On the simulation of extended TDHF theory

A novel method is presented for implementation of the extended mean-field theory incorporating two-body collisions. At a given time, stochastic imaginary time propagation of occupied states is used to generate a convenient basis. The quantal collision terms, including memory effects, are then computed by a backward mean-field propagation of these single-particle states. The method is illustrated in an exactly solvable model. Whereas the usual TDHF fails to reproduce the long time evolution, a good agreement is found between the extended TDHF and the exact solution.     

Geometrical derivation of collective operators and paths in TDHF

It is first found that the intrinsic parity of an operator under time reversal and the interpretation of the operator as coordinate- or momentum-like in a TDHF calculation are not simply related. This is because the TDHF particle-hole basis is, in general, complex. The TDHF equation is then reformulated in the plane tangent to the Slater determinant manifold. This plane is spanned by the particle-hole basis. The particle-hole matrix elements of the Hartree-Fock Hamiltonian define the energy gradient vector in this tangent plane. This gradient is real when the Slater determinant is real. A TDHF calculation initiated from a real determinant induces, during the first infinitesimal time step, a purely imaginary variation of this determinant along the gradient. The gradient is thus identified with the matrix elements of a boost operator. The next infinitesimal time step defines, in turn, a displacement operator. These operators are retained as collective if the TDHF path is stable under changes of velocities. Various criteria are found for this stability condition. The theory cannot be applied straightforwardly to translations and rotations for there is no energy gradient to generate coordinate operators. Particle-hole matrix elements of boost operators can be obtained, however, by a multiplication by i of the matrix elements of displacement operators, since the latter are known explicitly. It is finally found that the rotation of a wavefunction is contradictory with angular momentum conservation in general. Conservation can be ensured by a rotation of the density only and a more elaborate evolution of the velocity field.     

Damping of nuclear collective motion in the extended mean-field theory

The dissipation mechanism in slow nuclear collective motion is studied in the frame of the extended mean-field theory. The collective motion is treated explicitly by employing a travelling single-particle representation in the semi-classical approximation. The rate of change of the collective kinetic energy is determined by: (i) one-body dissipation, which reflects uncorrelated particle-hole excitations as a result of the collisions of particles with the mean field, (ii) two-body dissipation, which consists of simultaneous 2 particle-2 hole excitations via direct coupling of the residual two-body interactions, and (iii) potential dissipation due to the redistribution of the single-particle energies as a result of the random two-body collisions. In contrast to the first two processes the potential dissipation exhibits memory effects due to the large values of the local equilibration times.     

Memory effects in the energy dissipation for slow collective nuclear motion

The energy dissipation in slow collective nuclear motion is considered as a combined effect of the diabatic production of particle-hole excitations and the subsequent equilibration by two-body collisions. Memory effects due to the long mean free path of the nucleons are treated analytically for an interacting Fermi gas within moving walls leading to a friction kernel (frequency-dependent friction coefficient) in the classical equation of collective motion.     

Microscopic nuclear collective motion studied by classical equations of motion

The time-dependent variation principle is used to obtain generally non-canonical equations of motion from any class of quantum states which are parameterized by a set of continuous complex quantities. A class of states is presented whose associated classical dynamics is described by the five collective quadrupole degrees of freedom. Information about the classical dynamics of the system can be obtained from the non-canonical equations by finding physically interesting quantities which are coordinate independent and which characterize the low-energy collective motion. Approximate collective hamiltonians, of either a Bohr-Mottelson or an IBM type, can be found by insisting that the interesting physical quantities which describe the low-energy classical behavior of the many-body system are the same as those describing the classical behavior of the system given by the collective hamiltonian. The method is applied to two simple schematic models, one vibrational and one rotational, and IBM hamiltonians are obtained.     

The moment map and collective motion

Given a Hamiltonian action of a Lie group G on a symplectic manifold M there is an induced map Φ: $\text{M → g}{}^1$ where $\text{g}{}^1$ is the dual space to the Lie algebra, g, of G. The map Φ is called the moment map. Any function P on $\text{g}{}^1$ then gives rise to a function F = P ° Φ on M which is a “collective Hamiltonian” associated to the group action G. We show how the rigid rotor, liquid drop, and other collective models of the nucleus fit into this framework. We describe the steps involved in integrating collective equations of motion and indicate some principles involved in the choice of collective Hamiltonians, i.e., the functions P. We discuss these constructions in some detail for the case that G is a semidirect product.     

TDHF equations of motion for algebraic hamiltonians in a coherent state representation

Time-dependent Hartee-Fock (TDHF) equations are derived for nuclear systems with internal dynamical group U(r). The coordinates which appear in the TDHF equations are the coordinates which parameterize the U(r) coherent states. The TDHF orbits for the hamiltonian $\text{H}$ are identical with equations of motion for a classical system described by the hamiltonian function 〈$\text{H}$〉 obtained directly from the operator $\text{H}$. This quantum-classical correspondence facilitates interpretation of TDHF orbits. The phenomena of coexistence and critical elongation are discusses, as is the relation between the critical points of the function 〈$\text{H}$〉 and the spectral properties fo the operator $\text{H}$.     

Dissipation of nuclear collective motion

The collective transport theory provides a framework for understanding damped collective motion. The irreversibility of collective motion is traced to the fact that the nucleus is an open system. The finite lifetime of single-particle excitations causes the relaxation of the nuclear collective response. Both vibrational states and damped heavy-ion collisions can be understood quantitatively by computations without free parameters.     

Microscopic study of fermionic collective motion: (I). Functional representation of collective motion

Based on the shell structure of the finite nuclear many fermion system (FMFS), the coherent states related to the Spin(2r) group are defined. The global and local functional representations of the FMFS state-vectors and operators, defined on the coset space Spin(2r)/U(r), are constructed. The nonuniqueness of the coherent state functional representations is overcome by the imposition of a consistency condition on the wave functions. The influence of the boundary of the coset space Spin(2r)/U(r) on the local functional representation is physically removed only for the bound states of FMFS. The reason for the non-hermitian behavior of the local functional representation is exposed. Finally, using Bargmann's theory, the boson representation of FMFS are directly calculated from the local functional representation of FMFS. Thus, in this paper, we have demonstrated that the kinematics of the collective behavior of FMFS can be described in three non-equivalent representations: the fermion representation, the global functional representation and the local functional representation.     

Boson description of nuclear collective motion

Nuclear system with octupole-octupole interaction is studied by means of the boson expansion method. Expressions of the fourth-order collective Hamiltonian and third-order octupole moment operator are derived. For¹¹²Cd and¹⁴⁸Sm, characteristics of octupole vibrational spectra are discussed in comparison with the quadrupole vibration.     

Boson description of nuclear collective motion

Nuclear system interacting via quadrupole and octupole particle-hole forces is studied by the boson expansion technique. Energy spectra of the negative parity yrast band and the ground state band are calculated and compared with experiment for¹⁰⁰Ru,¹¹²Cd,¹⁵⁰Sm and¹⁵⁰Gd. ExperimentalB(E1)/B(E2) ratios show strong hindrance for E1 transitions, and are used to deduce the static polarizability of E1 transitions.     

Onset of collective and cohesive motion

We study the onset of collective motion, with and without cohesion, of groups of noisy self-propelled particles interacting locally. We find that this phase transition, in two space dimensions, is always discontinuous, including for the minimal model of Vicsek et al. [Phys. Rev. Lett. 75, 1226 (1995)]] for which a nontrivial critical point was previously advocated. We also show that cohesion is always lost near onset, as a result of the interplay of density, velocity, and shape fluctuations.     

The contribution of collective zero-point motion to mean-square charge radii

The contribution of collective quadrupole zero-point motion to changes in mean-square radii is calculated for transition nuclei in a long sequence of rubidium isotopes. The results of the calculation do not agree with the measured values. Some additional calculations are performed to show that the results are not sensitive to the choice of the collective inertial parameters.     

Large-scale collective motion in heavy-ion collisions

Heavy-ion fusion and deep inelastic reactions have been studied for symmetric systems in a classical dynamical model with deformation and necking as the collective shape coordinates. The calculated fusion excitation functions (for compound nucleus formation) are in good agreement with the experimental results from the evaporation residue measurements. It is observed that “nuclear molecules” are formed for not too heavy systems. The calculated reaction time for collisions of very heavy ions like ²³⁸U + ²³⁸U is found to be ~10⁻²¹ sec only and thus the width of the positron spectra observed in these reactions can not be explained in the light of quantum electrodynamics. The extra-extra push energies for fusion of heavy nuclei have also been studied.     

Shell-structure inertia for slow collective motion

The modified shell correction method is suggested for the calculation of transport coefficients in a slow nuclear collective dynamics. For the multipole low-lying vibrations near the spherical shape of a nucleus, the smooth transport coefficients corresponding to the extended Thomas-Fermi approach are used as a macroscopic background. The time-dependent mean field is approximated through the infinitely deep square-well potential for the calculation of the shell corrections. Significant shell effects in stiffness and inertia are found at small temperatures. These effects disappear approximately at the same large enough temperature as in the free energy. It is shown that the collective inertia is substantially larger than that of irrotational flow owing to the consistency condition of particle density and potential variations. The collective vibration energies and reduced friction and effective damping coefficients with accounting for the shell effects are in better agreement with allowance data than that found from the hydrodynamic model. The text was submitted by the authors in English.     

Chaotic motion and collective nuclear rotation

The regular and chaotic motion in the classical and quantal versions of a model hamiltonian with two degrees of freedom are investigated. This model contains a parameter which is identified with a conserved quantum number, the total spin. In particular, transitions between states differing in spin by one unit are studied. The transition is strongly collective for regular motion, and collectivity is destroyed with increasing stochasticity of the model.     

On the description of dissipative collective motion

We use a recently developed time-dependent projection method to describe the dissipation of collective motion coupled to an intrinsic system. The underlying physical picture is similar to that of the linear response approach. Our approach is, however, different from the conceptual point of view. We do not resort to a quasistatic picture but use instead a time-dependent projector. Furthermore, we project on a model space which includes the intrinsic hamiltonian in addition to the collective subspace. In this way we obtain a Fokker-Planck equation for the collective variables which is coupled to a transport equation describing the evolution of the temperature of the intrinsic system.     

Damping of nuclear collective and single-particle motion

The collective motion of the nuclear system is studied. In the independent-particle model, the motion is completely reversible. The neglected residual interactions couple the ph states to more complicated states. This coupling is taken into account by the optical model potential assuming independent decay of particle and hole states. Irreversibility is thereby introduced and damping of collective motion described in terms of the widths of the ph states. The validity of the assumption of independent decay is discussed. It is argued that spreading widths to low-frequency collective states are not part of the optical model, and do not contribute to damping of collective motion.     

Particle motion in guide potentials and the collective nuclear motion

The motion of a particle which is constrained by a guide potential to move on a curve is studied in the framework of the Generator Coordinate Method (GCM). In the limit of narrow guide potentials a differential equation for the wave function of the constrained motion is obtained which differs from the corresponding Schrödinger equation by an additional potential. This additional potential is due to the embedding of the curve in the space and depends on the form of the guide potential and on the curvature of the curve. Nonadiabatic transitions in the constrained motion are possible for finite widths of the guide potential. The coupling terms are given explicitly and it is shown that an adiabatic limit exists. Since the GCM can equally well describe the collective motion of nuclei, some insight into the more complicated problem of collective motion is obtained from its analogies to the studied problem of constrained particle motion: The collective motion of a nucleus can be considered as the motion of a particle with variable mass along a curve in a guide potential which is given by the interaction potential between the nucleons. It is shown that Schrödinger's quantized kinetic energy is correctly used in the cranking model and that the additional potential terms mentioned above are included there by the definition of the collective potential energy. Approximations to the idealized GCM used here are discussed and the connection with the method of Born, Oppenheimer and Villars is indicated.