12 Nuclear Models – 4

Sanjay Kumar Chamoli

1. Nuclear Shell Model

There are various nuclear models which have been proposed from time to time are able to explain only some limited features of the nucleus. The liquid drop model is able to explain the observed variations of the nuclear binding energy with the mass number and the fission of the heavy nuclei. However, this model predicts very closely spaced energy levels in the nuclei which are contrary to the observation at low energies. However, in actual it is observed that the low lying excited states in nuclei are actually quiet widely spaced, which cannot be explained on the basis of the liquid drop model. This and certain other observed properties of the nucleus would requires us to consider the motion of the individual nucleons in a potential well which would give rise to the existence of a nuclear shell structure , similar to the electronic shells in the atoms as shown in fig. 1.

As the atomic shell model has been successfully able to explain the finer details of atomic structure, in same way the nuclear shell model can also do the same thing.

The shell model of the nucleus is an attempt to account for the existence of magic numbers and certain other nuclear properties in terms of nucleon behavior in a common force filed. This model is the simplest and the most successful of the so called independent particle models of the nucleus. In this model, it is assumed that the protons and neutrons are considered to be moving in well-defined closed shells.

This model was first suggested by W.M. Elasser, in 1933. Later, it was finally solved by Maria Gopart Meyer in 1948 and independently by O Haxel, J.H.D. Jensen and H.E. Suess in 1949.

The electrons in an atoms may be considered as occupying positions in shells e.g. K, L, M, N etc. designated by the various principal quantum numbers n = 1, 2, 3, 4 etc. Each of these shells has a number of subshells characterized by different values of the azimuthal quantum number = 0, 1, 2, 3,…… (n-1). A subshell of given can contain a maximum of 2(2l + 1) electrons, which means that the s, p, d, f etc. subshells with = 0, 1, 2, 3 etc. can accommodate up to 2, 6, 10, 14 etc. electrons respectively. The degree of occupancy of the outermost shell is what determines certain important aspects of an atom’s behavior. For instance atoms with 2, 10, 18, 36, 54, and 86 electrons have all their electronic shells completely filled. Such electron structures have high binding energies and are exceptionally stable, which accounts for the chemical inertness of the rare gases.

The same kind of effect is observed with respect to nuclei. Nuclei that have 2, 8, 20, 28, 50, 82, and 126 neutron or protons are more abundant in nature than other nuclei of similar mass numbers, suggesting that their structures are more stable.

Fig. 1: Analogy of shell structures in the nuclei with the electronic shell structures in an atom.

The nuclear shell model is similar to the atomic model where electrons arrange themselves into shells around the nucleus. The atomic shell structure is due to the quantum nature of electrons and the fact that electrons are fermions. Since protons and neutrons are also fermions, the energy states of the nucleons are filled from the lowest to the highest as nucleons are added to the nucleus. In the shell model the nucleons fill each energy state with nucleons in orbitals with definite angular momentum. There are separate energy levels for protons and neutrons as shown in fig. 2.

Fig. 2: Nuclear energy levels neglecting Coulomb forces

The ground state of a nucleus has each of its protons and neutrons in the lowest possible energy levels as indicated in fig. 2. Excited states of the nucleus are then described as promotions of nucleons to higher energy levels. This model has been very successful in explaining the basic nuclear properties. As is the case with atoms, many nuclear properties (angular momentum, magnetic moment, shape, etc.) are dominated by the last filled or unfilled valence level.

This nuclear shell model explains the existence of magic numbers which are 2, 8, 20, 28, 50, 82, and 126 and are analogous to the atomic numbers of inert gases. It was found that the nuclei containing magic numbers of protons and neutrons or both shows extra stability and have larger binding energy per nucleon than neighboring nuclei, and when N and Z are both magic the binding energy per nucleon is especially large. This suggests a shell structure, similar to the shell structure in atomic physics, where the noble gases have especially large ionization energies.

2. How Does The Shell Model Work For Nuclei

The interaction between the nucleons in nuclei averages out that result in a potential which depends only on its position but does not depend on time. This potential is called the “nuclear mean field”. A nucleus, therefore, is an example of a self-organizing system in which the nuclear potential emerges from a large number of n-n interactions.

However there is a significant difference in multi-electron atoms the EM potential between the nucleus and an electron dominates the potential of electron-electron interactions. There is no equivalent dominating potential in the nucleus.

3. Experimental Evidences for Existence of Nuclear Shell Structure

Following are the main evidences to show the existence of shell structure within the nuclei.

(a) Nuclei that contain magic numbers of protons or neutrons have much higher binding energy compared to the neighboring nuclei. They also show very high stability, compared to the nuclei containing one more nucleon of the same kind.

Fig. 3: Variation of binding energy per nucleon with mass number.

(b) The naturally occurring isotopes, whose nuclei contain magic number of neutrons or protons, have generally greater relative abundances (> 60%). For example, the isotopes Sr-88 with N=50, Ba-138 with N=82 and Ce-140 with N=82 have relative abundances of 82.56%, 71.66% and 88.48% respectively as shown in fig. 4.

Fig. 4: Higher relative abundances of nuclei containing magic numbers of N, Z.

(c) The number of stable isotopes of an element containing magic number of protons is usually large compared to those others elements. For example, calcium (Z=20) has 6 stable isotopes compared to 3 for argon (Z=18) and 5 for titanium (Z=22). Tin (Sn) with Z=50 has the largest number of 10 stable isotopes compared to 8 for cadmium (Z = 48) and tellurium (Z = 52).

(d) The number of naturally occurring isotones with magic numbers of neutrons is generally large as compared to those in the immediate neighbourhood. For example, the number of stable isotones at N = 82, is 7 compared to 3 and 2 at N = 80 and N = 84 respectively. Similar is the case at N = 20, 28 and 50 which have respectively 5, 5 and 6 isotones. These numbers are greater than in the cases of the neighbouring isotones.

(e) The neutron absorbing cross-sections are very low for the nuclei having magic number of neutron number. Since the neutron shells are completely filled in these nuclei, this indicates that these nuclei are much less likely to absorb an additional neutron as shown in fig. 5.

Fig. 5: Variation of neutron absorption cross section with neutron number N of target.

Fig. 6: Variation of neutron capture cross section with N showing discontinuities at the magic numbers

(f) The stable elements coming at the end of the principal radioactive series are the three isotopes of lead (²⁰⁶Pb, ²⁰⁷Pb, and ²⁰⁸Pb) which all have a ‘magic number’ Z=82 of protons in their nuclei.

(g) Separation energy of a neutron and proton from a nucleus containing a magic number of neutron or proton is large compared to that for a nucleus containing one more neutron and proton. (Separation energy is the minimum energy required for separating one neutron or proton from a nucleus) as shown in fig. 7.

(h) Nuclei with N just one more than the magic number spontaneously emit a neutron. E.g., O-17, K-87 and Xe-137.

(i) Nuclei with magic numbers of neutrons and protons have their first excited state at higher energies than for other neighbouring nuclei.

(j) Excitation probabilities of the first excited state are low for nuclei with magic numbers of neutrons and protons.

(k) Electric Quadrupole moments should be zero for closed shell nuclei since they are spherically symmetric.

Fig.7: Variation of neutron separation energy with neutron number.

(l) Energy of alpha or beta particles emitted by N or Z magic radioactive nuclei is larger than from other nuclei.

(m)If the α- disintegration energies of the heavy nuclei are plotted as function of the mass number A for a given Z, then usually a regular variation is observed till the magic neutron number N = 126 is reached where there is sudden discontinuity. This confirms the magic character of the neutron number 126.

(n) Similar discontinuities are observes amongst the β-emitters at the magic neutron or proton numbers.

Fig.8: Figure indicating number of stable isotopes of an element containing magic number of N or Z.

The experimental results discussed above lend strong support to the existence of shell structures for the nucleus.

To develop a theory of the nuclear shell structure, it is necessary to assume the existence of a potential well within the nucleus. It is known from quantum mechanics that a bound physical system in an attractive potential well can exist in a number of discrete quantum states. This is the case for the electrons in an atom which are acted upon by the Coulomb field of the nucleus. If the interactions between the electrons are neglected then we can regard the field as spherically symmetric. Solving the Schrodinger equation with a potential giving rise to such a field it is possible to find the energy levels for different sets of quantum numbers which determines the electronic shells in the atoms

Summary

The nuclear shall model is an attempt to explain the various observed properties of the nucleus which cannot be explained by the liquid drop model. This model is based on the Atomic shell model. Many experimental evidences clearly show the existence of shell structures in nuclei. The nuclear shall model is successfully able to explain the existence of magic numbers. The nuclei having N or Z as magic number show extra stability and extra features compared to their neighbors.

References:

Introduction to Nuclear Physics – by Keneth S Krane.
Introductory Nuclear Physics – by Samuel S M Wong.
Nuclear Physics – by R R Roy & B P Nigam.
Elementary Nuclear Theory by Hans A. Bethe and Phillip Morrison.
Carl Friedrich von Weizsäcker: Pioneer of Physics, Philosophy, Religion … edited by Ulrich Bartosch
Hans Bethe and His Physics By Gerald Edward Brown, Chang-Hwan Lee
Introduction to Nuclear Physics, 2nd Edition, W.N.Cottingham & D.A. Greenwood.
Concept of Nuclear Physics by B L Cohen, McGraw Hill.
Nuclear Physics ; an Introduction by S.B. Patel.
Exotic Nuclear Excitation by S.C. Pancholi
Nuclear spectroscopy Part B, by Fay Ajzenberg- Selove
Basic Ideas & Concepts in Nuclear Physics – by K Heyde
5. Nuclear Physics by Irving Kaplan, Narosa Publishing House.

Web Links

Did you know ?

The nuclear shell model is partly analogous to the atomic shell model which describes the arrangement of electrons in an atom, in that a filled shell results in greater stability.
When adding nucleons (protons or neutrons) to a nucleus, there are certain points (magic numbers) where the binding energy of the next nucleon is significantly less than the last one.
The nuclei with magic numbers of nucleons: 2, 8, 20, 28, 50, 82, 126 are more tightly bound than the other nuclei.
The observations in support of magic nuclei brought about the origin of the shell model.
The shells for protons and for neutrons are independent of each other. Therefore, one can have “magic nuclei” where one nucleon type or the other is at a magic number, and “doubly magic nuclei”, where both are.
The upper magic numbers are 126 and, speculatively, 184 for neutrons but only 114 for protons, playing a role in the search for the so-called island of stability.
Some semi-magic numbers have been found, notably Z=40 giving nuclear shell filling for the various elements; 16 may also be a magic number.
In order to get these numbers, the nuclear shell model starts from an average potential with a shape something between the square well and the harmonic oscillator.
A more realistic potential is the Wood-Saxon potential with a round edge to avoid discontinuity and a Coulomb field for the charged protons.
To this potential an empirical spin orbit coupling term must be added (with at least two or three different values of its coupling constant, depending on the nuclei being studied) to reproduce all magic numbers.
So, the magic numbers of nucleons, as well as other properties, can be correctly described by approximating the model with a three-dimensional harmonic oscillator plus a spin-orbit interaction. A more realistic but also complicated potential is known as Woods–Saxon potentia
The effect of the spin-orbit interaction term is to split the levels by an amount depending on the orbital quantum number.
The multiplicity of states i.e. different possible orientations of angular momentum, is calculated by the formula 2j + 1, where j is the total angular momentum (orbit plus spin) quantum number designated as an subscript in the diagram.
Experimentally, the shell structure of nuclei was evident by the enhanced abundance of elements for for nuclei which Z or N is a magic number.
The fact that the stable elements at the end of the naturally occurring radioactive series all have a “magic number” of neutrons or protons, strengthen the shell structure in nuclei.
For magic nuclei, the neutron absorption cross-sections are much lower than surrounding isotopes.
For magic nuclei, experimentally, it is found that the binding energy for the last neutron is a maximum and drops sharply for the next neutron added.
Electric quadrupole moments (a measurement for deviation from spherical distribution) are near zero for magic number nuclei.
The excitation energy from the ground nuclear state to the first excited state is greater for closed shells.
The problem with the shell model is in the region of the rare-earth nuclei. The quadrupole moments predicted from the orbital motion of the individual protons are much smaller than those observed.
From the shell model point of view, the rare-earth nuclei lie about midway between the neutron magic numbers 82 and 126. This is the region for which shell model calculations are the most difficult since there are many particles outside a closed shell.

you can view video on Nuclear Models – 4

Biography: