Shell model deformable

The close-packed-spheron theory of nuclear structure may be described as a refinement of the shell model and the liquid-drop model in which the geometric consequences of the effectively constant volumes of nucleons (aggregated into spherons) are taken into consideration. The spherons are assigned to concentric layers (mantle, outer core, inner core, innermost core) with use of a packing equation (Eq. I), and the assignment is related to the principal quantum number of the shell model. The theory has been applied in the discussion of the sequence of subsubshells, magic numbers, the proton-neutron ratio, prolate deformation of nuclei, and symmetric and asymmetric fission. 

A number of different atom-centered multipole models are available. We distinguish between valence-density models, in which the density functions represent all electrons in the valence shell, and deformation-density models, in which the aspherical functions describe the deviation from the IAM atomic density. In the former, the aspherical density is added to the unperturbed core density, as in the K-formalism, while in the latter, the aspherical density is superimposed on the isolated atom density, but the expansion and contraction of the valence density is not treated explicitly. 

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. 

Figure 6.16 Schematic diagram of the splitting of the f7spherical shell model level as the potential deforms. Positive deformations correspond to prolate shapes while negative deformations correspond to oblate shapes.

Given the following shell model state, /ci7/2, show qualitatively how it might split as a function of increasing prolate deformation. Label each... 

Since most nuclei in the region of deformation at A 100 can only be produced with rather low yields which makes detailed spectroscopic studies difficult, we have examined possibilities of extracting nuclear structure information from easily measurable gross 13-decav properties. As examples, comparisons of recent experimental results on Rb-Y and 101Rb-Y to RPA shell model calculations using Nilsson-model wave functions are presented and discussed. 

Toward a Shell-Model Description of Intruder States and the Onset of Deformation... 

In this picture, the excitation of nucleons into shell-model intruder states leads to the coexistence of states with different deformations. Shell-model intruder states for the odd-mass Au isotopes, for example, are presented in fig. 2. Note that the h9 2 and i13y2 intruders drop rapidly as one goes more neutron deficient [ZGA80]. 

A theoretical calculation was carried out to see if all the observed features of the Tl isotopes will emerge from the deformed shell model. First the equilibrium shape of the intrinsic mean field was determined by the... 

In this talk we will briefly review magnetic moment results for excited states in nuclei around closed shells (group a), and their significance to the shell model. Then we will summarize the results for transitional nuclei, and discuss the systematics of g-factors of states at the onset of deformation around A=100 and A 150. 

There exist a variety of extensions of the basic shell model. One variation for molecular systems uses an anisotropic oscillator to couple the core and shell charges,thus allowing for anisotropic polarizability in nonspherical systems. Other modifications of the basic shell model that account for explicit environment dependence include a deformable or breathing shelF ° and shell models allowing for charge transfer between neighboring sites. 

In the shell model, the electron cloud is treated as a spherical shell connected to the center of mass of ion with a certain spring 2. In this method the deformation of the electron cloud is not reproduced, since the electron cloud is fixed as the spheric shape, although the real polarizability is brought from the deviation of the electron cloud from the spheric shape. 

Computations of minimum-energy configurations for some off-centre systems were first carried out on the basis of polarizable rigid-ion models, mainly devoted to KChLi" " [95,167-169]. Van Winsum et al. [170] computed potential wells using a polarizable point-ion model and a simple shell model. Catlow et al. used a shell model with newly derived interionic potentials [171-174]. Hess used a deformation-dipole model with single-ion parameters [175]. At the best of our knowledge, only very limited ab initio calculations (mainly Hartree-Fock or pair potential) have been performed on these systems [176,177]. 

The assumption of spherical symmetry is only reasonable for closed-shell systems, to which our discussion will apply. For open-shell systems, departures from sphericity occur due to the Jahn-Teller effect, and can be described by analogy with the deformed-shell model of nuclear physics [687], but lie beyond the scope of the simple theory described here. 

The N = 83, 85 and 87 isotones above the neutron-shell closure at N = 82 are described essentially by the f7 shell-model state. Calculations withing the particle-rotor model of Larsson et al. [77], assuming small nuclear deformations of the core, account qualitatively for the measured nuclear moments (cf. Fig. 3). The variation in the spectroscopic quadrupole moments reflects the successive filling of the f j neutron shell. This is realized from the formula [78] ... 

As to the ground-state configuratiais of the odd-A europium isott s, the predominant part of the wave function in Eu is due to the shell model proton state d5,2> whereas in the strongly deformed - Eu, there is a change to the Nilsson orbital [413 5/2]. The increasing spectroscopic quadmpole monents on each side of N = 82 indicate a positive projection factor in Eq. 5, and thus, for positive deformations. 

The doubly-odd europium isotopes in the range Eu are well described by the dj proton state coupled to the different neutron-shell model states, discussed above in connection ivith the odd-neution nuclei d in Eu, h Q in Eu and in i4 -i5ogjj gy usjug g-factors of the neighbouring odd-A nuclei and the additivity theorem, tiie magnetic moments of these doubly-odd europium isotopes are weU reproduced. The strongly deformed Eu and Eu are shown to be due to the configurations 3 (p[413 5/2] n[505 11/2]) and 0 (p[413 5 ] n[642 5/2]), respectively. 

Furthermore, the spin I = S/2,7/2 and 9/2 states discussed here are built up mainly of sublevels from dj/j, Z n and h, j, respectively. One may therefore conclude that the nuclear structure of mesc we y deformed nuclei are described by relatively pure shell-model states, as evidence also by the measured spins and magnetic moments, and that the variations in occupation of the corresponding sublevels with increasing deformation account for the observed changes in the spectroscopic quadrupole moments. 

---