14 Nuclear Models-6

Sanjay Kumar Chamoli

1. Woods-Saxon Potential

In order to explain the existence of magic numbers different forms of potentials have been used for the calculations namely the square well potential and the harmonic oscillator potential etc. These forms of potential could reproduce only first two magic numbers. This suggests that none of these correspond to the actual potential; it has probably a shape intermediate between the two. Hence the Woods-Saxon form of potential has also been assumed for the calculation. In the following section, we shall assume a Woods-Saxon potential.

Woods-Saxon potential has the form of

where V₀ is a potential depth of the order of 30 to 60 MeV and R is the radius of the nucleus R ≈ 1.2 A^1/3 fm, and a is the skin thickness and are chosen compatible with experimental results ≈ 0.5 fm.

A realistic form of the nuclear shell model potential for the Woods-Saxon is given in fig. 1, and corresponding energy levels are represented in fig. 2.

N	0	1	2	2	3	3	4	4	4
nl	1s	1p	1d	2s	1f	2p	1g	2d	3s
Degeneracy	2	6	10	2	14	6	18	10	2
States with E ≤ E_nl	2	8	18	20	34	40	58	68	70

It can be seen from the above table that the first three magic numbers (2, 8 and 20) can then be understood as nucleon numbers for full shells. This simple model however does not work for the higher magic numbers as other magic numbers could not be obtained with this potential. For them it is necessary to include the spin-orbit coupling effects which further split the nl shells.

Fig. 1: A realistic form for the shell-model potential for the Woods-Saxon potential.

Fig. 2: Different nuclear energy levels of Woods-Saxon potential

A comparison of energy levels of different types of potential for harmonic oscillator, infinite square well and Woods-Saxon are given in fig. 3.

Fig 3: Energy levels for different types of potential

2. Spin-orbit interaction

One can see from these results that a central force potential is able to account for the first three magic numbers only, 2, 8, 20, but not the remaining four, 28, 50, 82, 126. This situation does not change when other shapes of potential forms are used. For higher levels there are discrepancies in the magic numbers thus we need a more precise model to obtain a more accurate prediction. The implication is that something very fundamental about the single-particle interaction picture is missing in the description.

So we have to improve the previous potential forms in order to reproduce the magic numbers correctly. We should not want to make radical changes because this would destroy the physical content of this potential. Hence in order to predict the higher magic numbers, we need to take into account other interactions between the nucleons. The first interaction we analyze is the spin-orbit coupling.

In order to explain the disagreement at the higher magic numbers, M. G¨oppert Mayer, and independently D. Haxel J. Jensen and H. Suess in 1949 suggested to add a spin-orbit interaction term for each nucleon to the central potential V(r) which solved the problem of finding the magic numbers and gave the suitable separation between the shells.

The spin-orbit potential, which is non-central, can be written as

Here lћ and sћ are the azimuthal and spin angular momenta of the nucleon under consideration. f (r) is a spherically symmetric function giving the profile of the potential (generally Woods-Saxon shape). It is weaker than V(r). ?_?? is a constant.

The S-O (spin-orbit) term in nuclei reduces the energy of states with spin oriented parallel to the orbital angular momentum l while increasing the energy of states with spin oriented opposite to the orbital angular momentum. We can also see from the above equation that this is opposite of S-O interaction in atoms where states with spin oriented opposite to l are lower in energy.

In nuclei the spin-orbit effect comes from an attractive interaction between the orbital angular momentum and the intrinsic spin angular momentum of the nucleon. In case of atoms it arises from the EM interaction of the magnetic moment of an electron with the magnetic field resulting from orbiting a charged nucleus.

We assume strong coupling between the spin and orbital angular momenta of each individual nucleons giving rise to a total angular momentum j for each so that we can write

Since s =1/2 for each nucleon, the two possible values of j are j = l + ½ and j = l – ½. These two different levels now have different energies because of the strong spin-orbit coupling. The splitting of the two levels can be calculated by computing the expectation values of the spin orbit potential in the two states of the different j.

This results in energy splitting of individual levels for given j is shown in fig. 4.

Fig. 4: Energy splitting of individual levels for given j.

The corresponding energy splitting becomes

j degeneracy

The effect of adding this spin-orbit term is to split the subshells according to the j degeneracy. Each subshell now can contain up to 2j + 1 protons or neutrons.

2.1 Nuclear Potential with Spin-Orbit Term

The nuclear potential with spin-orbit term is

The spin-orbit term makes the nuclear potential well wider for nucleons with spin parallel to the orbital angular momentum and less wide for nucleons with spin opposite to the orbital angular momentum.

The wider well results in states of lower energy, and without S-O term the energy of state does not depend on total angular momentum.

The effect of spin-orbit interaction on the nuclear shell model is shown in fig. 5.

Fig. 5: The effect of spin-orbit interaction on the shell model potential.

Fig. 6: Level scheme for a harmonic oscillator with spin-orbit coupling.

The magic numbers now come out correct. In many cases, the angular momentum of the single-particle states also explains the nuclear angular momentum near magic nuclei. There are large deviations for nuclei between magic shells, so nuclear deformation has to be added.

2.2 The Harmonic Oscillator with S-O interaction term

When spin-orbit interaction term is considered then the energy levels are labeled as nlj. Here the radial quantum number n, orbital angular momentum l, and total angular momentum j.

The nlj level is (2j + 1) times degenerate.

The nuclear potential with S-O interaction term for harmonic oscillator can be written as

The energy of harmonic oscillator with S-O interaction term is

Fig. 7: Energy levels of nucleons in a smoothly varying potential well with a strong spin-orbit coupling.

3. Applications of Shell Model 3.1 Spin and Parity

One of the most successful applications of the single particle shell model is the explanation of the ground spins (I) and parity of odd-A nuclei (?). The model correctly predicts excitation energies, spin/parities, magnetic and quadrupole moments for the ground state and low-energy excited states.

The intrinsic spin s of each nucleon couples with its orbital angular momentum l to form the total angular momentum j = l + s. The total angular momentum I of the system of nucleons is then obtained by the coupling of the j vectors of the individual nucleons in the nucleus: I = ∑ j_i. This coupling is known as j-j coupling. In nuclear physics it is customary to denote the total angular momentum by the symbol I in place of J.

In case of nucleus to find out the ground state spins of the nuclei we make the following assumption: Only the last unpaired nucleon dictates the properties of the nucleus, and an even number of nucleons of any kind in the same state j always combines to give the resultant spin 0 and even parity.

On the basis of the above assumptions we get the following results:

For odd-A nuclei: The total angular momentum of any shell is determined by the angular momentum of the last nucleon in the species (proton or neutron) that is odd.
Even-A nuclei: For even-even nuclei, the ground state always have spin 0 (net angular momentum associated with even N & even Z is zero) and parity is positive, i.e. 0⁺.
Even- A nuclei: For odd-odd nuclei, the last neutron couples to the last proton with their intrinsic spins in parallel orientation.
Odd-A nuclei: In case of odd-A nuclei the total spin and parity determined by the single (unpaired) nucleon (valance or a hole).

Here 2 closed shells for protons & for neutrons + one unpaired neutron.

¹⁷₈OGround State spin-parity:

According to the shell model the filling of levels of ¹⁷O is as follows

Fig. 8: Filling of levels of 178O

The spin parity of ¹⁷O would be that of the unpaired neutron in the 1d5/2 subshell, i.e. 5+/2 and this value is also obtained experimentally.

Similarly ground state spin and parity for ¹⁵O is given in fig. 9.

Fig–.9: The filling is of Energy levels in ¹⁵O and ¹⁷O.

Some other examples of odd A nuclei given below

Summary

Experimentally the Woods-Saxon potential is found to be most realistic potential, however it is able to account for the first three magic numbers (2, 8, 20) only. This potential fails for higher magic numbers; the problem cannot be solved even by considering different shapes of potentials. This issue can be resolved if we include the spin-orbit interaction term in the potential which is crucial in producing nuclear magic numbers.

The sign of spin-orbit interaction term in case of nuclei is found to be opposite to that in atoms. On including the spin-orbit interaction term in the potential, it changes the width of nuclear potential. The nuclear shell model is able to predict the ground state properties in nuclei well. The energy of nucleons can to good approximation be calculated from an effective central potential + spin‐orbit coupling.

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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.
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.
The Origin of the Concept of Nuclear Force by L.M. Brown and Rechenberg.
Theoretical Nuclear Physics by John M. Blatt and Victor F. Weisskopf.
Experimental techniques in Nuclear Physics by Dorin N. Poenaru & Walter Greiner
Exotic Nuclear Excitation by S.C. Pancholi
Nuclear spectroscopy Part B, by Fay Ajzenberg- Selove
Theory and Problems of modern Physics (Schaum’s outline Series)
Basic Ideas & Concepts in Nuclear Physics – by K Heyde
The “Particles of Modern Physics” by J. D. Stranathan, Philadephia: Blakiston.
5. Nuclear Physics by Irving Kaplan, Narosa Publishing House.

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Did you know ?

As per the quantum mechanics, the spin–orbit interaction is nothing but the interaction of a particle’s spin with its motion.
The spin –orbit interaction is commonly called spin–orbit effect or spin–orbit coupling.
The consequence of the spin-orbit interaction is to modify the energy levels of the atom (or nucleus).
The energy splitting is detectable as a splitting of spectral lines, which can be thought of as a Zeeman Effect due to the internal field.
The energy splitting is proportional to the dot product of the orbital angular momentum (L) and the atomic (or nuclear) spin (S)
Due to the spin-orbit interaction, the coupled angular momentum J becomes good quantum number.
The effect of the spin-orbit interaction in nuclei is opposite to the effect in atoms due to the different signs of charge and hence the different direction of spin angular momentum.
Classically, the spin-orbit interaction is not enough for the level having l = 0. However, the quantum mechanics reveals that for describing the l = 0 level, the Fermi contact term is needed which does not have any classical analogue.
In order to predict the accurate energies, relativistic correction needs to be applied.
In the field of spintronics, spin–orbit effects for electrons in semiconductors and other materials are explored for technological applications.
The spin–orbit interaction in materials is one of the cause of magneto-crystalline anisotropy.

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