Schematic Shell Model

With a general understanding of the form of nuclear potentials, we can begin to solve the problem of the calculation of the properties of the quantum mechanical states that will fill the energy well. One might imagine that the nucleons will have certain finite energy levels and exist in stationary states or orbitals in the nuclear well similar to the electrons in the atomic potential well. This interpretation is 

This filling agrees with the enhanced stability of the lightest nuclei (4He, 160), taking the neutrons and protons in separate orbits, but does not agree with that of heavier nuclei. 

The addition of the spin-orbit term to the nuclear harmonic oscillator potential causes a separation or removal of the degeneracy of the energy levels according to their total angular momentum (j = l + s). In the nuclear case, the states with 

Place the three protons into the lowest available orbital. The protons in the lsi/2 state must be paired according to the Pauli principle, so we have a configuration (ls1/2)2(lp3/2)1- 

Place the four neutrons into their lowest available orbitals. The neutrons should be paired in the partially filled orbital (i.e., in contrast to the case for atomic electrons), giving a configuration of (1 s /2)2( 1P3/2)2- 

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Figure 6.3 Energy level pattern and spectroscopic labeling of states from the schematic shell model. The angular momentum coupling is indicated at the left side and the numbers of nucleons needed to fill each orbital and each shell are shown on the right side. From M. G. Mayer and J. H. D. Jenson, Elementery Theory of Nuclear Shell Structure, Wiley, New York, 1955.

Figure 6.5 Energy level pattern and filling for the exotic nucleus nLi in the schematic shell model.

The hyperactivity of, for example, lipases at low w -values (shown in Fig. 5) is explained by the water-shell-model [2]. The activity of the enzyme at w -values higher than 5 corresponds to its activity in bulk aqueous solutions. There exist two aqueous regions within a reverse micelle, schematically shown in Fig. 6. One is located in the inner part of the reverse micelle and has the same physical properties as bulk water the other is attached to the polar head groups of the surfactant and differs in its physical properties strongly from bulk water. 

Fig. 5. Water-shell-model schematically drafted location of two water parts in a reverse micelle. One is located in the inner part of the reverse micelle and has the same physical properties as bulk water, the other is attached to the polar headgroups of the surfactant...

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.

Fig. 1.17 Segment density distribution of a dendrimer molecule according to the dense-shell model (schematic)...

Shown in Fig 2 is a schematic representation of the shell model states for the region near 1 8Gd. There has been a great deal of interest in this mass region since it was proposed [0GA78] that a shell closure occurs for Z-64. 

Schematic representation of shelL model orbitals near...

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. 

Fig. 2-9. Schematic multizone models for ion solvation in solvents (a) with low degree of order such as hydrocarbons, consisting of solvation shell A and disordered bulk solvent B [98] (b) in highly ordered solvents such as water, consisting of solvation shell A with immobilized solvent molecules, followed by a structure-broken region B, and the ordered bulk solvent C (Frank and Wen [16]).

The whole procedure of the human blood-cell suspension study is presented schematically in Fig. 50. The TDS measurements on the cell suspension, the volume-fraction measurement of this suspension, and measurements of cell radius are excecuted during each experiment on the sample. The electrode-polarization correction (see Sec. II) is performed at flie stage of data treatment (in the time domain) and then the suspension spectrum is obtained. The singlecell spectrum is calculated by the Maxwell-Wagner mixture formula [Eq. (88)], using the measured cell radius and volume fraction. This spectrum is then fitted to the single-shell model [Eq. (89)] in the case of erythrocytes or to the double-shell model [Eqs (94)-(98)] to obtain flie cell-phase parameters of lymphocytes. 

Fig. 8 Schematic diagram of a cell represented by the shell model, together with a plot of the real and imaginary part of the Clausius-Mossotti factor for a single shelled object...

When an atom or ion is placed in an electrostatic field, its electrons are polarized, as shown schematically in Figure 1. The polarized species then exerts an altered effect on its surroundings in comparison with its unpolarized form. Attempts to account for the substantial polarizability of atoms and ions that exist in many solids have elicited the development of additional potential models. Most notable are the point polarizable ion model and the shell model. 

Figure 7 Schematic representation of the empirical method for deriving short-range potential energy function parameters p, are the computed properties for the system, observed properties, and S the sum of squared deviations from target values p and C,y are potential parameters of Eq. [13] K and fCy are shell model parameters, and qi and qj the charges of component species.

Polarization of ions can be included in one of two ways. The natural approach is to use point ion polarizabilities, which has been successfully explored by Wilson and Madden (1996). An alternative, which has been used for many decades, is the so-called shell model (Dick and Overhauser 1958) as illustrated schematically in Figure 1. This is a simple mechanical model, in which an ion is represented by two particles-a core and a shell-where the core can be regarded as the representing the nucleus and inner electrons, while the shell represents the valence electrons. As such, all the mass is assigned to the core, while the total ion charge (qt = qc + qs) is split between both of the species. The core and shell interact by a harmonic spring constant, Kcs, but are Coulombically screened from each other. The polarizability is then given by ... 

Figure 1. Schematic representation of the dipolar/breathing shell model for polarizability.

F1GURE12.7 Schematic representation of the ellipsoidal core/shell model. The hydrophilic shell contains the more or less twisted and hydrated ethoxy groups of the surfactant molecules. In the diffuse layer nonhydrated ethoxy groups and some hydrophobic parts of the surfactant molecules form an intermediate area. In the inner core only hydrophobic tails of the surfactant molecules are present. Due to the steric hindrance, not all hydrophobic tails can completely stick in this hydrophobic core. (Erom Preu, H., Zradba, A., Rast, S., Kunz, W., Hardy, E.H., and Zeidler, M.D., Phys. Chem. Chem. Phys., 1, 3321,1999. With permission.)... 

Fig. 5 Schematic representation of the states in the nuclear shell model. The oscillator shells on the left are first split into the individual subshells by deviations of the nuclear potential from the harmonic oscillator, before the spin-orbit interaction creates the groupings of states that produce the correct magic numbers above N = Z = 20. The diagram is schematic and not to scale...

Fig. 6 Schematic representation of the shell model potential and the spin-orbit interaction (top) usually taken as proportional to the derivative of the potential, illustrated via a Woods-Saxon potential (bottom) with a nuclear radius Ro and a surface diffuseness a. A few nuclear levels inside the potential are schematically indicated. It is then obvious that the spin-orbit interaction mainly acts near the surface of the nucleus...

The shell model describes the nucleus as a system of independent particles coupled by a residual interaction. This residual interaction is generally complicated, but in the case of particles with the same spin j it takes a particularly simple form. Figure 15 shows the schematic level scheme of a pair of 1 9/2 protons, compared with the experimentally observed level scheme of Po, which, in the shell model, is described as two 1 9/2 protons outside a closed jj, general... 

Fig. 14.5 The dual-shell model for the short- and long-range interactions in nanosolid ferroelectric nanosolid, a Schematic of the high-order exchange bonds loss of an atom in a spherical nanosolid with radius K. Kc is the critical correlation radius. The Vva loss (the shaded portion) is the difference between volumes of the two spherical caps ...

Fig. 12.7 Schematic representation of the simple core-shell model for description of the importance of the disordered surface layer in nanoparticle...

Figure 7. A schematic representation of the microscopic model for the metal/electrolyte solution interface. Shown from top to bottom are an ion that is contact adsorbed with partial loss of its hydration shell, an ion whose hydration shell partially consists of first layer of water molecules, and an ion that is not contact adsorbed.

Fig. 21 Schematic representation of strategies for spin alignment in D/A salts or complexes by application of spin conservation in different electron configurations of interacting molecular orbitals, (a) Typical D/A interaction between two closed-shell D and A, (b and b ) McConnell s proposal, (c) Breslow s extension, (d) Torrance s model, (e) Wudl s model, and (f) Chiang s model for further doping.

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