Single-particle shell model

Einteilchen-Schalenmodell, single particle shell model 7, 8. 

The single-particle shell model is used for inclusion of the residual nucleon interactions [60Br39, 71Scl7] (see papers by R. Casten, V. Zelevinsky, and others, references in I/19B1). Strengths of 7-ray transitions in A = 91 — 150 nuclei and decay of first excited states in even-even nuclei were considered in compilations by P. Endt [81En06] and S. Raman [01Ra27]. 

Nilsson (1955) extended the single-particle shell model to deformed potentials. The solutions of the Schrodinger equation then depend on deformation also. In the independent-partide model (Wagemans 1991) the sum of the single-particle energies of an even-even nucleus is given by... 

In the 1950s, many basic nuclear properties and phenomena were qualitatively understood in terms of single-particle and/or collective degrees of freedom. A hot topic was the study of collective excitations of nuclei such as giant dipole resonance or shape vibrations, and the state-of-the-art method was the nuclear shell model plus random phase approximation (RPA). With improved experimental precision and theoretical ambitions in the 1960s, the nuclear many-body problem was born. The importance of the ground-state correlations for the transition amplitudes to excited states was recognized. 

As we have seen, the nucleons reside in well-defined orbitals in the nucleus that can be understood in a relatively simple quantum mechanical model, the shell model. In this model, the properties of the nucleus are dominated by the wave functions of the one or two unpaired nucleons. Notice that the bulk of the nucleons, which may even number in the hundreds, only contribute to the overall central potential. These core nucleons cannot be ignored in reality and they give rise to large-scale, macroscopic behavior of the nucleus that is very different from the behavior of single particles. There are two important collective motions of the nucleus that we have already mentioned that we should address collective or overall rotation of deformed nuclei and vibrations of the nuclear shape about a spherical ground-state shape. 

Before discussing the recent developments of the model, let me remind you of the main components of the DDM (1) The starting point is the spherical shell model of Mayer and Jensen, where the single-particle level energies are taken from the experimental spectra of odd-A nuclei with one particle (or hole) outside a closed shell. There is a single level scheme for all nuclei. Our version can be found in Table I of [KUM77]. 

By studying the a decay of mass-separated nuclei in the region around Z=82 we have extended the knowledge of shell-model intruder states. The allowed a-decay branches of the odd-odd Bi nuclei connect across the Z=82 shell closure initial and final states of the same single-particle character. 

Shell model calculations predict a quasi-shell closure at 96Zr. Therefore, it is of interest to measure g-factors of states in 97Zr and test whether they can be described by simple shell model configurations. The 1264.4 keV level has a half-life of 102 nsec, and its g-factor was measured by the time-differential PAC method at TRISTAN [BER85a]. The result, g-0.39(4), is consistent with the Schmidt value of 0.43, which assumes no core polarization and the free value for the neutron g factor, g g free. This indicates that the 1264.4 keV level is a very pure single-particle state, thus confirming the shell model prediction of a quasi-shell closure at 96Zr. 

Apart from the demands of the Pauli principle, the motion of electrons described by the wavefunction P° attached to the Hamiltonian H° is independent. This situation is called the independent particle or single-particle picture. Examples of single-particle wavefunctions are the hydrogenic functions (pfr,ms) introduced above, and also wavefunctions from a Hartree-Fock (HF) approach (see Section 7.3). HF wavefunctions follow from a self-consistent procedure, i.e., they are derived from an ab initio calculation without any adjustable parameters. Therefore, they represent the best wavefunctions within the independent particle model. As mentioned above, the description of the Z-electron system by independent particle functions then leads to the shell model. However, if the Coulomb interaction between the electrons is taken more accurately into account (not by a mean-field approach), this simplified picture changes and the electrons are subject to a correlated motion which is not described by the shell model. This correlated motion will be explained for the simplest correlated system, the ground state of helium. 

Meldner, H. Predictions of New Magic Regions and Masses for Super-Heavy Nuclei from Calculations with Realistic Shell Model Single Particle Hamiltonians . In Ref. [5], pp. 593-598. 

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