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States of an atom in spectral notation

  1. Apr 1, 2007 #1
    1. The problem statement, all variables and given/known data

    1)Give in spectral notation the possible states of an atom which has a closed core plus one d electron.


    2. Relevant equations



    3. The attempt at a solution

    I solved in the following way:
    For d electron,
    Orbital angular momentum quantum number(L)=2
    Spin angular momentum quantum number(S)=1/2
    Possible values of total angular momentum quantum number(J)= L+S,| L+S-1|,…,|L+S|
    Hence,
    J= (2+1/2), |2+1/2-1|,|2-1/2|
    J= 5/2,3/2,1/2
    Possible states of the atom in spectral notation are,
    2 D 5/2, 2 D 3/2, 2 D ½
    But the answer given in my book is 2 D 5/2, 2 D 3/2.
     
  2. jcsd
  3. Apr 1, 2007 #2

    nrqed

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    thisis where your mistake is....

    (2+1/2) = 5/2

    |2+1/2-1| = 3/2

    |2-1/2| = 3/2
     
  4. Apr 1, 2007 #3
    Sorry,I meant J=|L+S-2| =|2+1/2-2|=1/2 instead of J=|L+S-1|=3/2
    Is there something to do with the multiplicity of states given by 2S+1 = 2 so that for a given value of J there are only 2 possible values L+S and L-S?
     
    Last edited: Apr 1, 2007
  5. Apr 1, 2007 #4

    nrqed

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    I think you misunderstand the rule. You calculate L+S and then you calculate |L-S|. J may take any value between those two extremes, differing by steps of one.

    In your example, L+S = 5/2 and |L-S| = |2-1/2| = 3/2.

    So the possible values of J are 3/2 and 5/2. J=1/2 is not possible.

    Of course, you can check that the number of states comes out right. L=2 has 5 states and S=1/2 has two states so the total number of states is 10.

    Now, J=5/2 has 6 states and J=3/2 has 4 states so the total number checks out.
     
  6. Apr 1, 2007 #5
    Thats cool!Thanx.
     
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