# Problem with shell model and magnetic moment of Lithium-6

• bznm
In summary, the configuration of protons and neutrons in Li-6 is 1p_{3/2}, and the magnetic moment of the nucleus is calculated by adding the magnetic moments of the uncoupled proton and neutron. However, the predicted magnetic moments from the shell model do not match experimental values, as the model does not take into account interconfiguration mixing. When considering the proton's J value, it should also be considered for the neutron, and the sign of the term should be taken into account.
bznm
I have a problem with the calculus of magnetic moment of Li-6.

The configuration of protons is $1p_{3/2}$, and the neutrons' one is the same.

I have to add the magnetic moment of uncoupled proton and uncoupled neutron.

I use the following formula for $J=l+\frac{1}{2}$ (J is the particle spin):
$\frac{\mu}{\mu_N}=g_lJ+\frac{g_s-g_l}{2}$

For the proton I have: $g_l=1; g_s=5.58 \rightarrow \frac{\mu}{\mu_N}=J+2.29=3.79$

For the neutron I have: $g_l=0; g_s=-3.82 \rightarrow \frac{\mu}{\mu_N}=-1.91$

So the total $\frac{\mu}{\mu_N}=3.79-1.91=1.88$, exactly 1 more than the correct value, 0.88!
What's wrong?

Your value for the proton's magnetic moment is off by 1: it should be 2.79, not 3.79. Not sure where you are getting the values you are using to calculate it.

PeterDonis said:
Your value for the proton's magnetic moment is off by 1: it should be 2.79, not 3.79. Not sure where you are getting the values you are using to calculate it.

It was told me that the proton's magnetic moment is 2,79 but, when I consider a proton in the nucleus, I have to consider the proton's J value.

In this case, the proton's J is 3/2 and, if you insert this value in the formula, you obtain 3.79.

Li-6 is an odd-odd nucleus, and therefore the magnetic moments predicted by the shell model are not in complete agreement with experiment.

Quoting from Preston, "Physics of the Nucleus", p323:

"Turning to odd-odd nuclides, the shell model would suggest simply adding the magnetic moments due to proton and neutron configurations, ignoring any interaction between the unfilled neutron and proton shells, except perhaps in the light nuclei in which neutrons and protons are filling the same shells and i-spin is a good quantum number. It may be argued that, in this latter case, the neutrons and protons have precisely the same spatial motion and orientation but g-factors of opposite sign, and therefore the corrections to their free-particle g-values are roughly equal and opposite. Hence, despite the occurrence of interconfiguration mixing and quenching the free-nucleon g-factors can be used, and μ is just the sum of the neutron and proton moments of the extreme single-particle model. For nuclides in which neutrons and protons are filling different shells, it would seem appropriate to take the values of gp and gn from neighboring odd nuclides, thus allowing for interconfiguration mixing. This works quite well, and whenever the value gemp obtained from empirical g's differs from gsp obtained from free-nucleon g's, the observed value is always much nearer gemp. Some cases are shown in Table 12-1, where both μemp and μsp are calculated from the following formula, the only difference being the g-values used:

$$\mu = \frac{1}{2}\left[(g_p + g_n) + (g_p - g_n)\frac{j_p(j_p + 1) - j_n(j_n + 1)}{J + 1}\right]$$

The table entry for Li-6 has μsp = 0.6, μemp = 0.4 and μobs = 0.8.

The conclusion which can be derived from our discussion is that, for the nearly spherical nuclei, which we have mainly considered, magnetic-moment values are consistent with the shell model, but it is essential to include interconfiguration mixing in the ground state."

(BTW, I think it would have been appropriate for you to mention that you were simultaneously posting this same question to both PF and stackexchange!)

Last edited:
bznm said:
In this case, the proton's J is 3/2

But the neutron's J is 3/2 as well, correct? If you're going to include the J term, it seems to me that you should include it for both the proton and the neutron; i.e., I don't see why you have ##g_l = 0## for the neutron. (And if you include both, as the quote Bill_K posted points out, you have to consider the sign as well; if the neutron's ##g_l## is opposite in sign to the proton's, the two J terms cancel each other.)

## 1. What is the shell model and how does it relate to the magnetic moment of Lithium-6?

The shell model is a theoretical framework used to describe the arrangement of protons and neutrons in an atomic nucleus. It predicts that the nucleus of Lithium-6 should have a magnetic moment of zero, but this is not observed experimentally.

## 2. Why is the magnetic moment of Lithium-6 a problem for the shell model?

The magnetic moment of an atomic nucleus is a fundamental property that is predicted by the shell model. The fact that the predicted value for Lithium-6 does not match experimental results raises questions about the accuracy and completeness of the model.

## 3. What factors could be contributing to the discrepancy between the predicted and observed magnetic moment of Lithium-6?

There are multiple possible explanations for the problem with the shell model and the magnetic moment of Lithium-6. These include the presence of additional interactions between the protons and neutrons in the nucleus, the effects of nuclear deformation, and the influence of quantum tunneling.

## 4. Are there any proposed solutions to this problem?

Scientists have proposed several solutions to address the discrepancy between the shell model and the magnetic moment of Lithium-6. These include adjusting the parameters of the model to better match experimental results, incorporating new theoretical frameworks, and conducting further experimental studies.

## 5. How does this problem impact our understanding of nuclear physics?

The problem with the shell model and the magnetic moment of Lithium-6 highlights the complexity of atomic nuclei and the limitations of our current theoretical models. It challenges scientists to continue exploring and expanding our understanding of nuclear physics to better explain and predict the behavior of these fundamental particles.

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