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Energy shift in excited state sodium

  1. Oct 18, 2016 #1
    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
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  3. Oct 18, 2016 #2

    blue_leaf77

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    Yes, that's right.
    What are the two possible values for ##j## in terms of ##l##?
     
  4. Oct 18, 2016 #3
    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
  5. Oct 18, 2016 #4

    blue_leaf77

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    If you do that sum, the result will have the unit of energy.
     
  6. Oct 18, 2016 #5
    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
  7. Oct 18, 2016 #6

    blue_leaf77

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    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.
     
  8. Oct 19, 2016 #7
    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.
     
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