13 Nuclear Models – 5

Sanjay Kumar Chamoli

1. Nuclear Shell Model

In the nuclear shell model the interaction between the individual 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”. The average nuclear potential arises from the short-ranged attractive nucleon-nucleon interaction and is determined by the shape of the nuclear density distribution. All nucleons move independent of one another in this mean field.

The basic assumption of the shell model is that each nucleon moves in the average force field created by all the other nucleons and the effects of inter-nuclear interactions can be represented by a single-particle potential called central potential as in case of atomic shell model which is then used in the Schrödinger equation to find the distribution of nuclear density. The theoretical understanding of the origin of the nuclear shell structure is based on the assumption of such a dominant spherically symmetric central field of force governing the motion of the individual nucleons in the nuclei. In the theory to be developed, it is assumed that no residual interaction exists between the nucleons.

2. Nuclear Potentials

The shell model of the nucleus presumes that a given nucleon moves in an effective attractive potential formed by all the other nucleons, if we assume that this average central potential V(r) gives rise to such a force, then it is possible to obtain a solution of the Schrödinger wave equation governing the motion of an individual nucleon in this field, making suitable assumption regarding the mathematical form of the potential.

There are two approaches to solve the central potential:

(1) Assume an empirical form of the potential that resembles the nuclear density, e.g. square well, harmonic oscillator, Woods-Saxon etc.

(2) The mean filed is generated self-consistently from the nucleon-nucleon interactions.

Fig. 1: Nucleon interactions in a nucleus and in a mean field of n-n nucleons.

2.1 Different forms of mean potential

Assumptions

To find the suitable mathematical form of the mean potential some assumptions are considered which are given as follows:

(1) Ignore detailed two body interactions, i.e. Interactions between nucleons are neglected.

(2) Each nucleon can move independently in the nuclear potential well.

(3) The mean field is the average smoothed-out interaction with all the other particles.

(4) An individual nucleon only experiences a central force.

(5) It is assumed that the no residual interaction exists between the nucleons.

Different forms of the potential have been used for the calculation of the nuclear energy levels, viz., the square well potential and the harmonic oscillator etc. None of these correspond to the actual potential which probably has a shape intermediate between the two as shown in fig. 2.

All the three different types of potential well candidates are given below.

Fig. 2: Different potential well candidates.

3. Square Well Potential

The square well potential is the simplest form of potential. In the following case, we shall assume a finite three dimensional square well potential whose potential is of the form

Fig. 3: Dependence of square well potential with distance r

In order to find the eigen functions and energy eigenvalues for this system, we have to solve the time-independent Schrödinger equation in a 3-dimensional potential well corresponding to the volume of the nucleus

Where is the energy of any nucleon (neutron or proton), m is its mass (=?_? ?? ?_?), ? (r) is its wave function, and V (r) is its potential energy.

The three dimensional Schrödinger equation for the square well potential can be solved using the spherical polar coordinates.

For spherical polar coordinates wave function can be written as

Since we have a spherically symmetric potential we can separate the wave functions of the particles in the nuclear potential into two parts angular (?_?? ) part and the radial (?_?? ) part. The radial part (?_??) depends only on the radius r, and an angular part ?_??(θ, ∅) depends only on the orientation.

? (?) = ?_?? ?_??(?,∅) (7)

The energy eigenvalues will depend on the principle quantum number, n, and the orbital angular momentum, l, but are degenerate in the magnetic quantum number m. These energy levels comes in ‘bunches’ called “shells” with a large energy gap just above each shell.

It can then be shown that the radial part and the angular part satisfy separate differential equations, which can each be solved by variable separable method.

For r > R the wave function must vanish, so R(r) = 0 in this region. For r < R, we will have the radial equation. On putting the value of potential given by equation 4 in the three dimensional Schrodinger equation, then a separation of variables yields the following radial equation:

where A₀ is a normalization constant and k is related to the energy E. We now find the value(s) of k from the general boundary condition that j_l(kr) = for r = R. This is so provided sin (kR) = 0, giving energy levels that correspond to

From the boundary condition at r = R this gives kR = 4.49, 7.73, (found numerically)

Solutions for l = 2 and larger can be found in a similar way. They all give a series of larger kR values, implying larger values for the energy E = (ħk)²/2m. All these values are given in table 1.

Table 1: Solutions of Bessel Functions

Fig. 4: Representation of solutions of Bessel functions.

Recalling that the degeneracy of each energy level is 2(2l +1), we can build up a table of energy levels and degeneracies – starting from the lowest levels which will be filled first. [Here n =1 denotes the first solution in a sequence, n =2 the second, and so on].

The main quantum number n – corresponds to nodes of the Bessel function: X_nl

The wave number k is

The energy of states is given as

4. Harmonic Oscillator Potential

Let’s assume an infinite three dimensional harmonic oscillator potential of the form

Here m is the nucleon mass, ω is the circular frequency of simple harmonic oscillations of the nucleon.

The time-independent Schrödinger equation is

On solving three dimensional Schrödinger equation for the harmonic oscillator in spherical polar coordinates. If the potential given by equation 13 is substituted in the three dimensional Schrödinger equation, then a separation of variables yields the following radial equation.

From the above equation we can see that energy depends only on N and not on nx, ny or nz. One should notice the degeneracy in the oscillator energy levels.

The quantum number N can be divided into radial quantum number n (1, 2, …) and orbital quantum numbers l (0, 1, 2 …).

In general, we have: ?_? = (? + 1) (? + 2)

Where

N = 2 n_r + l – 2

And n_r = n +1

Value of N turns out to be

N = 2n + l n can take value, n= 0, 1 2, 3,…..

n_r radial quantum number and l orbital quantum number

n_r = 1, 2, 3, 4, …..

l = 0, 1, 2, 3,…….

One can see from these results that a central force potential is able to account for the first three magic numbers, 2, 8, 20, but not the remaining four, 28, 50, 82, 126. This situation does not change when more rounded potential forms are used. The implication is that something very fundamental about the single-particle interaction picture is missing in the description.

Fig. 5: Energy levels of nucleons in (a) a square infinite well, (b) harmonic oscillator potential well.

5. Summary

In the nuclear shell model mean field is provided by the interaction among nucleons in nuclei and is determined by the shape of the nuclear density distribution. To get nuclear magic numbers, different forms of the potentials were suggested for the calculation of the nuclear energy levels, such as the square well potential and the harmonic oscillator well etc. None of these correspond to the actual potential as square well potential and harmonic oscillator potential could reproduce only first two magic numbers. It suggests that nuclear potential is neither perfect square well type not harmonic oscillator type it has a shape intermediate between the two.

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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.

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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.

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