# Finding a method to compute the magnetic moment of an even-odd nucleus

• JD_PM
In summary, the formula for computing the magnetic moment of an even-odd nucleus is given by μJ=gJ×j×μN, where j is the quantum number determined by the shell model and gJ is calculated based on the value of j. However, this method has led to incorrect results for several examples, indicating that something is missing in the calculation. Further guidance is needed to correctly compute the magnetic moment for even-odd nuclei.
JD_PM
Homework Statement
Compute the magnetic moment of 57-Ni and compare it with its experimental value of ##-0.8 \mu_N##. Note that I have computed two extra magnetic moments for odd-even nucleus since I wanted to understand why I was getting a wrong answer (I also got those wrong). This is extra work, since the original question is about ##\mu## of 57-Ni. I want to understand what I am doing wrong.
Relevant Equations
##\mu_J=g_J\times j\times \mu_N##
I am having difficulties computing the magnetic moment for an even-odd (proton-neutron) nucleus.

The formula is:

$$\mu_J=g_J\times j\times \mu_N$$

I checked this helpful post: https://physics.stackexchange.com/q...ic-moments-how-does-one-decide-between-l-frac

I worked out the magnetic moment for 57-Ni.

Based on the shell model we see that only the unpaired neutron contributes to a non-zero magnetic moment; all neutron shells can be filled up to ##2p_{3/2}##. Thus the quantum numbers are:

$$j=\frac{3}{2}, s=\frac{1}{2}, l=1.$$

Thus we know that ##j=l+\frac{1}{2}##, so to calculate ##g_J##:

$$g_J=\Big(1+\frac{1}{2j}\Big)g_l+\frac{1}{2j}g_s$$

For neutrons: g_l = 0 and g_s =-3.8260837.

Knowing that we get for 57-Ni case:

$$g_J = -1.275$$

Thus:

$$\mu_J = -1.913 \mu_{N}$$

This result is wrong. It is far from its experimental value; ##\mu_J = -0.8 \mu_{N}##. I have read (Krane page 126-127) that it is acceptable to get a theoretical value slightly different from the experimental one. That is not the case here of course.

I checked more similar examples:

a) 87-Sr (38 protons). ##J^\pi=\frac{9}{2}^+## and ##j=l+\frac{1}{2}## Applying the same method I get:

$$\mu_J = -1.913 \mu_{N}$$

Experimental value: ##\mu_J = -1.093 \mu_{N}##.

Again a significant difference; something is wrong.

b) 91-Zr (40 protons). ##J^\pi=\frac{5}{2}^+## and ##j=l+\frac{1}{2}## Applying the same method I get:

$$\mu_J = -1.913 \mu_{N}$$

Experimental value: ##\mu_J = -1.304 \mu_{N}##.

Again a significant difference; something is wrong.

Note that in all three cases I get the same mistaken theoretical value. There must be something I am missing.

To recap, this is the method that I've used:

1) Get ##j## based on the shell model (note that in some cases there are exceptions, but we are not concerned with that in this post).

2) Get ##g_J##.

2.1)If ##j=l+\frac{1}{2}## meets your case then use:

$$g_J=\Big(1+\frac{1}{2j}\Big)g_l+\frac{1}{2j}g_s$$

Finally calculate the magnetic moment:

$$\mu_J=g_J\times j\times \mu_N$$

2.2)If ##j=l-\frac{1}{2}## meets your case then use:

$$g_J=\frac{1}{j+1}\Big[\Big(j+\frac{3}{2}\Big)g_l-\frac{1}{2}g_s\Big]$$

Finally calculate the magnetic moment:

$$\mu_J=g_J\times j\times \mu_N$$

What's wrong with it?

Thanks.

PS: Let me know if something needs to be added and I will do it.

I've asked this question at PSE as well, but it's not catching a lot of attention.

https://physics.stackexchange.com/questions/496458/method-for-calculating-nuclear-magnetic-moments

Last edited:
Hi @mfb may you help me out with this one? I am basically stuck in how to compute the magnetic moment of an even-odd nucleus. I've found the above method but I am missing something (I am getting wrong answers for the three examples).

Thanks.

With g_l = 0 your calculation simplifies to ##\mu_J=g_J\times j\times \mu_N = \frac{1}{2j}g_s \times j\times \mu_N = \frac{1}{2} g_s \mu_N## and that is the same for every nucleus.
I'm not sure which step went wrong, I didn't look at that for years, @Orodruin should be able to help.

JD_PM

## 1. How is the magnetic moment of an even-odd nucleus computed?

The magnetic moment of an even-odd nucleus is computed using the nuclear shell model, which takes into account the number of protons and neutrons in the nucleus and their arrangement in energy levels. This model uses quantum mechanics principles to calculate the magnetic moment of the nucleus.

## 2. What factors affect the magnetic moment of an even-odd nucleus?

The magnetic moment of an even-odd nucleus is affected by the number of protons and neutrons, their spin orientations, and the nuclear spin of the nucleus. Additionally, the presence of an external magnetic field can also influence the magnetic moment of the nucleus.

## 3. Can the magnetic moment of an even-odd nucleus be experimentally measured?

Yes, the magnetic moment of an even-odd nucleus can be experimentally measured using techniques such as nuclear magnetic resonance (NMR) or electron magnetic resonance (EMR). These methods involve applying a magnetic field to the nucleus and measuring the resulting interactions.

## 4. How accurate are the calculations for the magnetic moment of an even-odd nucleus?

The accuracy of the calculations for the magnetic moment of an even-odd nucleus depends on the complexity of the nuclear shell model used and the level of detail included in the calculations. Generally, these calculations have a high level of accuracy, but they can be affected by experimental uncertainties and limitations.

## 5. Can the magnetic moment of an even-odd nucleus change over time?

Yes, the magnetic moment of an even-odd nucleus can change over time due to changes in the nuclear spin or the orientation of the spin of particles within the nucleus. External factors such as temperature and pressure can also affect the magnetic moment of the nucleus.

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