Shell Model

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 Nuclei

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 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 have a magic number of 50 protons, and the atom 54Fe has a magic number of 28 neutrons. Some nuclei, 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."

Shell Model of Atom

Shell atomic model was first proposed by physicists J Hans D. Jensen and Maria Goeppert Mayer. In the atomic shell model, the electrons in the atom, which are the negatively charged particles, are considered as the occupying diffuse shells within the space that is surrounding the dense, charged nucleus. In this model, the primary shell is seen to be the closest to the nucleus, whereas it is seen to be extending outwards from the nucleus, overlapping with each other. 

These shells are often referred to with the use of capital letters, where K is associated with the primary shell, L is used for the second cell, M for the third shell, and so on. The number of electrons that occupy the shell is present in the sequence of 2,8,18,32,50,72, and 98.  The lightest element, that is hydrogen, is seen to possess one electron in the first shell, while the heaviest elements present in their normal states are seen to have only the first four shells completely occupied with electrons, while the next three shells are only partially occupied. 

The atomic shell model will help in the understanding of the chemical properties of the atom. This chemical property is explained with the help of how the shells are occupied with the electrons. For instance, helium with the atomic number 2 has the first shell full. While neon with an atomic number 10, has a full first and second shell, with eight electrons in its outermost shell. Other atoms with eight electrons present in their outermost shell, even though it might not be full, go on to resemble helium and neon chemically in their relative stability and inactivity. 

The maximum number of atoms that can occupy the energy state in an atom can be figured using the given formula, 

Electron capacity= 2n₂, where n represents the principal quantum number. 

Given below is the table representing the shell levels and their electron capacity.

Table for Energy Level with Designation

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 that 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 stable 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 modelling 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 of a neutron, and this is because of Coulomb’s 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 reaction, 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 evidence of shell structure. For example, the nuclei around 208Pb.

Importance of Shell Model

When the single or paired nucleons get excited out of the bottom state, they tend to modify the momentum of the nucleus as well because of its parity and isospin projection quantum numbers. The shell model helps to describe the proportion of energy required in order to maneuver nucleons from one orbit to another. It also helps to understand the ways in which quantum numbers undergo change. 

Limitations of the Shell Model

• The first limitation that one notices is that there is a difference between shell-model wave functions and the real state of the nucleus.

• Apart from this, the large value of the quadrupole moment seen in many nuclei is hard to explain using this method. 

• The strong-spin orbit interaction cannot be applied using this method.

• The shell model has limited applications and cannot be applied to heavy nuclei.

This is all about the shell model of the atom and its nucleus. Understand how the neutron and protons are configured inside the nucleus. Also, learn how the electrons are distributed in the shells following a particular pattern.