Collective motion from various aspects

Three methods to describe collective motion, the Random Phase Approximation (RPA), Wigner Function Moments (WFM), and Green’s Function (GF) methods, are compared in detail and their physical content analyzed in the example of a simple model, a harmonic oscillator with quadrupole-quadrupole residual interaction. It is shown that they give identical formulae for eigenfrequencies and transition probabilities of all collective excitations of the model. The exact relation between the RPA and WFM variables and the respective dynamical equations is established. The transformation of the RPA spectrum into one of WFM is explained. The very close connection of the WFM method with the GF method is demonstrated. A differential equation describing the current lines of RPA modes is established and the current lines of the scissors mode are analyzed as a superposition of rotational and irrotational components. The orthogonality of the spurious state to all physical states is proved rigorously. The text was submitted by the author in English.     

Wigner Function Moments versus RPA in a simple model

Two complementary methods to describe the collective motion, RPA and Wigner Function Moments (WFM) method, are compared using an example of a simple model—harmonic oscillator with quadrupole-quadrupole residual interaction. It is shown that they give identical formulas for eigenfrequencies and transition probabilities of all collective excitations of the model, including the scissors mode, which is a subject of our special attention. The exact relation between the variables of the two methods and the respective dynamical equations is established. The transformation of the RPA spectrum into the WFM one is explained. The text was submitted by the authors in English.     

The nuclear scissors mode by two approaches (Wigner function moments versus RPA)

Two complementary methods to describe the collective motion, the RPA and the method of Wigner function moments, are compared using a simple model as an example—a harmonic oscillator with quadrupole-quadrupole residual interaction. It is shown that they give identical formulas for eigenfrequencies and transition probabilities of all collective excitations of the model, including the scissors mode, which is a subject of our special attention. The normalization factor of the “synthetic” scissors state and its overlap with physical states are calculated analytically. The orthogonality of the spurious state to all physical states is proved rigorously.     

From conventional (neutron-proton) nuclear scissors to spin nuclear scissors

Investigations of the nuclear scissors mode in the frame of the Wigner Function Moments (WFM) method leading to the discovery of the new types of the nuclear collective motion are reviewed. It is demonstrated how the generalization of WFM method to take into account spin degrees of freedom allows one to reproduce all earlier described qualitative features of the conventional (neutron-proton) nuclear scissors (deformation dependence of the energy and transition probabilities, connection with isovector GQR implying the Fermi surface deformation, flows) and allows one to reveal a variety of new collective modes: isovector and isoscalar spin scissors, the relative motion of the orbital angular momentum and spin, isovector and isoscalar spin-vector GQR, spin-flip excitations.     

Low-collective scissors mode

Realistic microscopic RPA calculations for¹⁵⁶Gd with a deformed Woods-Saxon mean field, quadrupole-quadrupole, spin-spin and symmetry-restoring residual interactions show that the purely collective scissors mode of the two-rotor model is fragmented over orbital isovector 1⁺ states, lying at 2–7 MeV. The strongest experimentally observed magnetic dipole state is interpreted as performing a low-collective scissors-type of geometrical motion. This conclusion evolves from the identification of the above state with the strongest RPA excitation, which reproduces well the experimental energy,B(M1) value and (e, e′) form factor, has the largest overlap with the scissors state and can be represented as a low-collective scissors type vibration.     

Phase space moments with pair correlations on the example of nuclear scissors

The coupled dynamics of the isovector and isoscalar giant quadrupole resonances and low lying modes (including scissors) are studied with the help of the Wigner Function Moments (WFM) method generalized to take into account pair correlations. Equations of motion for relevant collective variables are derived on the basis of the Time Dependent Hartree-Fock-Bogoliubov (TDHFB) equations. Especial care is taken of the continuity equation. The inclusion of pair correlations leads also to the appearance of the isoscalar low lying mode.     

Spin distributions in the vibrational excitations of closed-shell nuclei

Self-consistent Hartree-Fock and RPA calculations with the Skyrme-type interaction SGII are used for a systematic investigation of the 1⁺ and the triplet 0^?, 1^?, 2^? states in ⁴⁰Ca and ²⁰⁸Pb. Response functions to spin-dependent multipole operators are calculated and the particle-hole structure of the spin-dependent collective states is studied. Collective spin-dependent 0^? and 1^? states above the giant dipole resonance as well as a collective spin-independent 2^? state (twist mode) are identified. Transition spin and current densities are calculated for the collective excitations and found to be useful for the study of these excitation modes.     

Thermodynamic properties of ferromagnets with uniaxial and biaxial anisotropy

The thermodynamic properties of spin-one Heisenberg ferromagnets with uniaxial and biaxial anisotropy are investigated in the molecular field approximation (MFA) and random phase approximation (RPA). In MFA a full phase diagram comprising three lines of bicritical points, is obtained.Using the standard-basis operator method, the collective excitation spectrum is studied in detail; moreover the softening of the excitations at the phase boundaries is discussed. A self-consistent version of RPA with emphasis on the role of kinematic restrictions with regard to the standard-basis operators is analyzed.Moreover, the phase transitions are considered in the Ising model, where a line of tricritical points occurs, and the planar model, where a line of bicritical points, two lines of tricritical points and a line of triple points, occur.     

The Nuclear Spin Scissors Mode—Theory and Experiment

A new type of nuclear collective motion—the spin scissors mode—was predicted seven years ago. Promising signs of its existence in ²³²Th were found. We perform a systematic analysis of experimental data on M1 excitations in rare earth nuclei to find traces of the spin scissors mode in this area. Obvious signs of its existence will be demonstrated. We propose new criteria to attribute the observed 1⁺ states to the scissors mode, entailing that the agreement of the experimental data with the results of our calculations and with the sum rules is improved substantially.     

Giant resonances on excited states

We derive modified RPA equations for small vibrations about excited states. The temperature dependence of collective excitations is examined. The formalism is applied to the ground state and the first excited state of ⁹⁰Zr in order to confirm a hypothesis which states that not only the ground state but every excited state of a nucleus has a giant resonance built upon it.     

Collective excitations in the unitary correlation operator method and relativistic QRPA studies of exotic nuclei

The collective excitation phenomena in atomic nuclei are studied in two different formulations of the random-phase approximation (RPA): (i) RPA based on correlated realistic nucleon-nucleon interactions constructed within the unitary correlation operator method (UCOM) and (ii) relativistic RPA derived from effective Lagrangians with density-dependent meson-exchange interactions. The former includes the dominant interaction-induced short-range central and tensor correlations by means of unitary transformation. It is shown that UCOM-RPA correlations induced by collective nuclear vibrations recover a part of the residual long-range correlations that are not explicitly included in the UCOM Hartree-Fock ground state. Both RPA models are employed in studies of the isoscalar giant monopole resonance in closed-shell nuclei across the nuclide chart, with an emphasis on the sensitivity of its properties on the constraints for the range of the UCOM correlation functions. Within the relativistic quasiparticle RPA (RQRPA) based on the relativistic Hartree-Bogolyubov model, the occurrence of pronounced low-lying dipole excitations is predicted in nuclei towards the proton drip line. From the analysis of the transition densities and the structure of the RQRPA amplitudes, it is shown that these states correspond to the proton pygmy dipole resonance. The text was submitted by the authors in English.     

Spin scissors mode and the fine structure of M1 states in nuclei

The coupled dynamics of low lying modes, including the scissors mode, and various giant quadrupole resonances are studied with the help of the Wigner Function Moments method generalized to take into account spin degrees of freedom. Equations of motion for collective variables are derived on the basis of Time Dependent Hartree-Fock equations in the harmonic oscillator model including spin-orbit potential plus quadrupole-quadrupole residual interaction. Introducing spin allows one to consider new types of nuclear collective motion where the nucleons with spin ‘up’ oscillate against nucleons with spin ‘down’.     

RPA calculations with Gaussian expansion method

The Gaussian expansion method (GEM) is applied to calculations of the nuclear excitations in the random-phase approximation (RPA). We adopt the mass-independent basis-set that is successful in the mean-field calculations. The RPA results obtained by the GEM are compared with those obtained by several other available methods in Ca isotopes, by using a density-dependent contact interaction along with the Woods–Saxon single-particle states. It is confirmed that energies, transition strengths and widths of their distribution are described by the GEM with good precision, for the 1⁻, 2⁺ and 3⁻ collective states. The GEM is then applied to the self-consistent RPA calculations with the finite-range Gogny D1S interaction. The spurious center-of-mass motion is well separated from the physical states in the E1 response, and the energy-weighted sum rules for the isoscalar transitions are fulfilled reasonably well. Properties of low-energy transitions in ⁶⁰Ca are investigated in some detail.     

Collective excitations and a backbending phenomenon in <Superscript>156</Superscript>Dy

We propose a self-consistent practical method to study collective excitations in rotating nuclei within the cranking + random phase approximation approach. It consists in solving the cranking Hartree-Bogolyubov equations for the modified Nilsson potential + monopole pairing forces. Further, the mean field results are used to construct collective vibrations treated in the random phase approximation (RPA). Special attention is paid to fulfill all conservation laws in the RPA to separate spurious and physical solutions. We demonstrate that the backbending in ¹⁵⁶Dy can be explained as a result of the disappearance of collective γ vibrations of the positive signature in the rotating frame.     

Temperature dependence of collective states in the random-phase approximation

We investigate the temperature dependence of collective states in the framework of the random-phase approximation at finite temperature. We show that sum rules can be extended to collective energies at finite temperature. Numerical methods are developed to solve the RPA equations at finite temperature. Results are presented and discussed in the case of ⁴⁰Ca for isovector dipole and isoscalar octupole vibrations, using oscillator wave functions and a zero-range force. We show that the broadening of giant dipole resonances observed experimentally, appears as a natural consequence of the structure of the RPA equations. Comparison is made with the schematic model for which the temperature dependence of collective states can be worked out analytically.