Energy shift in excited state sodium

[Kara386]

Homework Statement

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. Homework Equations

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 $j$.

 

[blue_leaf77]

Is that related to the fact that s=12s=\frac{1}{2}?

Yes, that's right.

If that proves that for l>0l>0 there are 2 sub-levels then how do I derive expressions for ΔEj\Delta E_j in terms of ll and CC? Substitute in s=12s=\frac{1}{2}, but that still leaves a jj.

What are the two possible values for $j$ in terms of $l$?

 

[Kara386]

Yes, that's right.

What are the two possible values for $j$ in terms of $l$?

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...

 

[blue_leaf77]

If I calculated then $\Sigma_j \Delta E_j (2j+1)$, would that give me all possible states?

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

 

[Kara386]

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

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!

 

[blue_leaf77]

And then this ΔEjΔEj\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!

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.

So it represents the difference in the energy of the atom after splitting?

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.

 

[Kara386]

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.

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.