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2-6: The Nuclear Shell Model

As learned on the preceding page, the nuclei whose proton number Z or neutron number N is one of the magic numbers
2, 8, 20, 28, 50, 82, 126
are especially stable and they correspond to the noble gases in the atomic world. The periodic law and the periodic table are valid in the Atomic World. Similarly, the idea of the periodic table in the nuclear world was also proposed in a rather early time.

[Why the Periodic Law]
As discussed on the page in the present seminar, "Microscopic World -2-: 3-3: The Periodic Law of Elements", the periodic law of elements could be interpreted with the shell structure of the atoms.
Let us review the atomic shell structure again.
The fundamental properties of atoms are determined by the structure of electron distributions surrounding the nucleus at the center of atom. Namely, they are determined by the electron number and their arrangements (i.e. configurations).
The electrons in an atom make a layer structure starting from the central nucleus and going outward. Each of these layers is called a shell, and this is none other than the shell structure.
An electron in atoms is bound by the attractive average potential whose main part is the Coulomb force between the positive charge of the nucleus and the negative charge of the electron. The electrons bound in the atom by the average potential are staying in eigenstates with discrete energy eigenvalues. These eigenstates form the shell structure of which mechanism was explained in detail on the page, "Microscopic World -2-: 3-3: The Periodic Law of Elements". In this shell structure, each shell is occupied by a certain number of electrons according to the Pauli principle. When one of the shells is filled up, a noble gas is formed. This is the reason why the periodic law appears.

[The Nuclear Shell Model]
The idea of the nuclear shell model had been proposed at a rather early time on the analogy of the atomic shell structure. However, there was some doubt on this idea because there were differences between the cases of atoms and nuclei. The differences are as follows:
(1) In the case of an atom, there is a heavy nucleus at the central position of the system, and it attracts light electrons around by the Coulomb interaction. In contrast to this, there is nothing special at the nuclear center.
(2) In the case of a nucleus, it is clear that the whole system can be described in the image of a "raindrop". Namely, the liquid drop model is valid for nuclei. However, such an image is not valid for atoms.
Moreover, the biggest problem is the difference in the magic numbers of both the systems, atoms and nuclei. Although the magic numbers in atoms are
Z = 2, 10, 18, 36, 54, 86,
the magic numbers in nuclei are
Z or N = 2, 8, 20, 28, 50, 82, 126.
There is a big difference between these two kinds of systems, atoms and nuclei. Even if the shell structure is held in both the systems, the average potentials are presumed to be considerably different.
In the case of atoms, if we assume an average potential consisting of the Coulomb attractive potential acting from the +Ze charge of the nucleus (and the averaged Coulomb repulsive potential between electrons), it was rather easy to reproduce the atomic magic numbers, (Z = 2, 10, 18, 36, 54, 86). (See the previous page of the present seminar, "Microscopic World -2-: 3-3: The Periodic Law of Elements").
It was, however, very difficult to find out the average potential which can reproduce the nuclear magic numbers, Z or N = 2, 8, 20, 28, 50, 82, 126.

[Mayer-Jensen's shell Model]
In 1949, M. G. Mayer (USA, 1906 - 72) and H. D. Jensen (Germany, 1907 - 73) et al. independently proposed an average potential which can reproduce the nuclear magic numbers. Thereby the shell structure in nuclei was theoretically established and it became a firm foundation of the Nuclear Structure Theory.
The average potential proposed by Mayer, Jensen et al. includes a very special term called LS-potential. Because of this, Mayer-Jensen's shell model is sometimes called the jj-coupling shell model. The details on the LS-potential are somewhat expertized discussions, so that we omit them here.
In the following figure, the single-particle levels in both the models, the nuclear shell model and the atomic shell model, are compared.
[The Energy Levels in the Nuclear and Atomic Shell Models]
The red levels show the single-particle levels in the Mayer-Jensen shell model for nuclei and the blue levels are those in the atomic shell model.
In each of the red levels, specified numbers of protons and neutrons are filled one by one from the bottom, and in blue levels electrons are filled. The circled numbers indicate the particle number which can be filled up to the levels. They are just the magic numbers.
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