The Shell Model

The first steps toward the understanding of the nature of the chemical bond could not be taken until the composition and structure of atoms had been elucidated. The model of the atom that emerged from the early work of Thomson, Rutherford, Moseley, and Bohr was of 

Period Z Element n = / Number of Electrons in Each Shell 2 3 4 

The outer shell is called the valence shell because it is these electrons that are involved in bond formation and give the atom its valence. 

On the basis of the shell model, two apparently different models of the chemical bond were proposed, the ionic model and the covalent model. 

The electronic polarizability can be included in the simulation procedure by use of the shell model. In this formalism, the core of an ion is considered to be massive, and to take a charge z e. This is surrounded by a massless shell, with a charge z e, where 

The core and the shell are bound together by a spring. When an ion becomes polarized, the shell is displaced with respect to the core so that the center of the shell is no longer coincident with the core (Fig. 2.8b). The spring is assumed to 

The single pair interactions are now each replaced by four pair interactions core ion /-core ion j, shell ion /-shell ion j, shell ion /-core ion j, and core ion /-shell ion j (Fig. 2.8c). Equation (2.17) now becomes 

With this modification, simulations give good agreement with experimental quantities where these are available. 

The shell model can be modified to take into account further aspects of the solid being modeled. In particular, angular-dependent terms can be added. These are important in materials such as silicates where the tetrahedral geometry about the silicon atoms is an important constraint. These aspects can be explored via the sources listed in Further Reading at the end of this chapter. 

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Here,. Ai(X) is the partial SASA of atom i (which depends on the solute configuration X), and Yi is an atomic free energy per unit area associated with atom i. We refer to those models as full SASA. Because it is so simple, this approach is widely used in computations on biomolecules [96-98]. Variations of the solvent-exposed area models are the shell model of Scheraga [99,100], the excluded-volume model of Colonna-Cesari and Sander [101,102], and the Gaussian model of Lazaridis and Karplus [103]. Full SASA models have been used for investigating the thermal denaturation of proteins [103] and to examine protein-protein association [104]. 

THE CLOSE-PACKED-SPHERON MODEL OF ATOMIC NUCLEI AND ITS RELATION TO THE SHELL MODEL... 

To avoid confusion with the shells of the shell model of the nucleus we shall refer to the layers of spherons by special names the mantle for the surface layer, and the outer core and inner core for the two other layers of a three-layer nucleus. 

The Relation between the Shell Model and Layers of Spherons.—In the customary nomenclature for nucleon orbitals the principal quantum number n is taken to be nr + 1, where nr> the radial quantum number, is the number of nodes in the radial wave function. (For electrons n is taken to be nT + l + 1.) The nucleon distribution function for n = 1 corresponds to a single shell (for Is a ball) about the origin. For n = 2 the wave function has a small negative value inside the nodal surface, that is, in the region where the wave function for n = 1 and the same value of l is large, and a large value in the region just beyond this surface. 

The nature of the radial wave functions thus leads us to the following interpretation1 of the subshells of the shell model ... 

The Structural Basis of the Magic Numbers.—Elsasser10 in 1933 pointed out that certain numbers of neutrons or protons in an atomic nucleus confer increased stability on it. These numbers, called magic numbers, played an important part in the development of the shell model 4 s it was found possible to associate them with configurations involving a spin-orbit subsubshell, but not with any reasonable combination of shells and subshells alone. The shell-model level sequence in its usual form,11 however, leads to many numbers at which subsubshells are completed, and provides no explanation of the selection of a few of them (6 of 25 in the range 0-170) as magic numbers. 

The close-packed-spheron theory8 incorporates some of the features of the shell model, the alpha-particle model, and the liquid-drop model. Nuclei are considered to be close-packed aggregates of spherons (helicons, tritons, and dineutrons), arranged in spherical or ellipsoidal layers, which are called the mantle, the outer core, and the inner core. The assignment of spherons, and hence nucleons, to the layers is made in a straightforward way on... 

I assume that in nuclei the nucleons may. as a first approximation, he described as occupying localized 1. orbitals to form small clusters. These small clusters, called spherons. arc usually hclions, tritons, and dincutrons in nuclei containing an odd number of neutrons, an Hc i cluster or a deuteron may serve as a spheron. The localized l.v orbitals may be described as hybrids of the central-field orbitals of the shell model. 

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. 

Jacucci G, McDonald IR, Singer K (1974) Introduction of the shell model of ionic polarizability into molecular dynamics calculations. Phys Lett A 50(2) 141—143... 

Lindan PJD (1995) Dynamics with the shell-model. Mol Simul 14(4-5) 303-312... 

Scientists have known that nuclides which have certain "magic numbers" of protons and neutrons are especially stable. Nuclides with a number of protons or a number of neutrons or a sum of the two equal to 2, 8, 20, 28, 50, 82 or 126 have unusual stability. Examples of this are He, gO, 2oCa, Sr, and 2gfPb. This suggests a shell (energy level) model for the nucleus similar to the shell model of electron configurations. 

These shell closures have a profound influence on nuclear properties, in particular the binding energy (adding terms not accounted for in Eq. 2.2), particle separation energies and neutron capture cross-sections. The shell model also forms a basis for predicting the properties of nuclear energy levels, especially the ground... 

The shell model is used to take into account ... 

The effect of oxygen isotope exchange on the FE transition in STO has been also discussed by Bussmann-Holder using the shell model [13]. 

Thus we obtain for the second term in Eqn (35) the shell-model form of the shell correction, Eqn (30) ... 

The essence of the shell model is that the association constant of the first step is very much smaller than the association constants of all subsequent steps. It was assumed by de Kruif et al. (2002) that, since there is a considerable similarity in various properties of p- and K-caseins, then the k-casein micelle can also be described using the shell model. Nevertheless, Vreeman et al. (1981) had previously suggested that the structure of the... 

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