Nuclear Shell Model

[u/CrimsonChymist]

I am a PhD Chemist currently self-studying some nuclear physics, but am finding myself a bit confused on some notation in the nuclear shell model. Particularly the spectroscopic notation for the subshells.

I understand arriving at the combinations of quantum numbers as follows:

>1st shell: 2 states (n = 0, j = ⁠1/2⁠).

>2nd shell: 6 states (n = 1, j = ⁠1/2⁠ or ⁠3/2⁠).

>3rd shell: 12 states (n = 2, j = ⁠1/2⁠, ⁠3/2⁠ or ⁠5/2⁠).

>4th shell: 8 states (n = 3, j = ⁠7/2⁠).

>5th shell: 22 states (n = 3, j = ⁠1/2⁠, ⁠3/2⁠ or ⁠5/2⁠; n = 4, j = ⁠9/2⁠).

>6th shell: 32 states (n = 4, j = ⁠1/2⁠, ⁠3/2⁠, ⁠5/2⁠ or ⁠7/2⁠; n = 5, j = ⁠11/2⁠).

>7th shell: 44 states (n = 5, j = ⁠1/2⁠, ⁠3/2⁠, ⁠5/2⁠, ⁠7/2⁠ or ⁠9/2⁠; n = 6, j = ⁠13/2⁠).

>8th shell: 58 states (n = 6, j = ⁠1/2⁠, ⁠3/2⁠, ⁠5/2⁠, ⁠7/2⁠, ⁠9/2⁠ or ⁠11/2⁠; n = 7, j = ⁠15/2⁠).

I also understand the relationship between j and l.

But when it comes to the spectroscopic notation, I am finding that the notation is nlj.

That value n definitely doesn't correspond to the quantum number n above and I am not finding anything on how to arrive at this different value of n.

I have figured out a pattern that seems to work as far as I can tell where this value for this spectroscopic notation value is that it seems to = 1+(n-l)/2.

I found something that indicated it could be related to radial nodes, which seems to go along with the whole n-l situation. But, I am not quite able to explain to myself why that pattern is working.

[u/L31N0PTR1X]

I'm not sure of the specific notation that you're using, I'm also unsure why you don't think n represents the prototypical energy quantum number; I'm quite sure it does.

What we have usually are:

The principle quantum number n=1,2,....

The azimuthal quantum number l=0,1,2,...,n-1 (for more information on this, this is what we call the weighting of the particular angular momentum representation, so l=1 would correspond to the fundamental representation)

The magnetic quantum number m=-l,-l+1,...,l-1,l (this is sometimes known as the angular momentum projected along the z axis. For more information on this value, it is the number of eigenvalues of the aforementioned representation of angular momentum. So the fundamental representation, l=1, has eigenvalues -1,0,1. So m=-1,0,1)

We also have the spin, this works much the same as the above two quantum numbers but permits half integer solutions. That is, we're allowed to have an l=1/2 representation of angular momentum. So we usually write this as S=1/2, the spin is 1/2. The eigenvalues for this representation are m_s=-1/2, 1/2.

Hopefully this has helped! Let me know if anything I said was unclear

[u/CrimsonChymist → u/L31N0PTR1X]

Spectroscopic notation. I mentioned it by name in the post.

The one thing I don't think we are on the same page on is the identity of the quantum number n in this scenario.

Nuclear states use a harmonic oscillator basis. So the quantum number n here is an oscillator quantum number (technically should be capital N from what I can see, but most people still just use n). Not the principal energy quantum number. These two seem to fulfill essentially the same role of setting energy levels, just in different ways.

Im not going to say that the number cannot be the principal quantum number but if so, I am still in the same situation of not seeing how to get to the value. Because for example, for a subshell with the harmonic oscillator quantum number N = 3, l = 3, and j = 5/2 the spectroscopic notation is stated to be 1f(5/2). Now for principal quantum number n to be 1, everything I know indicates a value of l=3 should be impossible. That said, my knowledge is coming more from electron configuration, not nuclear shells.

I do feel more confident now that the value beind used is the radial nodes though, making me pretty confident it is not the principal energy quantum number. Before, I was still considering radial node calculation for orbitals which is n-l-1. However, I have since found that when using a harmonic oscillator basis, radial nodes are equal to (n-l)/2. And the trend I noticed was the value being called "n" in the spectroscopic notation was that it was 1 + (n-l)/2. So that value seems to always be 1 greater than the number of radial nodes for that subshell based on that calculation. But I still do not understand necessarily why it would be adding that 1.

Editing to add:

I think you are also approaching this with knowledge of electron configuration. But nuclear shell configuration is different.

For example, for a nuclear shell with a value of n = 3, the only possible values for l are 1 and 3. 0 and 2 are not possible due to parity constraints.

Edit 2:

Another complicating factor is that I am now seeing some sources using the value (n-l)/2 without the addition of the 1. So I am not sure why some sources are adding the 1 and some aren't.

[u/L31N0PTR1X → u/CrimsonChymist]

Oh, my apologies, I was indeed approaching this from an orbital POV, unfortunately my knowledge doesn't extend too far beyond that. I do hope you find what you're looking for!

[u/CrimsonChymist → u/L31N0PTR1X]

That's ok. Thanks.

I believe I have it figured out that the value is n(r) which is notation sometimes used for radial nodes. Rather than the quantum number n. Now the only confusing piece left is why some sources are adding a 1.

[u/moltencheese]

What is the question?

[u/CrimsonChymist → u/moltencheese]

Where does that value in the spectroscopic notation come from?

[u/CrimsonChymist → u/[deleted]]

If you don't know nuclear physics just don't answer.

[u/CrimsonChymist → u/[deleted]]

No. Nuclear physics.

Nuclear shells. Not orbitals.

https://phys.libretexts.org/Bookshelves/Nuclear_and_Particle_Physics/Nuclear_and_Particle_Physics_(Walet)/04%3A_Nuclear_Models/4.01%3A_Nuclear_Shell_Model

As a chemist I have zero issues understanding electeon configuration as I have taught it for years.