Collective dynamics in exotic nuclei

The nucleus is an interesting many-body system: although it contains only a very limited number of particles, it exhibits both individual-particle dynamics and collective dynamics at the same energy scale. The challenge to nuclear structure theory is to understand the many-body mechanisms governing the nature of nuclear collective motion. Exotic nuclei, because of their weak binding and abnormal neutron-to-proton ratios, offer unique opportunities for study. Examples of new collective modes expected to appear in nuclei from stability are: skin excitations, isoscalar (proton-neutron) pairing, clusters in the skin, exotic cluster decays, and magnetic deformations associated with nucleonic currents. Of course, the nature of collective modes in "normal" nuclei will also be different in the new territory. What happens to low- and high-frequency multipole modes when the neutron excess is unusually large? What is the nature of pygmy resonances? What is the role of isospin mixing in heavy N~Z nuclei? How are beta decay rates affected by weak binding? How do collective modes feed back into the weak nuclear binding? Below, we list a number of topics that we intend to discuss during the program.

A) The ground state: mean field theory and beyond

We can't construct a consistent theory of dynamics without first understanding the statics. Self-consistent mean-field theory, which we can also call density-functional theory, has been enormously successful. We want to build on that success to find better functionals for finite nuclei as well as for nuclear matter having an arbitrary ratio of neutrons to protons.

The major challenge here is to compute the correlation term in the density functional that corrects the mean field. While many of the correlations are short-range and therefore likely to be adequately treated by local approximations, the breaking of translational and sometimes rotational symmetry in the mean-field wave function signal long-range correlations outside the scope of ordinary density functionals. Yet, the attempt to calculate the effects as separate contributions to the energy raises serious conceptual issues, particular when the functional depends on the density in a way different than does an ordinary two-body interaction.

Vibrational and rotational corrections to binding energy are often calculated within the Gaussian overlap approximation to GCM, while particle number conservation is restored either by straightforward projection or by the approximate projection method of Lipkin and Nogami. For nuclei with N close to Z, isospin fluctuations become important, and a corresponding correlation term should be added. Clearly, for global mass calculations approximations must be worked out that avoid full-scale collective calculations. Our hope is that we can combine and improve on these methods to approach the Holy Grail: a global theory of nuclear binding energies in which both the mean-field mass and the dynamical corrections come from the same energy-density functional.

B) Collective strength in exotic nuclei: small-amplitude collective motion.

For a consistent description of nuclear excitations, one has to go beyond the static mean-field approximation. A powerful tool for understanding both low-lying collective states and giant resonances is the quasiparticle random-phase approximation (QRPA). The approximation, which should be good for collective vibrations as long as their amplitudes are small, is especially effective in conjunction with energy density functionals. The QRPA is a standard method for describing collective excitations in open-shell superconducting nuclei with stable mean-field solutions, either spherical or deformed. What is not standard, and at the same time is extremely important for weakly bound nuclei, is the treatment of the particle continuum. Continuum extensions of the random phase approximation (RPA) or QRPA are usually carried out in coordinate space, facilitating treatment of decay channels and guaranteeing correct asymptotics.

It is only during the recent years that fully-selfconsistent QRPAs have been developed. (Self-consistency is crucial because the assumptions behind QRPA are not valid without it.) A very limited number of QRPA studies have been carried out to address properties of exotic nuclei such as electromagnetic strength, nature of individual collective states, decay properties, and electroweak processes. The main challenge is the inclusion of symmetry breaking effects, associated with shape deformations and pairing, in the presence of strong coupling to the particle continuum. Once a deformed QRPA framework is developed, the whole range of open-shell neutron-rich nuclei will open up for exploration!

C) The main battleground: large amplitude collective motion

Microscopic understanding of nuclear collective dynamics is a long-term goal. Large amplitude collective motion (LACM), as seen in fission, fusion, cluster decay, shape coexistence, and phase transitions, provides a particularly important challenge. All those phenomena involve the mixing of mean fields with different symmetries. The transition from one stable mean field to another goes through one of several level crossings around which the original symmetry of the system is broken. We have yet to obtain a microscopic understanding of LACM that is comparable to what we have for ground states, excited states, and response functions.

The usual starting point for the LACM is the Time Dependent Hartree Fock Bogoliubov method (TDHFB). Although TDHFB nicely incorporates collective and single-particle effects, it has essentially never been applied; even the simpler TDHF method has only rarely been implemented. TDHF alone requires a tremendous amount of numerical effort. Extensions of TDHF are not only extremely complicated technically but, even worse, they are also conceptually unclear. The TDHFB wave function is a product state that behaves in nearly classical way. As a consequence, the superposition principle cannot be applied and the theory must be seriously modified to account for the configuration mixing caused by a residual interaction and for restoration of spontaneously broken symmetries. Another important deficiency is the impossibility of describing tunnelling, a particularly painful problem in the context of fission or cluster decay. Because of al this, there are now many approximations to TDHFB, most of them variants of the Adiabatic TDHFB method (ATDHFB).

ATDHFB is designed for slow processes, which obey the Born-Oppenheimer approximation. With ATDHFB one can derive a collective Schrodinger equation involving inertial parameters (collective masses) associated with collective variables. The choice of collective coordinates is by no means simple and one usually needs quite a few collective degrees of freedom to effectively represent the motion of the system. Usually, collective coordinates are associated with the center of mass, the Euler angles, two pairing gauge angles, and shape vibrations. ATDHF has been applied mainly to low-energy vibrational states and to spontaneous fission. Its chief deficiency is related to degeneracy, or level crossings, near which the motion cannot be separated into collective and non-collective parts. The adiabatic approximation then breaks down and the the notions of the collective potential and collective mass lose their meaning.

A useful microscopic tool for describing LACM is the Generator Coordinate Method (GCM). The GCM wave function is usually taken as a combination of many (projected) intrinsic states, calculated self-consistently within constrained HFB theory. The constraining operators define collective degrees of freedom. The GCM wave function is rich enough to accommodate correlations absent in the mean field and is not limited to the adiabatic regime. Moreover, GCM is based on the variational principle. The most sophisticated GCM calculations on the market, applied to the problem of shape coexistence, shape mixing in transitional nuclei, and a decay of superdeformed states, involve 2-3 collective coordinates (such as quadrupole and octupole degrees of freedom) and a wave function projected onto good proton and neutron number, angular momentum, and sometimes parity.

Spontaneous fission is one the oldest decay modes known, but is still not fully understood. It represents an extreme example of the LACM, the tunneling of a many-body system. The pairing interaction plays a large role here because it causes a dramatic smoothing of single-particle crossings, thus improving the adiabatic approximation. Many calculations of spontaneous fission (lifetimes, sometimes mass/charge splits) are based on the adiabatic assumption. While the majority have used microscopic-macroscopic methods, a very few self-consistent applications have beeen carried out in the framework of ATDHF and its variants.

The future challenges, both numerical and conceptual, related to practical applications of GCM are numerous. First among these is the choice of generator coordinates, which are usually selected in an arbitrary way that depends on the problem and our physical intuition. While some theoretical methods allow for a self-consistent determination of collective coordinates, very few have been applied. Second, many problems, such as fission or cluster decay, require the use of several collective degrees of freedom and an immense computational effort. Third, it is not clear how to apply GCM to weakly bound nuclei with nearly vanishing chemical potentials. Finally, as mentioned earlier, there are fundamental problems with incorporating GCM into density-functional theory.

One interesting method, barely applied so far, is imaginary-time mean-field theory (similar to the instanton method in relativistic field theory), which allows for a TDHF treatment of tunneling. Early applications show a dramatic difference between the results of the static constrained HF (or ATDHF) and TDHF; both the collective paths and the tunneling probabilities are different. It will certainly be worthwhile to re-examine the usefulness of the imaginary-time method.

Summary

Limited by the speed and memories of our computers, we have not taken full advantage of existing theoretical tools, such as TDHFB and its imaginary-time extensions, the multidimensional GCM method, projection operator techniques, and other theoretical treatments of LACM. But developments in many-body theory, powerful new numerical algorithms, and better computers hold out the promise of significant advances in the our understanding of collective dynamics. At the same time high-resolution data in exotic nuclei and superheavy elements are providing new phenomena for us to tackle: new kinds of deformations associated with spins and currents, spectacular phase transitions, coupling between coexisting states, and new examples of fission.