Orbitals shell model

Empirical log /rvalues for some P decays are presented in O Table 2.10. The first column shows the P transition and the probable valence-orbit shell-model configurations in the mother and daughter nuclei. The next columns give the log ft value, the spin (A/ = If Jil), and parity (Atu) changes in the P decay. The table shows that the P transitions can be... 

Figure Cl. 1.2. (a) Mass spectmm of sodium clusters (Na ), N= 4-75. The inset corresponds to A = 75-100. Note tire more abundant clusters at A = 8, 20, 40, 58, and 92. (b) Calculated relative electronic stability, A(A + 1) - A(A0 versus N using tire spherical electron shell model. The closed shell orbitals are labelled, which correspond to tire more abundant clusters observed in tire mass spectmm. Knight W D, Clemenger K, de Heer W A, Saunders W A, Chou M Y and Cohen ML 1984 Phys. Rev. Lett. 52 2141, figure 1.

For two and three dimensions, it provides a erude but useful pieture for eleetronie states on surfaees or in erystals, respeetively. Free motion within a spherieal volume gives rise to eigenfunetions that are used in nuelear physies to deseribe the motions of neutrons and protons in nuelei. In the so-ealled shell model of nuelei, the neutrons and protons fill separate s, p, d, ete orbitals with eaeh type of nueleon foreed to obey the Pauli prineiple. These orbitals are not the same in their radial shapes as the s, p, d, ete orbitals of atoms beeause, in atoms, there is an additional radial potential V(r) = -Ze /r present. However, their angular shapes are the same as in atomie strueture beeause, in both eases, the potential is independent of 0 and (j). This same spherieal box model has been used to deseribe the orbitals of valenee eleetrons in elusters of mono-valent metal atoms sueh as Csn, Cun, Nan and their positive and negative ions. Beeause of the metallie nature of these speeies, their valenee eleetrons are suffieiently deloealized to render this simple model rather effeetive (see T. P. Martin, T. Bergmann, H. Gohlieh, and T. Lange, J. Phys. Chem. 6421 (1991)). 

Although not strictly part of a model chemistry, there is a third component to every Gaussian calculation involving how electron spin is handled whether it is performed using an open shell model or a closed shell model the two options are also referred to as unrestricted and restricted calculations, respectively. For closed shell molecules, having an even number of electrons divided into pairs of opposite spin, a spin restricted model is the default. In other words, closed shell calculations use doubly occupied orbitals, each containing two electrons of opposite spin. 

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 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. 

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. 

Certain numbers of neutrons and protons were recognized by Elsasser (75) as conferring increased stability on nuclei. These numbers are 2, 8, 20, 50, 82, and 126. (The set is sometimes considered to include 28 also.) It was in part their effort to account for these numbers that led Mayer and Haxel, Jensen, and Suess to propose their shell model with spin-orbit coupling. 

These are derived frum a. lensui force resulting from a coupling between individual pairs of nucleons and from the coupling between spin and orbital angular moments of the individual nucleus, as described by the shell model of the nucleus. 

Quantum Number (Orbital). A quantum number characterizing the orbital angular momentum of an electron in an atom or of a nucleon in the shell-model description of the atomic nucleus. The symbol for the orbital quantum number is l. 

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.

Au is a nucleus with 79 protons and 119 neutrons. Filling in the shell model energy level diagram we should find that the highest partially filled orbitals are... 

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. 

Working in the fermion space, even in a considerably reduced shell model space like the S-D subspace, is quite complicated if nucleons are supposed to move in many j-orbits. In recent calculations performed in truncated shell model spaces, for nucleons in a single j-orbit (CAT85), a computational procedure has been set up to evaluate overlaps and matrix elements. The procedure has used recursion formulas which have been... 

Calculations were performed at LLNL using the vectorized shell model code, VLADIMIR [HAU76]. The model space for these calculations included five orbits 1g7/2> 2d5/2> 2d3/2> 33 /2>... 

Conventional spherical shell model calculations have been undertaken to describe 90 88zr and 90 88y in these large scale calculations valence orbitals included If5/2 2P3/2 2Pl/2 and 199/2 The d5/2 orbital was included for 98Y and for high-spin calculations in 98Zr. Restrictions were placed on orbital occupancy so that the basis set amounted to less than 2b,000 Slater determinants. Calculations were done with a local, state independent, two-body interaction with single Yukawa form factor. Predicted excitation energies and electromagnetic transition rates are compared with recent experimental results. 

In principle, the same conclusion on the Nilsson orbital of the 9 Sr g.s. as obtained by the above mentioned T-spectroscopic work can already be drawn from a comparison of easily measurable gross 13-decay features (requiring 5 h measuring time for 13- and n-multiscaling, compared to 3 weeks for -y-spectroscopy) to predictions of our RPA shell model. In the... 

Tab.l. Comparison of experimental J /2 and Pn for Sr to shell model predictions for different odd-particle Nilsson orbitals near the Fermi level... 

Fig. 4. Shell model predictions of Sg distributions of decay (middle part) involving different odd-particle Nilsson orbitals (left part). In the right part, the corresponding model-T1/2 and Pn are listed and compared to experimental data.

---