The nuclear scissors mode from various aspects

Three methods to describe collective motion, random phase approximation (RPA), Wigner function moments (WFM) and the Green’s Function (GF) method are compared in detail and their physical content analyzed on an example of a simple model, the 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, including the scissors mode, which is the subject of our special attention. The exact relation between the RPA and WFM variables and the respective dynamical equations is established. The transformation of the RPA spectrum into the one of WFM is explained. The very close connection of the WFM method with the GF one is demonstrated. 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. A differential equation describing the current lines of RPA modes is established and the current lines of the scissors mode analyzed as a superposition of rotational and irrotational components.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Plasma oscillations in a one-dimensional many-fermion system

Following the Shevchik technique, a model Hamiltonian of collective oscillations (plasmons) in a one-dimensional system of complete degenerate fermions is obtained in terms of the Tomonaga boson operators. This Hamiltonian is diagonalized by means of the Mattis and Lieb canonical transformation and the plasma frequency is derived. The equation-of-motion method is applied in the RPA in order to include the coupling between the collective and individual degrees of freedom. The generalization to finite temperatures is performed and connection with the Tomonaga model is discussed.     

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.     

Langevin dynamics of the glass forming polymer melt: Fluctuations around the random phase approximation

In this paper the Martin-Siggia-Rose (MSR) functional integral representation is used for the study of the Langevin dynamics of a polymer melt in terms of collective variables: mass density and response field density. The resulting generating functional (GF) takes into account fluctuations around the random phase approximation (RPA) up to an arbitrary order. The set of equations for the correlation and response functions is derived. It is generally shown that for cases whenever the fluctuation-dissipation theorem (FDT) holds we arrive at equations similar to those derived by Mori-Zwanzig. The case when FDT in the glassy phase is violated is also qualitatively considered and it is shown that this results in a smearing out of the ideal glass transition. The memory kernel is specified for the ideal glass transition as a sum of all “water-melon” diagrams. For the Gaussian chain model the explicit expression for the memory kernel was obtained and discussed in a qualitative link to the mode-coupling equation. Received: 9 January 1998 / Revised: 24 April 1998 / Accepted: 2 July 1998     

An extended Lipkin-type model with residual proton-neutron interaction

The Lipkin-Meshkov-Glick (LMG) model is extended to explicitly take into account the proton and neutron degrees of freedom. The proton and neutron Hamiltonians are taken to be of the LMG form, and, in addition, a residual proton-neutron interaction is include. Exact solutions in an SU(2) ⊗ SU(2) basis as well as the RPA solutions for the energy spectrum of the model Hamiltonian are obtained. The spectrum of the exact solutions is degenerate in the limit of no proton-neutron residual interaction, but this degeneracy is totally removed when this type of residual interaction is turned on. The spectrum obtained with RPA is compressed or expanded, as compared to the LMG model with the same total number of nucleons (N), depending of the relative magnitudes of the like- and unlike-particle residual interactions. Furthermore, the behavior of the first collective excited state is studied as function of the interaction parameters of the model using the exact and RPA methods. At a given N, it was found that the RPA result is closer to the exact one when the like- and unlike-particle residual interactions are both taken into account in the model Hamiltonian, as compared to the case when residual interaction of only one type (e.g. either like- or unlike-particle type) is involved.     

空间目标可见光散射特性建模方法研究

针对空间目标的可见光散射特性提出一种建模方法.在分析空间目标所处的背景辐射环境基础上建立了空间目标背景辐射物理模型.对目标表面进行面元划分后,基于辐射理论引入双向反射分布函数模型来描述目标表面面元的光散射特性,将目标各个表面所有面元散射分量叠加建立了目标可见光散射特性的数学模型.建立目标本体坐标系,通过坐标变换确定目标、背景辐射源与探测器的相对位置关系,利用矢量坐标法确定目标对观测系统的“可视表面”.根据给定的目标几何结构尺寸和物性参量仿真获得了目标在轨光学特性.计算结果验证了建模的有效性.     

Electron-hole excitations in NiO: LSDA+U-based calculations vs. inelastic X-ray scattering and ellipsometry measurements

The performance of the LSDA+U functional—in particular, the quality of the ground-state—is tested via calculations of the electron-hole excitations of NiO, which are compared with (non-resonant) inelastic X-ray scattering (IXS) and ellipsometry measurements. The dynamical density-response calculations are performed within the random-phase approximation (RPA), defining an LSDA+U/RPA density-response method. A significant success of this method is the insight it provides into the main loss present in the IXS data above the NiO optical gap, namely, a peak lying at ∼7.5 eV. This excitation, which is shown to be collective in nature, and to be induced by e_g-e_g transitions, provides a direct link between the correlated e_g-states and the IXS data. This finding illustrates the power of IXS, combined with correlated-band-structure theory (here, LSDA+U theory), for the investigation of the electronic structure of strongly correlated materials. At the same time, our results indicate that the LSDA+U/RPA response method does not represent a complete theory.     

Faddeev random phase approximation applied to molecules

The Faddeev Random Phase Approximation (FRPA) is a Green’s function method which couples collective degrees of freedom to the single particle motion by resumming an infinite number of Feynman diagrams. The Faddeev technique is applied to describe the two-particle-one-hole (2p1h) and two-hole-one-particle (2h1p) Green’s function in terms of non-interacting propagators and kernels for the particle-particle (pp) and particle-hole (ph) interactions. This results in an equal treatment of the intermediary pp and ph channels. In FRPA both the pp and ph phonons are calculated on the random phase approximation (RPA) level. In this work the equations that lead to the FRPA eigenvalue problem are derived. The method is then applied to atoms, small molecules and the Hubbard model, for which the ground state energy and the ionization energies are calculated. Special attention is directed to the RPA instability in the dissociation limit of diatomic molecules and in the Hubbard model. Several solutions are proposed to overcome this problem.