Energy levels in closed shell nuclei

General features. It is necessary to distinguish here between the double closed shell nuclei, i.e. the few nuclei for which both neutron and proton shells are closed, and nuclei in which only one shell is closed. In both cases the energies of the first excited states are exceptionally high, those of double closed shell nuclei being the highest the latter nuclei are also exceptions to the rule (Sect. 68) that the spins of the first excited states of even-even nuclei are 2 units. 

There are a few nuclei for which the neutron shell is closed at iV = 50 or 82 but for which only a proton subshell is closed. Strictly these nuclei are not double closed shell nuclei however, they are of interest in that the level structure of some of them is exceptional. For N = 50, there are two such nuclei, e.g. Sr for which the proton subshell is closed, and Zr , for which both p and p subshells are closed. In Sr the only noteworthy feature is the second excited state which is odd and has spin 31. Zr has some exceptional features which are discussed in Sect. 87. For N = S2, the proton shell is full at Ce (which is excited by the decay of La ). According to Kelly and Wiedenbeck the first and second excited states in this nucleus follow the spin sequence 0—2—4 and, except for their spacing, are not exceptional. The existence of a 0 state in this nucleus at an energy below that of the 1.6 Mev state might well have escaped detection. 

Levels in the Ca isotopes. The levels in the Ca isotopes have been explored with great care by Braams with the aid of the [d, p) reaction and inelastic proton scattering some of the results confirm data derived from /3-decay. 

35 Mev in good agreement with the position of the level found by Braams. This is confirmed by the detection of annihilation radiation by Day from inelastic neutron scattering in Ca, when the neutron energies were above 

36 Mev. A state at 3 73 Mev is strongly excited by the decay of Sc . The decay energy of K is too low to excite Ca . Little else is known about the higher states of this nucleus. 

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Harveys Experiment, energy levels in closed shell nuclei, Harvey s experiment 36I. 

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]. 

The meaning of the quantities evaluated in this way is quite clear. With each occupied one-electron wave-function is associated an energy s which for closed shell systems represents the ionisation potential from that level s) and this ionisation potential is, using the technique of photoelectron spectroscopy (93) measurable in principle and often, already in practice. The total energy can also be calculated, and represents the energy of formation of the system from infinitely separated nuclei (or nuclei with cores) and electrons. Net orbital populations, bond populations and gross populations are readily defined. 

The poor behavior of restricted closed-shell Hartree-Fock calculations upon dissociation to open-shell products does not detract from their utility in the region of equilibrium. The calculated equilibrium geometry is that at which tot is a minimum with respect to the coordinates of the nuclei. Table 3.2 shows the value of this energy for intemuclear distances in the vicinity of the experimental bond length of 1.4 a.u. The calculated minimum energy occurs at 1.346 a.u. This is in error by 4% and errors of similar magnitude can be expected for equilibrium geometries of other molecules at this level of approximation. 

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