Damping of high-lying single-particle modes in heavy nuclei

The recent experimental and theoretical results on the damping of high-lying single-particle modes in heavy nuclei are reviewed. In one-nucleon transfer reactions these states manifest themselves as broad “resonance”-like structures superimposed on a large continuum. The advantages and the limitations of the transfer reaction approach will be presented using the results from neutron and proton pick-up and stripping reactions. The problem raised by the subtraction of the underlying background, the assumptions made to describe the reaction process and the method used to extract the strength distributions are presented. The existing empirical systematics is summarized for nuclei ranging from ⁹⁰Zr to ²⁰⁸Pb.The theoretical approaches used to explain the damping of the high-lying single-particle modes are based on the coupling between collective and single-particle degrees of freedom. In a first step the bare single-particle mode is spread over several doorway collective states due to the interaction with surface vibrations. In a second step the doorway states spread their strengths over many other degrees of freedom. These two steps of the damping mechanism are discussed in detail within the framework of the quasiparticle-phonon nuclear model. A large-scale comparison between the measured and calculated average energies, spreading widths and spectroscopic strengths of the high-lying single-particle (hole) states in heavy nuclei is presented. The systematic features of the damping (energy, angular momentum and isotopic dependence) are discussed. Recent advances of the experimental approaches, such as the γ-decay of the high-lying states or the use of heavy-ion transfer reactions at intermediate energies, are outlined.The detailed study of the damping mechanism of high-lying single-particle modes reveals new features and leads us to a new field in nuclear structure: “the spectroscopy of inner and outer subshells”.     

Estimating the occupation probabilities of single-particle orbits in nuclei

Occupation probabilities of neutron and proton single-particle orbits are estimated for a number of spherical and close to spherical nuclei with 20 ≤ Z ≤ 50 and 20 ≤ N ≤ 82 near the Fermi energy. The estimates are made according to the formula of the BCS theory with single-particle energies calculated using the mean-field model with dispersive optical potential. The closeness of the occupation probabilities to 0 and 1 is demonstrated for nuclei with traditional and new magic numbers.     

Separable single-particle Hamiltonian and its local equivalent

By using a method developed by Coz, Arnold and MacKellar the local equivalent of a single-particle potential discussed in a previous work is constructed and analyzed. The accuracy of the method is checked in the present case. The numerical examples refer to the low-energy elastic scattering of neutrons from nuclei described by a hole in a closed shell. The single-particle bound states are approximately described by harmonic oscillator wave functions. The results are similar to those derived by Coz, Arnold and MacKellar from non-local Hartree-Fock potentials.     

Diabatic aspects of single-particle motion within the time-dependent Hartree-Fock theory

Time-dependent Hartree-Fock (TDHF) calculations for central collisions of⁸⁶Kr on¹⁶⁶Er are investigated with respect to quasi-stationary properties for various energies in the center-of-mass system. A quasi-stationary potential as well as quasi-stationary single-particle levels are extracted as functions of the relative distance between the two nuclei. A quantitative comparison with the diabatic two-center shell-model yields surprising agreement with the results from TDHF-calculations for both the nucleus-nucleus potential and the single-particle levels. From this we conclude that the approach phase of a nucleus-nucleus collision can be well described within a diabatic approximation for the single-particle motion.     

Relativistic description of single-particle resonances via phase shift analysis

Single-particle resonant states in spherical nuclei are studied by the real stabilization method in coordinate space within the framework of self-consistent relativistic mean field theory. Taking 122Zr as an example, the resonant parameters, including the energies and widths are extracted by fitting energy and phase shift. Good agreement with the previous calculations has been found. The details of single-particle resonant states are analysed.     

Relativistic description of single-particle resonances via phase shift analysis

Single-particle resonant states in spherical nuclei are studied by the real stabilization method in coordinate space within the framework of self-consistent relativistic mean field theory. Taking 122Zr as an example, the resonant parameters, including the energies and widths are extracted by fitting energy and phase shift. Good agreement with the previous calculations has been found. The details of single-particle resonant states are analysed.     

Redundancy-free single-particle equation-of-motion method for nuclei: (I). Intermediate coupling calculations for odd-mass near-spherical nuclei

The problem of coupling an odd nucleon to the collective states of an even core is considered in the intermediate-coupling limit. It is now well known that such intermediate-coupling calculations in spherical open-shell nuclei necessitate the inclusion of ground-state correlation or backward coupling which gives rise to an overcomplete basic set of states for the diagonalization of the hamiltonian. In a recent letter¹), we have derived a technique to free the single-particle equation-of-motion method of redundancy. Here we shall apply this redundancy-free equation-of-motion method to intermediate-coupling calculations in two regions of near-spherical odd-mass nuclei where forward coupling alone has not been successful. It is shown that qualitative effects of backward coupling previously reported are not spurious effects of double counting, although they are significantly modified by the removal of redundancy. We also discuss what further modifications of the theory will be needed in order to treat the dynamical interplay of collective and single-particle modes in nuclei self-consistently on the same footing.     

On the magnitude of single-particle proton and neutron separation energies in the chains of isotopes and isotones near closed shells

Single-particle separation energies of protons or neutrons in nuclei near the closed shells are examined in the chains of isotopes or isotones. From the observed dependences we determine spins and parities of the single-particle orbitals and their sequence in nuclei far from the stability line, where direct experimental information on single-particle characteristics is not available at present.Received: 18 November 2003, Published online: 26 May 2004PACS: 21.10.Dr Binding energies and masses - 21.60.Cs Shell model - 21.10.Pc Single-particle levels and strength functions     

Mean distribution of single-particle levels in thin-skinned potential wells and the macroscopic level density parameters of nuclei

A general leptodermous expansion for the density of single-particle levels in thin-skinned potential wells is derived and is used to study the finite size corrections to the macroscopic level density parameters of nuclei. With droplet model values for the potential parameters, the calculated level density parameters for nuclei along the β-stability line show systematic deviations from the experimental estimates. Possible reasons for these deviations are also discussed.     

Level densities for nuclei nearA=60 based on realistic single-particle level spectra

We report on microscopic level density calculations for (1f, 2p)-shell nuclei. The agreement with experimental data can be improved considerably if the underlying single-particle levels are adjusted to their observed positions near the Fermi surface.     

The systematics of nuclear single-particle states: (III). Widths and rearrangement energies

Widths and rearrangement energies are obtained from fragment distributions of single-particle states determined from one-nucleon pickup reactions on nuclei in the range 6 ≦ A ≦ 65. These show systematic features that are compared with the results of Brueckner-Hartree-Fock calculations.     

Role of the nuclear surface in a unified description of the damping of single-particle states and giant resonances

The damping widths of single-particle states and of giant resonances are estimated in spherical nuclei, based on the excitation of surface modes.A Skyrme III interaction with an effective mass consistent with that resulting from infinite nuclear matter calculations with “realistic” forces $\text{(m}{}^1\text{/m = 0.76)}$, was utilized. The single-particle basis needed to construct the unperturbed nuclear response function for each multipolarity was obtained, treating this force in the Hartree-Fock approximation. Diagonalizing a schematic interaction in this basis, the surface modes were calculated. They are used to dress the single-particle and single-hole states and to renormalize the vertex interaction, taking into account the proper energy dependence of the couplings.The essential new feature of the present calculation as compared to the calculations reported in ref.¹) is that the energy dependence of the real and imaginary part of the self-energy is taken into account. This is done utilizing a strength function model.About 70 % of the damping widths arise from the coupling to specific intermediate states containing one low-lying collective surface vibration. The rest, from the coupling to many nonspecific states.Qualitative agreement is found with the experimental data for spherical nuclei throughout the mass table for both the single-particle states and the giant resonances. The model seems however to predict widths which are smaller than those experimentally observed.     

The systematics of nuclear single-particle states (II). the region A < 35

The energies of single-particle states in nuclei with A < 35 are obtained as eigenvalues of a local Saxon-Woods potential with depth depending linearly on A and on the nuclear symmetry parameter.     

An improved single-particle basis for nuclear structure studies far from stability

A new single-particle basis is proposed for use in weakly bound nuclei far from the valley of beta stability.The basis, obtained by applying a local-scaling point transformation to the states of a harmonic oscillator potential, can be tailored to have the correct asymptotic properties for weakly bound systems.We first present a test of the basis and then apply it in Hartree-Fock-Bogolyubov calculations of the even Mg isotopes, from the proton drip line to the neutron drip line.     

The systematics of nuclear single-particle states (I). The region 35 ≦ A ≦ 65

The energies of single-particle states in nuclei with 35 ≦ A ≦ 65 are obtained as eigenvalues of a local Saxon-Woods potential with depth depending linearly on A and on the nuclear symmetry parameter.     

Collective octupole vibration and proton single-particle levels in zirconium nuclei

Self-energy correction to the shell model single-particle motion, arising from the excitation of octupole vibration in the intermediate state, accounts quite well for the energy shifts of the 2p _1/2 and 1g _9/2 proton orbits in zirconium nuclei.     

复标度方法对原子核单粒子共振态的研究

在相对论平均场理论框架下, 利用复标度方法研究了 Zr 同位素的单粒子共振问题. 以 ¹²²Zr 为例, 演示了复标度方法确定共振态的具体步骤, 确定了 ¹²²Zr 所有可能共振态的能量和宽度, 以及相应共振态的复标度波函数, 并与耦合常数的解析延拓方法进行了比较.在此基础上, 进一步系统研究了 Zr 同位素的共振问题, 获得了与散射相移方法一致的结果.     

On the influence of the shell structure of single-particle levels in dissipative heavy-ion collisions

The effect of the shell structure of the single-particle levels on energy damping due to nucleon exchange processes in deep-inelastic heavy-ion collisions is investigated. From a study in different closed- and open-shell nuclei, it is inferred that the shell structure does not play a significant role in the experimentally-observed energy loss per nucleon transfer.     

Quadrupole moments of highly deformed structures in the A approximately 135 region: probing the single-particle motion in a rotating potential

The latest generation gamma-ray detection system, GAMMASPHERE, coupled with the Microball charged-particle detector, has made possible a new class of nuclear lifetime measurement. For the first time differential lifetime measurements free from common systematic errors for over 15 different nuclei ( >30 rotational bands in various isotopes of Ce, Pr, Nd, Pm, and Sm) have been extracted at high spin within a single experiment. This comprehensive study establishes the effective single-particle transition quadrupole moments in the A approximately 135 light rare-earth region. Detailed comparisons are made with theoretical calculations using the self-consistent cranked mean-field theory which convincingly demonstrates the validity of the additivity of single-particle quadrupole moments in this mass region.