking: it consists of linear transformation of phonon operators in time, which is induced by the classical subsystem (mode). Due to this transformation the number of phonons increases while the energy of the local mode decreases in time. It is found that there exist some critical energies of the local mode when relaxation rate is strongly enhanced. Thereby relaxation jumps take place, which are accompanied by generation of bursts of phonons. The values of the critical energies do not depend on temperature. The final stage of relaxation is exponential, in agreement with perturbation theory. Numerical calculations of the relaxation jumps are presented.

A quantum field theory of localized modes in amorphous solids with nonlinear potentials at finite temperatures

Using the Ward-Takahashi relation, we establish the effective interaction Hamiltonian and the Heisenberg equation for phonon operators in amorphous solids with nonlinear potentials at finite temperatures. Applying the one loop approximation to the nonlinear potential in the Heisenberg equation, we obtain a dynamical equation with a self-consistent potential created by a spatially localized mode. Under the assumption that the mode is well-localized at a lattice site, we solve the dynamical equation. The localized modes show that when the atom at the lattice site oscillates, the surrounding atoms oscillate in opposite phase due to the conservation of the center of mass.

The Heisenberg equation for phonon operators and soliton solutions in nonlinear lattices

The Heisenberg equation for phonon operators in nonlinear lattices is derived establishing the interaction Hamiltonian included higher powers of particle-hole pairs in nonlinear lattices. A phonon operator consists of a particle-hole pair in the harmonic potential approximation in the two band model; it represents an up or down transition of atoms between two levels. Applying the boson transformation method to the Heisenberg equation for phonon operators, we obtain the classical dynamical equation and a linear equation with the self-consistent potential created by the extended objects in nonlinear lattices. The boson transformation leads to soliton solutions in the long wavelength limit. The linear equation can be used to obtain scattering states, bound states and translational modes for phonons.

A quantum field theory of localized and resonant modes in 3D nonlinear lattices at finite temperatures I

The effective interaction Hamiltonian in 3D nonlinear lattices is established taking into account the repetitions of the up and down transition of an atom between two levels at the same site. The effective interaction Hamiltonian leads to the Heisenberg equation for phonon operators, which yields the conventional dynamical equation for displacements of atoms in 3D nonlinear lattices in the tree approximation by the boson transformation method. Making the one-loop approximation to the nonlinear potential in the Heisenberg equation, we obtain a dynamical equation with a self-consistent potential created by a localized or a resonant mode. In paper II, we show that the dynamical equation yields solutions for localized and resonant modes at finite temperatures.

A quantum field theory of resonant modes in 3-d nonlinear lattices at finite temperatures

We establish the Heisenberg equation for phonon operators in 3-d nonlinear lattices at finite temperatures. Applying the one loop approximation to the nonlinear potential in the Heisenberg equation, we obtain a dynamical equation with a self-consistent potential created by a spatially localized mode. Under the assumption that the mode is well-localized at a lattice site, we obtain an eigenvalue equation with the frequency dependent couplings corresponding to that for the force constant defects in linear lattices. However, the frequency dependent couplings allow only frequencies near the middle of the phonon bands, while the constant couplings allow low frequencies.

On the low energy limits of inelastic molecule–surface scattering

The zero energy scattering limit of inelastic molecule–surface scattering is studied within the context of a multiphonon expansion of the molecule–bath wave function. By assuming that at low scattering energies the expansion may be truncated at first order in the phonon operators, we derived a closed form solution to the Lippmann Schwinger equation for the scattering wave function which includes a nonlocal and energy dependent self-energy term which correctly incorporates virtual phonon transitions in the elastic channel. The closure relation results from the use of a discrete spectral (ℒ2) form of the inelastic channel Greens functions. We compute the zero energy limit of these wave functions and discuss the trapping and reflection of cold atoms from ultracold surfaces. Our results indicate that for realistic atom surface interactions the low energy limit of the sticking coefficient, s, can deviate markedly from the expected s∝E1/2 behavior and is shown to approach a constant nonzero limiting value. This trend is consistent with recent experimental work involving the sticking of spin polarized hydrogen atoms on liquid He films.

A quantum field theory of localized and resonant modes in a 1-d nonlinear lattice

The Heisenberg equation for phonon operators in a 1-d nonlinear lattice with a quartic potential is solved in the one-loop approximation. We obtain a stationary localized mode and a resonant mode. A localized mode refers to a positive quartic potential and locates above the top of the phonon band. A resonant mode refers to a negative quartic potential and locates within the band. Both of these modes are symmetric.