Large Amplitude Collective Motion Research Articles

Equations of Collective Submanifold for Large Amplitude Collective Motion and Its Coupling with Intrinsic Degrees of Freedom. III

From the viewpoint defferent from that of Part I, the time-dependent Harteree-Fock theory is formulated for describing large amplitude collective motion and its coupling with intrinsic degrees of freedom. The basic idea consists of the introduction of an additional degree of freedom for the collective motion and the use of Dirac’s canonical theory with constraints. The formalism is essentially equivalent to the ordinary time-dependent Hartree-Fock theory presented in Part I. The result given in Part I is completely reproduced.

Application of a self-consistent theory of large amplitude collective motion to the generalized meshkov-glick-lipkin model

A recent formulation of the theory of large amplitude collective motion in the adiabatic limit is applied to a generalized monopole shell model. Numerical calculations are carried out for the three-level model, approximately equivalent to a classical system with two degrees of freedom. Our results go somewhat beyond previous treatments of this system and provide substantiation for the validity of the method, in suitable parameter ranges, as a way of recognizing and decoupling the collective and the non-collective degrees of freedom.

Treatment of Nucleon-Number Conservation in the Selfconsistent Collective-Coordinate Method: --Coupling between Large-Amplitude Collective Motion and Pairing Rotation--

The selfconsistent collective-coordinate (See) method is extended in order to microscopically describe large-amplitude collective motions in superconducting nuclei. To restore the nucleon-number conserva­ tion violated by the use of the quasiparticle representation, the collective variables representing the pairing rotation are explicitly introduced. The coupling between the large-amplitude collective motion and the pairing rotation is treated in such a way that unphysical deviation of the average nucleon number from the ground-state value is eliminated automatically. It is shown that the basic equations of the extended version of the see method can be solved by an iterative method similar to the well-known (7/,1/) expansion. Applicability of the proposed method is investigated for an 0(4) model.

Quantum Theory of Dynamical Collective Subspace for Large-Amplitude Collective Motion

By placing emphasis on conceptual correspondence to the theory which has been developed within the framework of the time· dependent Hartree·Fock theory, a full quantum theory appropriate for describing large-amplitude collective motion is proposed. A central problem of the quantum theory is how to determine an optimal representation called a dynamical representation; the representation is specific for the collective subspace where the large-amplitude collective motion is replicated as satisfactorily as possible. As an extension of the classical theory where the concept of an approximate integral surface plays an important role, the dynamical representation is properly characterized by introducing a concept of an approximate invariant subspace of the Hamiltonian.

Concept of Dynamical Collective Submanifold for Large-Amplitude Collective Motion in the TDHF Theory

Concept of Dynamical Collective Submanifold for Large-Amplitude Collective Motion in the TDHF Theory

Microscopic Description of Anharmonic Gamma-Vibrations by Means of the Selfconsistent-Collective-Coordinate Method. II

We investigate, in a fully microscopic way, the double gamma·vibrational states in l68Er by means of the ·selfconsistent-collective-coordinate (See) method proposed by Marumori, Maskawa, Sakata and Kuriyama. In applying the see method, we propose a new prescription to derive a quantized collective Hamiltonian. The collective potential is defined in terms of the Wigner transform of the quantized collective Hamiltonian. It is different from the potential defined by the energy expectation values with respect to the time-even Slater determinants, and we find that it acquires a much stronger tendency toward the triaxial equilibrium shape than the latter. Numerical calculations show that 168Er is situated just in the transitional region toward the triaxial equilibrium shape. The nuclear anharmonicities associated with low-frequency vibrational modes have been an outstanding subject in nuclear many-body problems. We encounter genuine non-linear vibrations, for instance, in transitional nuclei lying in the intermediate region between the spherical and deformed equilibrium shapes. For the aim of constructing a fully microscopic theory capable of describing the large-amplitude collective motions such as the non-linear nuclear vibrations, Marumori, Maskawa, Sakata and Kuriyama 1 ) have proposed a new theory called the selfconsistent-collective-coordinate (See) method. The quadrupole collective modes associated with the y-degree of freedom have been thought to constitute another typical example of large-amplitude, nuclear non-linear phenomena. 2 ) But, it is only recently that significant anharmonic character of the y vibrations is demonstrated in a clear-cut way.3) In this paper, we shall analyse the microscopic structure of the anharmonicities associated with the y-vibrational modes by means ofthe see method. ll For this aim, we shall propose a new prescription to quantize the classical collective Hamiltonian derived by the see method. We also propose a new procedure to relate the resulting quantized Hamiltonian with the phenomenological Bohr Hamiltonian. 4

Equations of Collective Submanifold for Large Amplitude Collective Motion and Its Coupling with Intrinsic Degrees of Freedom. II

A theory based on the time·dependent Hartree·Fock theory is proposed with the aim of describing large amplitude collective motion and its coupling with intrinsic degrees of freedom. With the use of time·dependent canonical transformation, the whole system is separated into two types of degrees of freedom. Under the condition of unique separation, two kinds of equations of collective submanifold are obtained. One js for the collective motion and of the same form as that previously given by the present authors. The other is for the intrinsic degrees of freedom and consists of a set of linear partial differential equations. It is reduced to the old one, the form of which is non-linear.

Equations of Collective Submanifold for Large Amplitude Collective Motion and Its Coupling with Intrinsic Degrees of Freedom. I

Equations of Collective Submanifold for Large Amplitude Collective Motion and Its Coupling with Intrinsic Degrees of Freedom. I

Dissipative adiabatic mean field theory for large amplitude collective motion

Using the projection approach to nonequilibrium statistical mechanics we investigate the concept of a maximally decoupled adiabatic collective motion for quantum many-body system. Choosing the collective variables as the relevant observables of the system we derive transport equations for the collective motion which properly contain the coupling of the collective motion to the internal excitations. These equations allow for the description of dissipative effects during the collective motion caused by internal excitations. Criteria for the determination of the maximally decoupled collective mean field are derived and investigated. For this mean field it can be shown, that, if in addition certain validity conditions are fulfilled, the one-body friction vanishes.

Dissipative adiabatic mean field theory for large amplitude collective motion: P. G. Reinhard, Institut für Theoretische Physik, Universität Erlangen-Nürnberg, D-8520 Erlangen, and Institut für Kernphysik, Universität Mainz, D-6500 Mainz, West Germany; H. Reinhardt, Zentralinstitut für Kernforschung, Rossendorf, DDR-8051 Dresden, East Germany; and K. Goeke, Institut für Kernphysik, Kernforschungsanlage, D-5170 Jülich, West Germany; and Institut für Kernphysik, Universität Bonn, D-5300 Bonn,

Dissipative adiabatic mean field theory for large amplitude collective motion: P. G. Reinhard, Institut für Theoretische Physik, Universität Erlangen-Nürnberg, D-8520 Erlangen, and Institut für Kernphysik, Universität Mainz, D-6500 Mainz, West Germany; H. Reinhardt, Zentralinstitut für Kernforschung, Rossendorf, DDR-8051 Dresden, East Germany; and K. Goeke, Institut für Kernphysik, Kernforschungsanlage, D-5170 Jülich, West Germany; and Institut für Kernphysik, Universität Bonn, D-5300 Bonn,

Concept of Dynamical Collective Submanifold for Large-Amplitude Collective Motion

Concept of Dynamical Collective Submanifold for Large-Amplitude Collective Motion

Microscopic Description of Anharmonic Gamma-Vibrations by Means of the Selfconsistent-Collective-Coordinate Method. I

We investigate, in a fully microscopic way, the double gamma·vibrational states in l68Er by means of the ·selfconsistent-collective-coordinate (See) method proposed by Marumori, Maskawa, Sakata and Kuriyama. In applying the see method, we propose a new prescription to derive a quantized collective Hamiltonian. The collective potential is defined in terms of the Wigner transform of the quantized collective Hamiltonian. It is different from the potential defined by the energy expectation values with respect to the time-even Slater determinants, and we find that it acquires a much stronger tendency toward the triaxial equilibrium shape than the latter. Numerical calculations show that 168Er is situated just in the transitional region toward the triaxial equilibrium shape. The nuclear anharmonicities associated with low-frequency vibrational modes have been an outstanding subject in nuclear many-body problems. We encounter genuine non-linear vibrations, for instance, in transitional nuclei lying in the intermediate region between the spherical and deformed equilibrium shapes. For the aim of constructing a fully microscopic theory capable of describing the large-amplitude collective motions such as the non-linear nuclear vibrations, Marumori, Maskawa, Sakata and Kuriyama 1 ) have proposed a new theory called the selfconsistent-collective-coordinate (See) method. The quadrupole collective modes associated with the y-degree of freedom have been thought to constitute another typical example of large-amplitude, nuclear non-linear phenomena. 2 ) But, it is only recently that significant anharmonic character of the y vibrations is demonstrated in a clear-cut way.3) In this paper, we shall analyse the microscopic structure of the anharmonicities associated with the y-vibrational modes by means ofthe see method. ll For this aim, we shall propose a new prescription to quantize the classical collective Hamiltonian derived by the see method. We also propose a new procedure to relate the resulting quantized Hamiltonian with the phenomenological Bohr Hamiltonian. 4

Diabatic Quasiparticle Representation for Rotating Shell Model

We propose a new powerful method to construct a quasiparticle to be used in the rotating shell model (RSM) 1),2) for high-spin states in the yrast region. Use of the diabatic representation enables us to unambiguously specify individual rotational bands in which internal structure of the quasiparticle state vectors smoothly changes as functions of the rotational frequency w. Our approach is based on the selfconsistent collective-coordinate (SCC) method which is proposed by Marumori et al. ) as a microscopic theory of large-amplitude collective motions. We start from the time-dependent HartreeBogoliubov state vector of the following form:

Is there a consistent theory of large-amplitude collective motion?

Villars's equations, first derived from the adiabatic limit of time-dependent Hartree-Fock theory, are presented here as a set of exact conditions for the decoupling (in the adiabatic limit) of a sub-system from a classical Hamiltonian system. It is emphasized that these conditions do not incorporate all the required decoupling conditions. By combining the additional requirements with Villars's equations, we obtain a mathematically complete system, which will yield exact solutions where such exist, but which can also be applied to cases of approximate decoupling.

General semi-classical method for collective motion and its connection with the Rowe-Basserman and marumori theories, and adiabatic limits

In recent work, we have shown that in the adiabatic limit (large amplitude, small momentum), time-dependent Hartree-Fock theory (TDHF) yields a well-defined theory of large-amplitude collective motion which provides an essentially unique construction for a collective hamiltonian. An alternative theory, put forward by Rowe and Basserman and by Marumori is, apparently, not restricted to small momenta. We describe a general framework for the study of collective motion in the semi-classical limit without limitation on the size of coordinates or momenta, which includes all previous methods as limiting cases. We find it convenient, as in the past, to consider two general systems: first, a system with n degrees of freedom and no special permutation symmetry, and, second, a system of fermions described in TDHF. For both systems the problem can be formulated as a search for a hamiltonian flow confined to a finite-dimensional hypersurface in a phase space, which itself may be finite- or infinite-dimensional. Though, in general, there are no exact solutions to this problem, we can formulate consistent approximation schemes corresponding to both the adiabatic and Rowe-Basserman, and Marumori limits. We also show how to extend the momentum expansion, which underlies the adiabatic approximation, to higher orders in the momentum. We thereby confirm the structure of the theory found in our previous work.

A Schematic Model of Large Amplitude Collective Motions with an Exact Classical Solution. II: Classical Analysis of Two-Level Pairing Hamiltonian

It is investigated how classical pictures of the whole phase space are connected with the original quantal system by using the two-level pairing model. Discontinuous shape changes of classical trajectories are shown to be related with the sudden changes of patterns in the transition strength between energy eigenstates. By introducing the collective and intrinsic coordinates representing pairing rotations and vibrations, respectively, these discontinuities can be interpreted as the breakdown of the excited pairing ·rotational band scheme.

A Schematic Model of Large Amplitude Collective Motions with an Exact Classical Solution. II : Classical Analysis of Two-Level Pairing Hamiltonian

A Schematic Model of Large Amplitude Collective Motions with an Exact Classical Solution. II : Classical Analysis of Two-Level Pairing Hamiltonian

Equations-of-motion approach to a quantum theory of large-amplitude collective motion

The equations-of-motion approach to large-amplitude collective motion is implemented both for systems of coupled bosons, also studied in a previous paper, and for systems of coupled fermions. For the fermion case, the underlying formulation is that provided by the generalized Hartree-Fock approximation (or generalized density matrix method). To obtain results valid in the semi-classical limit, as in most previous work, we compute the Wigner transform of quantum matrices in the representation in which collective coordinates are diagonal and keep only the leading contributions. Higher-order contributions can be retained, however, and, in any case, there is no ambiguity of requantization. The semi-classical limit is seen to comprise the dynamics of time-dependent Hartree-Fock theory (TDHF) and a classical canonicity condition. By utilizing a well-known parametrization of the manifold of Slater determinants in terms of classical canonical variables, we are able to derive and understand the equations of the adiabatic limit in full parallelism with the boson case. As in the previous paper, we can thus show: (i) to zero and first order in the adiabatic limit the physics is contained in Villars' equations; (ii) to second order there is consistency and no new conditions. The structure of the solution space (discussed thoroughly in the previous paper) is summarized. A discussion of associated variational principles is given. A form of the theory equivalent to self-consistent cranking is described. A method of solution is illustrated by working out several elementary examples. The relationship to previous work, especially that of Zeievinsky and Marumori and coworkers is discussed briefly. Three appendices deal respectively with the equations-of-motion method, with useful properties of Slater determinants, and with some technical details associated with the fermion equations of motion.

Collective mass parameters and linear response techniques in three-dimensional grids

We discuss four prescriptions for evaluating a collective mass parameter suitable for translations, rotations and large amplitude collective motions. These are the adiabatic time dependent Hartree-Fock theory (ATDHF) and the generator coordinate method (GCM), both with and without curvature corrections. As practical example we consider the16O+16O collision using a recently developed density dependent interaction with direct Yukawa and Coulomb terms. We present a fast iteration scheme for solving the linear response equation in a three-dimensional coordinate or momentum space grid. As test cases we consider the rotational and translational inertia parameters for various distances between the heavy ions. We find that curvature corrections are negligible whereas the difference between ATDHF and GCM may amount to about 35% in case of rotation.

Dynamics of fission and heavy ion reactions

Abstract We discuss recent advances in a unified macroscopic-microscopic description of large-amplitude collective nuclear motion such as occurs in fission and heavy ion reactions. With the goal of finding observable quantities that depend upon the magnitude and mechanism of nuclear dissipation, we consider one-body dissipation and two-body viscosity within the framework of a generalized Fokker-Planck equation for the time dependence of the distribution function in phase space of collective coordinates and momenta. Proceeding in two separate directions, we first solve the generalized Hamilton equations of motion for the first moments of the distribution function with a new shape parametrization and other technical innovations. This yields the mean translational fission-fragment kinetic energy and mass of a third fragment that sometimes forms between the two end fragments, as well as the energy required for fusion in symmetric heavy-ion reactions and the mass transfer and capture cross section in asymmetric heavy-ion reactions. In a second direction, we specialize to an inverted-oscillator fission barrier and use Kramers' stationary solution to calculate the mean time from the saddle point to scission for a heavy-ion-induced fission reaction for which experimental information is becoming available.