15 Nuclear Models-7

Sanjay Kumar Chamoli

Applications of Shell Model

1. Spin

We have seen in the last section that shell model can successfully predict ground state spin and parity of odd-A nuclei. The shell model also accounts spin of Even-A nuclei fairly well in general.

Spin of Even –A nuclei (with N = odd, Z = odd)

W L Nordheim in 1950 gave a rule to calculate the most probable value of spin in odd-odd nuclei which is known as Nordheim rule. According to this rule the most probable value of spin in odd-odd nuclei is given as

The Nordheim number (N_N) is defines as

?_? = ?_? – ? _? + ?_? − ?_? (1)

where

J_p = angular momentum of odd-proton

J_n = angular momentum of odd-neutron

Rules for calculating spin

(a) If N_N = 0, then the value of j is: = | j_p − j_n|. This rule is known as “Strong rule”.

(b) If N_N = ± 1, then the value of j is: = j_p + j_n or j = | j_p − j_n |. This rule is known as “Weak rule”.

A comparison of predicated spin ( j_???? ) values and experimentally obtained spin ( ?_???) values for some even-A nuclei is given in table 1. It can be seen from the table that predicated values are in well agreement with experimentally obtained values.

Table 1: Spin values for Even-A nuclei

In addition to the prediction of ground state spins and parities of nuclei, the shell model can also predict spins and parities of excited states as well, as shown in fig. 1.

Fig. 1: Spins and parities of excited states

2. Nuclear Magnetic Moment

The Shell model in principle can be used to predict the magnetic moment (μ) of nuclei, which can then be compared with the experimental values. Thus, the last unpaired nucleon determines the magnetic moment of the entire nucleus.

The magnetic moment of a nucleus is the vector sum of the spin magnetic moment μ_s ⃗⃗⃗⃗ and orbital magnetic moment μ_L ⃗⃗⃗⃗ :

Where μ_s ⃗⃗⃗⃗ is the vector sum of the intrinsic magnetic moments of the individual nucleons in the nucleus. The intrinsic moments for proton and neutrons are given as

?_? = ?_? ?_?/? and ?_? = ?_? ?_?/? (3)

Where = eћ/ 2Mp, is the nuclear magneton which is analogous to the Bohr magneton in atoms but with the electron mass replaced by the proton mass, M_p being the proton mass, ?_? and ?_? are the gyromagnetic ratios for the proton and the neutron. It has numerical values equal to

?_? = 2 × 2.7927 and ?_? = -2 × 1.9131 (4)

The magnetic moment of a nucleus of spin I (total angular momentum) can be written as

In order to measure the magnetic moments, a magnetic field is applied and it is the component of ?_? in the filed direction of which determines magnetic moment.

Where ?_? is the magnetic quantum number which can take values ?_? = I, I -1, …. – I. B is the magnetic induction filed. The component of along the z direction corresponding to ?_?= I usually gives the measured magnetic moment where value of ? is the largest.

As stated earlier ?_N is the nuclear magnetron, g is the gyromagnetic factor (or g-factor) & I is the spin.

For even-even nuclei: An even number of nucleons of any kind always gives the resultant spin of ground state I= 0⁺. Hence magnetic moment of an even-even nucleus will be 0.
For odd-A nuclei: For odd- A nuclei magnetic moment is only due to the last odd nucleon (proton or neutron).
For odd-odd nuclei: For odd-odd nuclei it the last unpaired odd nucleon which determines the magnetic moment. For such a nucleus the magnetic moment is the vector sum of magnetic moments due to odd-proton & odd-neutron ( i.e. ? = ?_p + ?_n).

Total magnetic moment is obtained by adding the intrinsic magnetic moment ( ?_? ) and the magnetic moment due to its orbital motion ( ?_? ).

As the neutron is an uncharged particle its orbital motion does not produce any magnetic moment ( ?_? = 0) so that

(?_? )_n = 0 (10)

For proton (?_? = 1) so that the orbital contribution is

Odd-A

For odd-A nuclei, either the proton number is odd (in the o-e nucleus) or the neutron is odd (in the e-o nucleus). So there are two different possibilities corresponding to odd proton and odd neutron.

Numerical values of g for proton and neutrons are ?_? = + 5.5856, ?_? = − 3.8262.

The above equations (19) and (20) gives the magnetic moments of odd A nuclei as functions of the nuclear spin I which is taken as equal to the j value of the last odd nucleons. The above values of the nuclear magnetic moments are known as Schmidt values. These Schmidt values as a functions of I = j are plotted in figure 2.

(a) (b)

Fig. 2: Schmidt line for (a) odd proton case, (b) odd neutron case

Fig. 2 (a) shows the Schmidt plots for the odd proton case for j = l ± 1/2 giving the two lines as shown. In the same diagram, the experimental values of the magnetic moments for some nuclei are also shown. Similarly, fig. 2(b) indicates the Schmidt lines for the odd neutron case for j = l ± ½. From the above diagrams we conclude that the experimental values do not in general agree with the Schmidt values.

The values of magnetic moment for some nuclei is given below

3. Quadrupole Moment

The electric quadrupole moment shows the deviation from spherical symmetry. Neutrons have no charge, so do not induce quadrupole moment. The electric quadrupole moment Q of a nucleus is the average of the quantity (3z² − r²) for the charge distribution in the nucleus. For a spherically symmetric charge distribution this average is zero and hence Q = 0 for even-even nuclei which have ground state spin I = 0 in the nucleus.

In case of a nucleus with single unpaired proton the quadrupole moment is given as

〈 r^r 〉 is the mean square radius of the charge distribution which in the present case is equal to the mean square distance of the proton from the nuclear centre.

So, quadrupole moments give

The negative sign indicates that orbital motion of the proton in the equatorial plane makes the charge distribution an oblate spheroid. On the other hand an odd hole in the case of j would make the charge distribution a prolate spheroid for which Q>0. Thus both positive and negative values of Q are expected.

Ideally, a nucleus with a single odd-neutron should have no quadrupole moment.

However, in actual the neutrons in nucleus interact with the nucleons of core to polarize it and generate a small quadrupole moment. The value of quadrupole moment for neutron is much smaller than the value for proton as indicated in figure 3.

Fig. 3: Variation of quadrupole moment for odd proton and odd neutron case

Summary

The nuclear shell model is successfully able to explain the spin and parities of states in nuclei. It can also explain the observed magnetic moment in nuclei. Shell model can explain the observed quadrupole moment in nuclei. Shell model can predict the excited state spin and parities in nuclei. Shell model can predict electromagnetic moments in nuclei well.

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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.
Advances in Nuclear Physics, Vol. 27, edited by J.W. Negele, Erich Vogt.
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 ?

The nuclear shell model predicts the ground state spins of most of the nuclei well.
For even- even nuclei, the ground state spin is always zero whereas for the odd-A nuclei, the spin is determined by the spin of the orbital occupied by the odd nucleon.
If a nucleus has odd numbers of protons and also odd number of neutrons, its ground state spin is given by the Nordheim theorem.
When compared with the experimental data Nordheim rules are found to work well for odd-odd nuclei.
The Nordheim rules reveal a tendency of alignment of intrinsic spin, like in case of deuteron (j = 1).
Almost all odd-odd nuclei are unstable except 2H, 6Li, 10B, 14N
All four stable odd-odd nuclei have J = 1 while among the unstable odd-odd nuclei there are a few that have J = 0.
Most of the unstable nuclei with J = 0 have short half-life (less than 1 sec.).
The longest half-life among unstable odd-odd nuclei is for 170Lu which has a half-life of ~ 2 days.
The unstable odd-odd nuclei can have states with higher angular momentum, so that J = 1,2, 3, etc. but these excited states decay very quickly.
The most unusual odd-odd nucleus is 180mTa because its decay has never been observed.
The 50V has a very long half-life (> 1017 years). Its decay has never been observed directly and the lifetime is inferred from geochemistry.

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