**Model with future **

*
Fifty years of the nuclear shell model -
Present state and future trends*

*
*
The fiftieth birthday of the nuclear shell model with strong spin-orbit
coupling was celebrated in a conference in one of the two birthplaces. Otto
Haxel, Hans Jensen and Hans Suess
had sent their manuscript about the nuclear shell structure to Physical
Review in April 1949.
It was published in the same volume with the work of Maria Goeppert-Mayer,
who had
simultaneously and independently come to equivalent conclusions. The common
book of
Goeppert-Mayer and Jensen "Elementary theory of nuclear shell structure"
appeared six years later. It was mainly concerning the ground-state
properties of nuclei; the study of reactions
with the removal of nucleons from occupied shells was discussed subsequently.

The main ingredients for the success of the nuclear shell model are the idea of independent-particle motion in nuclei, and strong spin-orbit coupling, which results in a sizeable splitting of the angular-momentum levels and yields the correct sequence of energy levels and shell closures. "Magic" numbers occur when the gaps between subsequent shells are big. The properties of the spin-orbit force are also decisive for the prediction of closed shells in superheavy nuclei and thus, for the synthesis of the heaviest elements in Darmstadt, Berkely and Dubna.

Even today a complete theoretical understanding of the nuclear spin-orbit force - which is much stronger, and of opposite sign than the atomic spin-orbit force - has not been achieved. In nonrelativistic models such as the Skyrme-Hartree-Fock method one uses phenomenological parametrizations; in relativistic mean-field theories the structure of the spin-orbit force can be derived, but its strength depends on an effective mass as a parameter. Models on the quark level account for certain qualitative features such as the much weaker spin-orbit force in hypernuclei.

In the introductory talk of the conference which was organized by Hans Weidenmüller and J. Hufner, Ben Mottelson presented the physical basis and the historical development of the concept of single-particle motion in nuclei. How can one reconcile the picture of independent particles with long mean free path (the basis of the shell model), and the rapid equilibration through the strong nuclear force in the compound nucleus? The conventional explanation of the long mean free path through the Pauli-principle was contrasted here with an investigation of Bose-systems which also exhibit independent-particle motion. The present state of the relativistic foundation of the shell model was discussed by Peter Ring. He showed that on the basis of the Walecka linear mean-field theory, additional nonlinear terms, the isospin-dependence in the spin-orbit term and an adjusted effective mass good agreement with the data can be achieved.

Nonrelativistic shell-model calculations with various effective interactions were presented in several talks and posters. They allow to interpret many single-particle, and also collective phenomena in nuclei. With a realistic G-Matrix that allows for a correct calculation of the spherical mean field, and additional multipol-terms is is possible, for example, to properly calculate the backbending-behavior of medium-heavy nuclei such as Chromium-48 and Chromium-50 (second backbending). Monte-Carlo-Shell-Model (MCSM-) techniques in large configuration spaces with many active particles leading to a wide variety of states have mainly been performed at Caltec (also for finite temperatures), but in other places such as Tokyo (T. Otsuka et al.) as well. These methods can be adapted to the architecture of parallel computers, and they may well contribute to the further development of such machines in the future.

Hans Weidenmüller talked about localization in self-bound many-body systems, starting from ordinary localization theory as formulated by Anderson in 1958 for electron propagation in disordered media. Based on local random matrix theory, an analogous theory for localization in many-body systems such as polyatomic molecules or quantum dots can be formulated. Nuclei, however, do not display localization in Fock-space, but rather randomization (GOE-statistics of level spacings, etc), which corresponds to non-localized states, and can be described by random two-body interactions. However, it may well be that at higher excitation energies - albeit below neutron threshold -localization may occur in heavier nuclei. Signatures for this may be difficult to detect, especially in view of the rapidly increasing density of states.

To classify the states of a system of identical particles, Wigner (1936) and Racah (1940s) had introduced algebraic methods, which were developed further by Arima and Iachello in the 1970s to obtain the Interacting Boson Model (IBM, or Boson Shell Model). These methods allow for a complete classification of nuclear spectra on the basis of symmetries up to relatively high excitation energies - for example, up to 4 MeV in Cadmium-112, as was shown by Francesco Iachello in his talk. The algebraic methods can meanwhile also be used to describe spectra of complex molecules, and possibly also for the classification of vibrational spectra in polymers.

New research branches of traditional nuclear physics were also present at the conference: Such as the determination of the limits of stability, proton radioactivity, new doubly-magic nuclei (Sn-132), experiments with radioactive beams, Halo nuclei (neutron halo Be-11, proton halo P-26), the investigation of short-range correlations through elastic electron scattering (about 70% of the nucleons are typically in single-particle states, 30% in higly correlated states), the investigation of superdeformed states, and their decay via tunneling through a doorway-state to a normal deformed state (Dy-152), spectroscopy in the second and third minimum of actinides (for the first time, the very deep third minimum in U-234 could be measured). In nuclear astrophysics, the efforts to include the microphysical processes as described by the shell model in a consistent fashion into the codes for macrophysical processes such as stellar evolution (element synthesis in the s-process) and supernova explosions (r-process) are continuing. In particular, better effective operators are being developed which should allow for consistent shell-model calculations for all the elements generated through the s- and r-processes.

Referring again to the origins of the shell model, Sharon McGrayne talked about Maria Goeppert-Mayer, her work in the US - which was not always favored by the external conditions -, her discussions with contemporary physicists such as E. Fermi, the evolution of the shell model and her cooperation with Jensen. In a lively talk based on his own recollections, Berthold Stech presented complementary stories about Hans Jensen.

Just in time for the celebrated birthday of the shell model, a collaboration at the Lawrence Berkeley National Laboratory's 88-inch-cyclotron reported the synthesis of element Z=118 in cold fusion reactions of Krypton-86 and Lead-208. This would have been the first member of the elusive "island" of superheavy elements (The result was retracted in July 2001 as being due to a so far undisclosed error in the analysis, see also the corresponding LBL statement ). The successful synthesis of heavy elements with proton numbers 107 to 112 by Peter Armbruster and his colleagues in Darmstadt could already be considered to be a great success of the shell model, because these nuclei exist only due to quantum-mechanical order. The unexpected synthesis of element 118 probably has the consequence that the closed proton shell expected at Z=114 (due to the spin-orbit splitting of the 2f state) is more likely to be at Z=120 - as has been predicted by relativistic mean-field calculations and other models, which calculate a depression of central density in this region.

Parallels between nuclei and metal clusters were investigated by G. Bertsch in a theoretical talk. Clusters are less accessible by spectroscopy than nuclei, but they allow to study completely new phenomena. For example, the electronic shells (with "magic numbers" as in nuclei) are replaced by geometric shells in sufficiently large clusters - in sodium, 1500 atoms. Many other research topics such as photoelectron spectroscopy can be considered.

Finally, D. Wilkinson gave a very detailed survey of the historical development of the shell model, and indicated the research directions that are likely to be relevant in the future.

G. Wolschin

**
Figure**

Maria Goeppert-Mayer and Hans Jensen working on the nuclear shell model (Source: Shell99, Heidelberg, cover)