Properties of heavy ion interaction potentials calculated in the energy density formalism

Properties of the real part of the interaction potential between two colliding ions are studied. The calculations are performed in the framework of the energy density formalism, using the sudden approximation. In particular, we show that the potential can be written as the product of a “universal” function and of a geometrical factor. The universal function itself can be parametrized in a simple way. Connections with the proximity theorem of Randrup, Swiatecki and Tsang are discussed in detail. Comparisons with other available potentials are made for a typical example.     

Oscillator parameters in nuclei

The dependence on nucleon numbers of the harmonic oscillator length parameter b or energy spacing ℏω in nuclei is determined, using an extensive tabulation of nuclear charge radii and an empirical expression for the difference between proton and neutron radii. Various possible constraints on the parameters or radii are discussed. Alternative parametrizations are compared and a preferred “universal” parametrization is proposed. It is argued that the established Blomqvist–Molinari formula is perfectly adequate for determining the oscillator parameter for specific applications.     

An improved proximity force approximation for electrostatics

A quite straightforward approximation for the electrostatic interaction between two perfectly conducting surfaces suggests itself when the distance between them is much smaller than the characteristic lengths associated with their shapes. Indeed, in the so called “proximity force approximation” the electrostatic force is evaluated by first dividing each surface into a set of small flat patches, and then adding up the forces due two opposite pairs, the contributions of which are approximated as due to pairs of parallel planes. This approximation has been widely and successfully applied in different contexts, ranging from nuclear physics to Casimir effect calculations. We present here an improvement on this approximation, based on a derivative expansion for the electrostatic energy contained between the surfaces. The results obtained could be useful for discussing the geometric dependence of the electrostatic force, and also as a convenient benchmark for numerical analyses of the tip–sample electrostatic interaction in atomic force microscopes.     

Calculation of the real part of the interaction potential between two heavy ions in the sudden approximation

We have calculated the interaction potential between two heavy ions using the energy density formalism and Fermi distributions for the nuclear densities. The experimental fusion barriers are rather well reproduced. The conditions for the observation of fusion between two heavy ions are discussed. As far as the nuclear part of the interaction potential is concerned, the proximity scaling is investigated in detail. It is found that the proximity theorem is satisfied to a good extent. However, as far as the neutron excess is concerned, disagreement with the proximity potential is observed.     

Low-energy neutron-deuteron reactions with N 3 LO chiral forces

Based on the preformed cluster model (PCM), we have extended our earlier study on cluster decays of heavy parent nuclei to analyze the effects of different nuclear proximity potentials in the ground-state clusterization of superheavy nuclei with Z = 113, 115 and 117. In order to look for the possible role of deformations, calculations are performed for spherical as well as β ₂-deformed choices of fragmentation. The relevance of “hot compact” over “cold elongated” configurations due to orientations is also explored, in addition to the role of Q value and angular momentum ℓ effects. As the PCM is based on collective clusterization picture, the preformation and penetration probabilities get modified considerably, and hence do so the decay constants and half-lives of the clusters, with the use of different nuclear proximity potentials. The comparative importance of nuclear proximity potentials Prox-1977 and Prox-2000 is analyzed and the calculated decay half-lives in the framework of PCM are compared with the recent predictions of the analytical super-asymmetric fission model (ASAFM). The possible role of shell corrections is also investigated for understanding the dynamics of heavy particle radioactivity. Finally, the potential energy surfaces are compared for different proton and neutron magic numbers in superheavy mass region.

     

Geometric dynamics on quantum Nano-surfaces and low-energy spectrum in a homogeneous magnetic field

Low-energy dynamics and spectrum of the Schrödinger operator on curved surfaces in a magnetic field are considered. Nano-and micro-algebraic structures generate a quantization of the surface. On this quantum space, the dynamics has a “surfing” behavior described via curvature invariants. Two types of surface deformations correspond to trapping and transistor types of dynamics.We observe the “negative mass” effect in the low-energy spectrum under trapping deformations. We also compute, in geometrical terms, the energy levels near the spectral bottom (these levels are below the minimal Landau level). The surfing dynamics and the spectrum are independent of the magnitude of the magnetic field.     

Non-Abelian Stokes theorems in the Yang-Mills and gravity theories

We discuss the interpretation of the non-Abelian Stokes theorem or the Wilson loop in the Yang-Mills theory. For the “gravitational Wilson loops,” i.e., holonomies in curved d=2, 3, 4 spaces, we then derive “ non-Abelian Stokes theorems” that are similar to our formula in the Yang-Mills theory. In particular, we derive an elegant formula for the holonomy in the case of a constant-curvature background in three dimensions and a formula for small-area loops in any number of dimensions.     

Nonlinear canonical transformations in the coupling between degenerate high and low energy excitations

In microscopic many-body physics the coupling between the motion of fast particles (electrons) and slow particles (nuclei) is universal. The standard Born-Oppenheimer decoupling procedure breaks down, if the energy separation in the “fast” system is of the same order as the elementary excitation in the “slow” system. In this case “dynamical resonance” effects are to be expected. In the present investigation a model system of a coupling between a doubly degenerate high energy excitation and doubly degenerate low energy oscillator is handled by a non-linear canonical transformation which is shown to be quasi-exact in the sense that it diagonalizes the Hamiltonian in both extremal coupling cases. The transformation has some flexibility, so that the diagonalization regions can be enlarged. It is employed to calculate the “zero-phonon” optical response, which indeed displays aresonance effect. Likewise, another nonlinear transformation is devised, which only in the strong coupling limit yields diagonalization. This latter transformation in a natural way leads to the conventional semi-classical approaches to the dynamical Jahn-Teller problem. The results gotten with it are identical with those from our transformation in the strong coupling limit. On the basis of our results some remarks are made concerning the possible impact of the breakdown of the adiabatic approximation in other regions.     

Mass-surface predictions near the doubly magic nuclide 78Ni

The mass surface of nuclei close to the doubly magic nuclide ⁷⁸Ni is calculated by two methods. The first relies on the multiparticle shell model based on an effective interaction and a mean nuclear potential. The second employs the concept of so-called “magic crosses” and enables us to determine the masses of odd-odd nuclei close to ⁷⁸Ni by using similarity of the shell structure and neutron-proton interaction in the region of nuclei under consideration and in the region of heavy magic nuclides. The energies of the separation of one and two neutrons from nuclei close to ⁷⁸Ni and the energies of the β decay of these nuclei—recall that these quantities of astrophysical interest—are presented.     

Folded Solitary Waves and Foldons in （2＋1） Dimensions

A general type of local/zed excitations, folded solitary waves and foldons, is defined and studied both analytically and graphically. The folded solitary waves and foldons may be “folded“ in quite complicated ways and possess qnite rich structures and abundant interaction properties. The folded phenomenon is quite universal in the real natural world. The folded solitary waves and foldons are obtained from a quite universal formula and the universal formul is valid for some quite universal (2 1)-dimensional physical mode/s. The “universal“ formula is also extended to a more general form with many more independent arbitrary functions.     

Two-centre Dirac-Coulomb operators: regularity and bonding properties

Assuming the Born-Oppenheimer (clamped nuclei) approximation, basic properties of the Dirac operator for homonuclear two-centre one-electron systems are studied. This includes a rigorous analysis of the regularity of the potential energy curves as a function of the internuclear distance; in particular, for all values of the nuclear charge parameter that guarantee existence of a physically reasonable self-adjoint extension of the corresponding atomic Dirac-Coulomb operator, the continuity and differentiability of the electronic contribution to the potential energy curves are shown also to hold in the united atom limit. Furthermore, for nuclear charges not too large, the ground state united atom energy is demonstrated to provide a universal lower bound to all molecular energies within the discrete spectrum. Together with a generalization of the virial theorem to diatomic Dirac-Coulomb operators, this leads to a lower bound on the equilibrium separation between the nuclei. In addition, by employing appropriate variational arguments, the possibility of molecular bonding (for sufficiently large nuclear mass) is proved for all systems with charges up to those of the hydrogen molecular ion H₂⁺.     

On the deviation of small nuclear systems from saturation

Below a definite thickness, the structure of nuclear slabs — treated in the energy density formalism — deviates from saturation, and in particular the surface tension decreases rapidly. These effects, seemingly a consequence of the finite range of nuclear forces, cannot be accounted for by the droplet-model concept. A (negative) correction term is proposed for the nuclear mass formula which becomes important for small nuclei in addition to the (positive) curvature energy. Its consideration may remove ambiguities in theoretical predictions and empirical values of the curvature energy.     

Derivation of a proximity interaction between nuclei from the Hartree-Fock functional with Skyrme interactions

The potential between heavy or medium heavy spherical nuclei is derived on the basis of the proximity approximation and the assumption of undisturbed spherical density distributions. The HF-theory with Skyrme forces is used to write the energy as a functional of the density. The resulting potential consists of an isoscalar and a (much smaller) isospin-dependent part. Depending on the choice of the Skyrme parameters calculated interaction between heavy ions are close to the double folding potential or to the proximity potential. Our interaction can be factorized with a very good approximation into scaling factors and universal functions of the proximity distance in units of the surface thickness.     

One- and two-body dissipation in heavy-ion collisions

A microscopic theory has been formulated for one- and two-body dissipation in collisions between two heavy nuclei. With a nucleon-nucleon interaction as the basic perturbation in a density matrix approach with “linear response” approximations, the one- and two-body nuclear friction coefficients for the ⁴⁰Ca + ⁴⁰Ca system have been calculated and their dependence on relative kinetic energy and smearing of nuclear single-particle states was obtained. The results of our calculation show that: (a) the combined one- and two-body friction coefficients compare favourably with phenomenological values, (b) the one-body dissipation is more effective than two-body in kinetic energy damping, while both the mechanisms are comparable for the damping of relative angular momentum, (c) the importance of the two-body friction compared to one-body increases at higher relative kinetic energies and (d) the effect of introducing a smearing in nuclear levels appears as a lowering of nuclear friction.     

Clustering aspects of nuclei with octupole and superdeformation

Extreme nuclear shapes are considered from the clustering point of view. General aspects of light nuclei which possess very exotic, elongated shapes (“α-strings”) are given. The relation between superdeformation and di-molecules is considered, and it is shown that nuclear di-molecules have the same magic numbers and orbits filled as the ellipsoidal superdeformations for light nuclei, while for heavier nuclei they are expected in different mass-regions. The structure of possible octupole-unstable 3∶1 nuclear shapes (“hyperdeformations”) is discussed. Indications and consequences of a four-fold symmetry of the nuclear hamiltonian are briefly discussed.     

Analytical description of heavy ion potentials for collisions between nuclei of same shell

Using Skyrme energy density formalism, we present an analytical formula of ion-ion potential (including spin-density part) in terms of the masses of colliding nuclei. The parametrization of the spin-independent part of the ion-ion potential is based on the proximity theorem whereas the spin-dependent potential is parametrized in terms of “the masses of colliding nuclei and their associated particle strength”. The particle strength accounts for the number of valence particles outside the closed core. Adding Coulomb interaction, this parametrization of ion-ion potential introduces a great simplification for the calculation of fusion barriers and cross-sections analytically. Our parametrized potentials are in good agreement with other theoretical potentials and the fusion cross-sections calculated with this potential are in good agreement with experimental data. Received: 2 October 1997 / Revised version: 15 December 1997     

Multi-linear Variable Separation Approach to Solve a （1＋1）-Dimensional Coupled Integrable Dispersionless System

The multi-linear variable separation approach （MLVSA ） is very useful to solve （2＋1）-dimensional integrable systems. In this letter, we extend this method to solve a （1＋1）-dimensional coupled integrable dispersion-less system. Namely, by using a Backlund transformation and the MLVSA, we find a new general solution and define a new ＂universal formula＂. Then, some new （1＋1）-dimensional coherent structures of this universal formula can be found by selecting corresponding functions appropriately. Specially, in some conditions, bell soliton and kink soliton can transform each other, which are illustrated graphically.     

Deduction of the Quasi-linear Transport Equations of Hydro-thermodynamics from the Gyarmati Principle

From the universal form of Gyarmati's variational principle of thermodynamics the differential equations governing the internal energy and impulse transport of one component hydro-thermodynamic systems are derived. In our particular case Gyarmati's “supplementary theorem” is confirmed, by which the validity of the universal form of Gyarmati's variational principle is guaranted also in non-linear cases. Finally some problems of the Gyarmati principle and of non-linear thermodynamics are discussed.     

Self-consistent hartree description of finite nuclei in a relativistic quantum field theory

Relativistic Hartree equations for spherical nuclei are derived from a relativistic nuclear quantum field theory using a coordinate-space Green function approach. The renormalizable field theory lagrangian includes the interaction of nucleons with σ, ω, ρ and π mesons and the photon. The Hartree equations represent the “mean-field” approximation for a finite nuclear system. Coupling constants and the σ-meson mass are determined from the properties of nuclear matter and the rms charge radius in ⁴⁰Ca, and pionic contributions are absent for static, closed-shell nuclei. Calculated charge densities, neutron densities, rms radii, and single-nucleon energy levels throughout the periodic table are compared with data and with results of non-relativistic calculations. Relativistic Hartree results agree with experiment at a level comparable to that of the most sophisticated non-relativistic calculations to date. It is shown that the Lorentz covariance of the relativistic formalism leads naturally to density-dependent interactions between nucleons. Furthermore, non-relativistic reduction reveals non-central and non-local aspects inherent in the Hartree formalism. The success of this simple relativistic Hartree approach is attributed to these features of the interaction.