# States of an atom in spectral notation

## Homework Statement

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

## 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.

nrqed
Science Advisor
Homework Helper
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## Homework Statement

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

## 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
thisis where your mistake is....

(2+1/2) = 5/2

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

|2-1/2| = 3/2

thisis where your mistake is....

(2+1/2) = 5/2

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

|2-1/2| = 3/2

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:
nrqed
Science Advisor
Homework Helper
Gold Member
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?

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.

Thats cool!Thanx.