Nuclear Collective Motion Research Articles

Fine structure of the nuclear scissors mode (new type of collective motion)

The coupled dynamics of low-lying modes (including scissors) and various giant quadrupole resonances are studied with the help of 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 model of harmonic oscillator with spin orbital interaction plus quadrupole-quadrupole residual interaction. Enlisting the spin allows one to introduce into consideration new types of nuclear collective motion.

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’.

Role of shape dependence of dissipation on nuclear fission

We examine the validity of extending Kramers' expression for fission width to systems with shape-dependent dissipations. For a system with a shape-dependent dissipation, Kramers' width obtained with the presaddle dissipation strength is found to be different from the stationary width obtained from the corresponding Langevin equations. It is demonstrated that the probability of a hot compound nucleus undergoing fission depends on both the presaddle and the postsaddle dynamics of collective nuclear motion. The predictions for prescission neutron multiplicity and evaporation residue cross section from statistical model calculations are also found to be different from those obtained from Langevin dynamical calculations when a shape-dependent dissipation is considered. For systems with shape-dependent dissipations, we conclude that the strength of ``presaddle dissipation'' determined by fitting experimental data in statistical model calculations does not represent the true strength of presaddle dissipation.

The long journey from ab initio calculations to density functional theory for nuclear large amplitude collective motion

At present there are two vastly different ab initio approaches to the description of the many-body dynamics: the density functional theory (DFT) and the functional integral (path integral) approaches. On one hand, if implemented exactly, the DFT approach can allow in principle the exact evaluation of arbitrary one-body densities. However, when applied to large amplitude collective motion (LACM), this approach needs to be extended in order to accommodate the phenomenon of surface-hopping, when adiabaticity is strongly violated and the description of a system using a single (generalized) Slater determinant is not valid anymore. The functional integral approach on the other hand does not appear to have such restrictions, but its implementation does not appear to be a straightforward endeavor. However, within a functional integral approach one seems to be able to evaluate in principle any kind of observable, such as the fragment mass and energy distributions in nuclear fission. These two radically different approaches can likely be brought together by formulating a stochastic time-dependent DFT approach to many-body dynamics.

Soft X-ray radiation damage in argon clusters

The photofragmentation dynamics of argon clusters excited with soft x rays is studied for various photon energies near the Ar 2p threshold. This allows localization of the initial energy to specific atoms on the cluster surface or in the bulk, or preparation of multiply-charged species in a controlled way. By changing the cluster sizes, we bridge the gap between sub-nano and nano-scaled objects and compare our observation to a radiation damage model for a simple fluid with the help of a classical approach relying on a statistical dissipation of the initial energy. These simple systems provide a benchmark to understand the underlying mechanisms due to soft X-ray in the weak interaction regime, pointing out the role of collective nuclear motions.

Work with Maskawa on Microscopic Theory of Nuclear Collective Motion

Development of theoretical study originated from a pioneering work on large-amplitude nuclear collective motion proposed at the end of the 1970’s by members of a theory group, Institute for Nuclear Study, University of Tokyo is reviewed by putting special emphasis on an important contribution of Toshihide Maskawa. Thanks to his broad and deep knowledge on mathematical physics, and owing to his friendly cooperation in the theory group, a basic idea of the work was further developed, and successfully formulated in a general and lucid form, which was presented in a paper entitled Self-Consistent Collective-Coordinate Method for the Large-Amplitude Nuclear Collective Motion. Since the theory was formulated in a widely applicable, plain and terse way, it has been developed by retaining a large overlap with theoretical forefront in various fields of science. Its recent developments for exploring an evolution of matter in connection with nonlinear dynamics, non-equilibrium statistical physics and bio-chemical sciences are also discussed. Subject Index: 210

The thermo field theory of nuclear collective motion

The dynamic equation of the Liquid Drop Model of collective motion is derived from the statistical quantum field theory of finite Fermi systems under the assumption that there is only one collective mode, whose characteristic frequencies are small compared to single-particle frequencies. The time evolution of the nucleus is characterized by the statistical matrix ρ χ , describing thermal equilibrium at fixed values of the expectancies of the operators Q, P, M of coordinate, momentum and mass of slow collective mode. Explicit expressions for ρ χ , P, M are obtained using the idea that collective motion in hot nuclei emerges because the randomly distributed phase of the nucleon field operator acquires a regular component χ x . The time evolution equations for the expectancies of Heisenberg operators Q t , P t , M t are converted into the dynamic equation for the collective coordinate q t = tr ( Q t ρ χ ) of the same form, as the dynamic equation of the Liquid Drop Model. This allows us to identify the microscopic expressions for collective mass, deformation force, and friction coefficient. To make these expressions tractable, the thermally equilibrated particle motion is modelled in terms of the temperature-dependent Hartree–Fock equations with the Skyrme force.

COLLECTIVE FLOW IN HEAVY-ION COLLISIONS FOR Eb=0.25-1.15GeV/NUCLEON

The study of the various collective flows of nuclear matter is one of the main subjects in heavy ion physics. The interest in collective nuclear motion under extreme conditions, like high density and/or high temperature originates from the equation of state (EOS) of nuclear matter. We investigate the collective transverse flow in heavy ion collisions at incident energies of 0.25 A GeV to 1.15 A GeV for Au + Au system within the quantum molecular dynamical (QMD) model. Some preliminary results are given and discussed.

A stabilized pairing functional

We propose a modified pairing functional for nuclear structure calculations which avoids the abrupt phase transition between pairing and non-pairing states. The intended application is the description of nuclear collective motion where the smoothing of the transition is compulsory to remove singularities. The stabilized pairing functional allows a thoroughly variational formulation, unlike the Lipkin-Nogami (LN) scheme which is often used for the purpose of smoothing. First applications to nuclear ground states and collective excitations prove the reliability and efficiency of the proposed stabilized pairing.

Nuclear collective excitations using correlated realistic interactions: The role of explicit random-phase approximation correlations

We examine to what extent correlated realistic nucleon-nucleon interactions, derived within the unitary correlation operator method (UCOM), can describe nuclear collective motion in the framework of first-order random-phase approximation (RPA). To this end we employ the correlated Argonne V18 interaction in calculations within the so-called ''Extended'' RPA (ERPA) and investigate the response of closed-shell nuclei. The ERPA is a renormalized RPA version which considers explicitly the depletion of the Fermi sea due to long-range correlations and thus allows us to examine how these affect the excitation spectra. It is found that the effect on the properties of giant resonances is rather small. Compared to the standard RPA, where excitations are built on top of the uncorrelated Hartree-Fock (HF) ground state, their centroid energies decrease by up to 1 MeV, approximately, in the isovector channel. The isoscalar response is less affected in general. Thus, the disagreement between our previous UCOM-based RPA calculations and the experimental data are to be attributed to other effects, namely to residual three-body terms in the Hamiltonian and to higher-order configurations. Single-particle energies and occupation probabilities obtained within the ERPA are compared with corresponding HF and perturbation-theory results and are discussed as well. The ERPA formalism ismore » presented in detail.« less

Classical chaos effect on nuclear collective motion

Here, the problem of the classical chaos effect on dissipation in nuclear many-body systems is addressed. To investigate this problem, the energy response on parametric driving of external confinement of the simplest interacting many-body system is studied: two identical like-charged particles moving on the plane in the field of a deformed harmonic oscillator, which shows a rich structure of the classical phase space. The parametric driving is represented by the slow periodic modulation of the deformation of confinement. Here, the focus is on how the driven dynamics of particles is defined by the structure of the unperturbed phase space of the system. It has been found that the rate at which the energy of the external driving is pumped into the energy of the interacting two-body system strongly depends on the driven dynamics of the regular orbits. A role of interaction for the energy absorption rate is also discussed.

Nuclear collective motion with a coherent coupling interaction between quadrupole and octupole modes

A collective Hamiltonian for the rotation-vibration motion of nuclei is considered, in which the axial quadrupole and octupole degrees of freedom are coupled through the centrifugal interaction. The potential of the system depends on the two deformation variables $\beta_2$ and $\beta_3$. The system is considered to oscillate between positive and negative $\beta_3$-values, by rounding an infinite potential core in the $(\beta_2,\beta_3)$-plane with $\beta_2>0$. By assuming a coherent contribution of the quadrupole and octupole oscillation modes in the collective motion, the energy spectrum is derived in an explicit analytic form, providing specific parity shift effects. On this basis several possible ways in the evolution of quadrupole-octupole collectivity are outlined. A particular application of the model to the energy levels and electric transition probabilities in alternating parity spectra of the nuclei $^{150}$Nd, $^{152}$Sm, $^{154}$Gd and $^{156}$Dy is presented.

Quantum and thermodynamic properties of spontaneous and low-energy induced fission of nuclei

It is shown that A. Bohr’s concept of transition fission states can be matched with the properties of Coriolis interaction if an axisymmetric fissile nucleus near the scission point remains cold despite a nonadiabatic character of nuclear collective deformation motion. The quantum and thermodynamic properties of various stages of binary and ternary fission after the descent of a fissile nucleus fromt he outer saddle point are studied within quantum-mechanical fission theory. It is shown that two-particle nucleon-nucleon correlations—in particular, superfluid correlations— play an important role in the formation of fission products and in the classification of fission transitions. The distributions of thermalized primary fission fragments with respect to spins and their projections onto the symmetry axis of the fissile nucleus and fission fragments are constructed, these distributions determining the properties of prompt neutrons and gamma rays emitted by these fragments. A new nonevaporation mechanism of third-particle production in ternary fission is proposed. This mechanism involves transitions of third particles from the cluster states of the fissile-nucleus neck to high-energy states under effects of the shake-off type that are due to the nonadiabatic character of nuclear collective deformation motion.

Description of nuclear octupole and quadrupole deformation close to the axial symmetry and phase transitions in the octupole mode

The dynamics of nuclear collective motion is investigated in the case of reflection-asymmetric shapes. The model is based on a new parameterization of the octupole and quadrupole degrees of freedom, valid for nuclei close to the axial symmetry. Amplitudes of oscillation in other degrees of freedom different from the axial ones are assumed to be small, but not frozen to zero. The case of nuclei which already possess a permanent quadrupole deformation is discussed in some more detail and a simple solution is obtained at the critical point of the phase transition between harmonic octupole oscillation and a permanent asymmetric shape. The results are compared with experimental data of the Thorium isotopic chain. The isotope Th-226 is found to be close to the critical point.

Nuclear collective motion with full nonlinearity

The periodic solutions previously considered for nuclear time-dependent Hartree-Fock problems are recalculated by a new method which is more accurate and reliable. It starts from a first guess of the entire periodic trajectory, instead of a first guess of the initial state. An exact solution of the discretized nonlinear equations is then obtained in three to five iterations involving, once again, the entire trajectory. The resulting families of solutions, and in particular the relationship between the period and the energy, are much more complicated than previously thought. The meaning of this outcome is discussed.

THE NUCLEAR BORN–OPPENHEIMER METHOD APPLIED TO NUCLEAR COLLECTIVE MOTION

We deal with the application of the nuclear Born–Oppenheimer (NBO) method to the study of nuclear collective motion. In particular, we look at the description of nuclear rotations and vibrations. The collective operators are specified within the NBO method only to the extent of identifying the type of collective degrees of freedom we intend to describe; the operators are then determined from the dynamics of the system. To separate the collective degrees of freedom into rotational and vibrational terms, we transform the collective tensor operator from the lab fixed frame of reference to the frame defined by the principal axes of the system; this transformation diagonalizes the tensor operator. We derive a general expression for the NBO mean energy and show that it contains internal, collective and coupling terms. Then, we specify the approximations that need to be made in order to establish a connection between Bohr's collective model and the NBO method. We show that Bohr's collective Hamiltonian can be recovered from the NBO Hamiltonian only after adopting some rather crude approximations. In addition, we try to understand, in light of the NBO approach, why Bohr's collective model gives the wrong inertial parameters. We show that this is due to two major reasons: the ad hoc selection of the collective degrees of freedom within the context of Bohr's collective model and the unwarranted neglect of several important terms from the Hamiltonian.

Memory effects on descent from the nuclear fission barrier

Non-Markovian transport equations for nuclear large amplitude motion are derived from the collisional kinetic equation. The memory effects are caused by the Fermi surface distortions and depend on the relaxation time. It is shown that the nuclear collective motion and the nuclear fission are influenced strongly by the memory effects at the relaxation time $\tau \geq 5\cdot 10^{-23}{\rm s}$. In particular, the descent of the nucleus from the fission barrier is accompanied by characteristic shape oscillations. The eigenfrequency and the damping of the shape oscillations depend on the contribution of the memory integral in the equations of motion. The shape oscillations disappear at the short relaxation time regime at $\tau \to 0$, which corresponds to the usual Markovian motion in the presence of friction forces. We show that the elastic forces produced by the memory integral lead to a significant delay for the descent of the nucleus from the barrier. Numerical calculations for the nucleus $^{236}$U shows that due to the memory effect the saddle-to-scission time grows by a factor of about 3 with respect to the corresponding saddle-to-scission time obtained in liquid drop model calculations with friction forces.

A Macsyma program for computing analytic Clebsch–Gordan coefficients of [formula omitted

Part 1 of the program generates the reduced matrix elements of the spherical tensor operator Q from which the matrix elements of all U(3) generators can be obtained. Part 2 computes the isoscalar factors for the coupling of two U(3) irreps. The basis used is symmetry adapted to SO(3)⊃SO(2) , as is desired for describing collective nuclear motion. The output of the matrix elements as well as the isoscalars is analytic (i.e. exact) for the input of the highest weights of U(3).

Non-Unitary Boson Mapping and Its Application to Nuclear Collective Motions

First, the general theory of boson mapping for even-number many-fermion systems is surveyed. In order to overcome the confusion concerning the so-called unphysical or spurious states in the boson mapping, the correct concept of the unphysical states is precisely given in a clear-cut way. Next, a method to apply the boson mapping to a truncated many-fermion Hilbert space consisting of collective phonons is proposed, by putting special emphasis on the Dyson-type non-unitary boson mapping. On the basis of this method, it becomes possible for the first time to apply the Dyson-type boson mapping to analyses of collective motions in realistic nuclei. This method is also extended to be applicable to odd-number-fermion systems. As known well, the Dyson-type boson mapping is a non-unitary transformation and it gives a non-Hermitian boson Hamiltonian. It is not easy (but not impossible) to solve the eigenstates of the non-Hermitian Hamiltonian. A Hermitian treatment of this nonHermitian eigenvalue problem is discussed and it is shown that this treatment is a very good approximation. Using this Hermitian treatment, we can obtain the normal-ordered Holstein-Primakoff-type boson expansion in the multi-collective-phonon subspace. Thereby the convergence of the boson expansion can be tested. Some examples of application of the Dyson-type non-unitary boson mapping to simplified models and realistic nuclei are also shown, and we can see that it is quite useful for analysis of the collective motions in realistic nuclei. In contrast to the above-mentioned ordinary type of boson mapping, which may be called a “static” boson mapping, the Dyson-type non-unitary selfconsistent-collective-coordinate method is discussed. The latter is, so to speak, a “dynamical” boson mapping, which is a dynamical extension of the ordinary boson mapping to be capable to include the coupling effects from the non-collective degrees of freedom selfconsistently. Thus all of the Dyson-type non-unitary boson mapping from A to Z is summarized in this paper.

Stability and instability of a hot and dilute nuclear droplet

The diabatic approach to collective nuclear motion is reformulated in the local-density approximation in order to treat the normal modes of a spherical nuclear droplet analytically. In a first application the adiabatic isoscalar modes are studied and results for the eigenvalues of compressional (bulk) and pure surface modes are presented as function of density and temperature inside the droplet, as well as for different mass numbers and for soft and stiff equations of state. We find that the region of bulk instabilities (spinodal regime) is substantially smaller for nuclear droplets than for infinite nuclear matter. For small densities below 30% of normal nuclear matter density and for temperatures below 5 MeV all relevant bulk modes become unstable with the same growth rates. The surface modes have a larger spinodal region, reaching out to densities and temperatures way beyond the spinodal line for bulk instabilities. Essential experimental features of multifragmentation, like fragmentation temperatures and fragment-mass distributions (in particular the power-law behavior) are consistent with the instability properties of an expanding nuclear droplet, and hence with a dynamical fragmentation process within the spinodal regime of bulk and surface modes (spinodal decomposition).