# Energy shift in excited state sodium

1. Oct 18, 2016

### Kara386

1. The problem statement, all variables and given/known data
The spin-orbit interaction in sodium results in fine structure with energy shifts given by:

$\Delta E_j = \frac{C}{2}[j(j+1)-l(l+1)-s(s+1)]$

If $l$, $s$ and $j$ describe the single outer electron in sodium. Show that if this electron is excited to a state where l>0, a pair of sub-levels is generated. Find expressions for the energy shifts in terms of $C$ and $l$.

2. Relevant equations

3. The attempt at a solution
For any electron $s=\frac{1}{2}$. Is there something else about sodium that gives me more information? Should I substitute in $j = l+s$? Except of course $j$ is also $l+s-1$, $l+s-2$... and so on, so I'm not sure that gets me anywhere. For $l=0$ I can see that the energy shift would be zero.

I think I need to prove there are two possible values of j, which would correspond to a pair of sub-levels. Is that related to the fact that $s=\frac{1}{2}$? Because then $l+s-1 = |l-s|$ so the only two possible values would be $l+s$ and $l-s$?

If that proves that for $l>0$ there are 2 sub-levels then how do I derive expressions for $\Delta E_j$ in terms of $l$ and $C$? Substitute in $s=\frac{1}{2}$, but that still leaves a $j$.

Last edited: Oct 18, 2016
2. Oct 18, 2016

### blue_leaf77

Yes, that's right.
What are the two possible values for $j$ in terms of $l$?

3. Oct 18, 2016

### Kara386

Ah, $j=l+\frac{1}{2}$ and $j=l-\frac{1}{2}$. And those are the only two values it can take.

So then I'd get the energy shifts to be
$\Delta E_j = \frac{C}{2}l$ and $-\frac{C}{2}(l+1)$.

If I calculated then $\Sigma_j \Delta E_j (2j+1)$, would that give me all possible states? Because for every combination of L, S and J there are 2J+1 quantum states. So multiplying them by the different changes in energy would give the new total number of quantum states available? Except that makes me think my values of $\Delta E_j$ must be wrong because one of them is negative and that seems to make the sum negative overall...

Last edited: Oct 18, 2016
4. Oct 18, 2016

### blue_leaf77

If you do that sum, the result will have the unit of energy.

5. Oct 18, 2016

### Kara386

So it represents the difference in the energy of the atom after splitting? I don't know... That 2j+1 term has come up in a couple of contexts. The Zeeman effect splits degenerate energy levels in 2j+1 equally spaced ones I think, and as I said it's also the number of possible states for a given combination of L, S and J. And then this $\Delta E_j$ is the shift in the energy levels from a no-spin state I think, for the sort of internal Zeeman effect here, or from a no-field state in the Zeeman effect. I don't know what combining them does though!

Last edited: Oct 18, 2016
6. Oct 18, 2016

### blue_leaf77

The energy splitting depends on the strength of the magnetic field: weak field or strong field, the latter is also referred to as Paschen-Back effect. The wikipedia page on Zeeman effect covers the related discussion.

That form you propose above? Not necessarily, since states with different $m_j$ but same $j$ are degenerate. I don't where you got that formula from, it seems like you came up with it.

7. Oct 19, 2016

### Kara386

Ok, so I've been confusing spin-orbit interaction with the Zeeman effect. I thought the thing where spin and orbit interact to generate a magnetic potential was like an internal Zeeman effect, or the 'anomalous' Zeeman effect. Reading that, I suppose the anomalous Zeeman effect is something different. The next part of the question asks you to calculate the sum which I did, and it's $Cl^2$, and says comment on the significance of this. I can't because I don't know what the quantity represents, but if I did I would guess the absence of $s$ and $j$ is significant in some way.