The Shell Model I

Electrons in atoms are attracted to the nucleus by a Coulombic force. Thus, energy must be supplied (by some means) if the electron is to be pulled away from the nucleus, thereby creating a positively charged species, or cation, and a free electron. For real atoms, the ionization energy (IE) of an element is the minimum energy required to remove an electron from a gaseous atom of that element. 

How much total energy would it take to remove the electrons from a mole of H atoms Write this energy in MJ/mole. 

In CA 1, the electrons were distributed around the nucleus at various distances. 

Predict the relationship between lEi and atomic number by making a rough graph of lEi vs. atomic number. DO NOT PROCEED TO THE NEXT PAGE UNTIL YOU HAVE COMPLETED THIS GRAPH. 

Based on our previous examination of ionization energies, it is expected that the ionization energy of an atom would increase as the nuclear charge, Z, increases. In addition, the ionization energy of an atom should decrease if the electron being removed is moved farther from the nucleus (that is, if d increases). 

---

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]. 

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. 

That this value is reasonable is shown by comparison with the magnitude increase of e with thermal expansion of the a-Fe203 lattice, i.e., (Ae/e)/(Aa/a) 80 (154). Finally, Vaughan and Drickamer (155) found the magnitude of e to decrease by a factor of two by increasing the external pressure to 200 kbar. Thus, the increased value of e (and also the lattice expansion) of 5-nm a-Fe203 particles may be represented in terms of an internal pressure of -200 kbar, as an alternative to that interpretation in terms of the shell model. 

I have not described the calculation of the eigenvalues, which requires the solution of the equations of motion and therefore a knowledge of the force constants. The shell model for ionic crystals, introduced by Dick and Overhauser (1958), has proved to be extremely useful in the development of empirical crystal potentials for the calculation of phonon dispersion and other physical properties of perfect and imperfect ionic crystals. There is now a considerable literature in this field, and the following references will provide an introduction Catlow etal. (1977), Gale (1997), Grimes etal. (1996), Jackson et al. (1995), Sangster and Attwood (1978). The shell model can also be used for polar and covalent crystals and has been applied to silicon and germanium (Cochran (1965)). 

This approach yields the shell model of the atom in which, under the restrictions of the Pauli principle and according to the aufbau principle, the electrons i are placed in the spin-orbitals (r, ms). For example, the shell structure of the magnesium atom is sketched schematically in Fig. 1.1. 

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. 

In the first sum, indices i and denote shells and cations, the second term runs over all point charges in the system, and the third term accounts for interactions of the shells with their cores. The shell model takes into account the polarization of the anions by the crystal field of the solid, which is an important feature. To better reproduce the characteristics of systems with partly covalent bonds, such as zeolites, Eqs. [15] and [16] are supplemented with a term... 

These facts are explained by the shell model, which is based on the following principles. First, one assumes that the electrons are confined by an approximately spherical effective potential. The simplest realistic potential for clusters is of the Woods-Saxon type (a hypothesis which can be further justified - see section 12.8.1 below), i.e. has the form... 

Fig. 7 Nilsson diagram for odd protons, including sublevels of the shell model states gy,2, d 2 and The Fermi levels of jl, Cs andgfii, g Fr are shown to be located at the lower parts of the gj,2, dg and h sublevels, respectively. The theoretical results on the spectroscopic quadrupole moments as a function of d ormation are given to the right. The potential parameters used are k=0.066, p = 0575for I and Cs, and k = 0.060, p = 0.630for Bi and Fr.

Strutinsky developed an extension of the liquid drop model which satisfactorily explains the fission isomers and asymmetric fission. For such short half-lives the barrier must be only 2-3 MeV. Noting the manner in which the shell model levels vary with deformation ( 11.5, the "Nilsson levels"), Strutinsky added shell corrections to the basic liquid-drop model and obtained the "double-well" potential energy curve in Figure 14.14b. In the first well the nucleus is a spheroid with the major axis about 25 % larger than the minor. In the second well, the deformation is much larger, the axis ratio being about 1.8. A nucleus in the second well is metastable (i.e. in isomeric state) as it is unstable to y-decay to the first well or to fission. Fission from the second well is hindered by a 2 - 3 MeV barrier, while from the first well the barrier is 5 - 6 MeV, accounting for the difference in half-lives. 

The Wheeler-Robell analysis envisions the main reaction to be dilfusion-con-trolled, but not the poisoning reaction. Whether or not this is so is a question of relative dimensions of molecules, but dual diffusion control would seem more typical. The analysis has been extended to diffusion-controlled poisoning by Haynes [H.W. Haynes, Jr., Chem. Eng. Sci., 25, 1615 (1970)], who used a shell model as an approximation for rapid poisoning in a Type I system. The Thiele modulus for the poisoning reaction is and shell model assumption to be valid. For a spherical catalyst particle the fraction of original activity, s, is related to the radius of the poison-free zone by... 

Within the shell-model of the electronic structure of clusters of monovalent metals, the ionization potential drops to a low value between sizes and N(, -I-1, where N. indicates a closed-shells cluster. The electron affinity, on the other hand, drops between Nj — 1 and Nc, since the cluster with size N — 1 easily accepts an extra electron to close its nearly-filled external shell. Consequently, the cluster of size N has a large ionization potential and a low electron affinity and will be inert towards reaction. One then expects peaks in a plot of 1 — A versus N for closed shell clusters. The shell effects arc clearly displayed in a Kohn-Sham density functional calculation. Figure 10 shows the results of such a calculation for jellium-like Sodium clusters using the non-local WDA description of exchange and correlation. This calculation employed the Przybylski-Borstel version of the WDA see reference 30 for details). The peaks in I — A occur at the familiar magic clusters with N = 2, 8,18, 20,34,40 and 58. It is well... 

Because one is usually only interested in solving for low-lying states, one first divides the Hilbert space spanned by Wp into two parts, using the projection operators P and Q. The truncated (or shell-model) space is defined by P, while Q defines the space outside the shell-model space consequently, P p = <>p in Eq. (2). It is assumed that the P and Q spaces are non-overlapping, i.e., PQ = 0. 

Equation (8) introduces notation, in which a labels the specific matrix element of rank R, SR(jij ) is a single-particle matrix element for the transition j-, - jf in the impulse approximation, and the quenching factor qJUiJf) corrects SrO jV) for the finite size of the model space and some effects of the nuclear medium so that the effective value of Spijijf) is qa jijf)SR jijf) = SKy ijf, eff). The DR jij[) are the one-body-transition densities that are the result of the shell-model calculations. For the special case of the axial-charge matrix element Mj the defining equation has the emedjijf) of Table 1 incorporated into the sum, i.e.. 

---