# Shell Model

## Shell Model in Nuclear Physics

The nuclear shell model for the nucleus of an atom describes the working structure in terms of energy levels by using Pauli's Exclusion Principle. It has its application in both nuclear physics and nuclear chemistry.

Dmitri Ivanenko and E. Gapon proposed the first nuclear shell model in 1932. This model was developed by the combined work of notable physicists Eugene Paul Wigner, J. Hans, D. Jensen, and Maria Goeppert Mayer in the year 1949.

The nucleus shell model is similar to the atomic shell model in a way that it depicts the structure of electrons in an atom for greater stability. When nucleons (protons or neutrons) are added to a nucleus, the binding energy of the upcoming nucleon becomes significantly less as compared to the previously added nucleon, at certain points.

This observation that there are some magic numbers of nucleons (2, 8, 20, 28, 50, 82, 126) being tightly bound in a nucleus as compared to the next higher number, gave rise to the shell model.

### Shell Model of the Atom

Shell model in nuclear physics is similar to the atomic shell model, which describes the arrangement of electrons for attaining greater stability.

When a shell is filled, a nucleus of unusual stability is formed.

Atomic Number = The number of electrons present in an atom

This concept is similar to that of an atom, in which a set of quantum numbers for electrons produces unusual stability–an inert gas. When all the shells of a nucleus are completely filled by protons or neutrons, the number of protons or neutrons is called a "magic number” (2, 8, 20, 28, 50, 82, and 126.)

For example, 116Sn atoms has a magic number of 50 protons, and the atom 54Fe has a magic number of 28 neutrons. Some nuclei, for e.g. 40Ca and 208Pb, have magic numbers of both neutrons and protons; due to that factor, these nuclei have exceptional stability and are called "doubly magic."

### Magic Numbers of Protons and Neutrons

A magic number is actually the number of nucleons present in a nucleus. It corresponds to complete shells within the nucleus of an atom. The nucleus of atoms, which consists of such magic numbers, usually has higher average binding energy per nucleon than that of the predictions based on the mass formula of von Weizsaecker.

Nuclei which have closed shells are more tightly bound in the atom as compared to the next higher number. The closing of shells occurs at Z or N = 2, 8, 20, 28, (40), 50, 82, 126.

It is found that the nuclei, which have an even number of protons and neutrons, are comparatively stablier than those with an odd number of protons and neutrons. The Nuclei with either a neutron number or proton number equal to one of the magic numbers is called doubly magic“, and such nuclei are found to be stable.

Doubly Magic: 2He4 8O16 20Ca40 20Ca48 82Pb208

### Shell Model Energy Levels

This concept leads to the quantization of energy in a similar manner, similar to the square well and harmonic oscillator potentials.

Since the study of these details well determines the energies, much effort has been put into constructing a potential well to observe the modeling of the nuclear energy levels. Solving for the energies for such potential results in a series of energy levels mentioned below.

The labels on the energy levels are quite different from the corresponding symbols of atomic energy levels. The energy levels increase with an increase in orbital angular momentum, quantum number l, and the orbits of s, p, d, f... symbols are used for l = 0,1,2,3... just like in the atomic case.

However, there is no similarity between the principal quantum number n and the energy levels. So, the numbers related to the level start at n=1 for the lowest level and increase with an increase in energy levels.

The quantum number of orbital angular momentum is not limited, like in the atomic structure.

There is a spin-orbit interaction that splits the energy levels by an amount, which increases with an increase in orbital quantum number. This leads to an overlapping of the energy levels.

The subscript on the symbol indicates the value of the total angular momentum j, and the multiplicity of the state is given by 2j + 1.

Proton's contribution to the energy level is different from that by a neutron, and this is because of Coulomb repulsion. However, there is a little difference in the set of energy levels.

### Evidence for Shell Structure

Evidence of shell structure can be seen in two ways:

1. Some nuclear reactions either add a nucleon or remove a nucleon from the closed shell nucleus. The most sensitive of this type of reaction is the electron knockout reactions, in which an electron comes in and an electron, or a proton or a neutron escapes from the shell. This is usually denoted by (e,e′p) (e,e′n) reactions. These reactions have clear evidence of peaks at the single-particle energies.

If we observe nuclei one particle away from a doubly magic nucleus, we can get the evidence of shell structure. For example, the nuclei around 208Pb.

Q1. Why is the Nuclear Shell model important?

Ans- The nuclear shell model works due to the tight binding and simplicity of closed shells. The working of the nuclear shell model is based on angular momentum and Pauli principle. The shell model describes important features of the nucleus with a strong nuclear force. Nuclei have tightly bound closed shells for protons and neutrons.

Q2. What are the limitations of the Shell Model?

Ans- The Large value of quadrupole moment ‘Q’ cannot be explained with this model. The strong spin-orbit interaction is also not applicable in this model.

Q3. Why is it called the Nuclear Model?

Ans- It was the first model of Rutherford to feature a nucleus at its model, and hence it is named so.

Q4. Difference between Shell structures of Nuclei and Atoms.

Ans-

 Shell Structure of Nuclei Shell Structure of Atom Energy levels start from zero state Energy levels start from the first state Magic numbers are, Z = 2, 8, 20, 28, 50, 82, 126 etc Magic numbers are, Z = 2, 10, 18, 36, 54, 86 etc The average potential of nucleus is different from that of an atom Average potential varies Particles of shell structure of a nuclei are neutrons and protons The shell structure particle of atom are electrons
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