Hund's Rules and L-S Coupling: Understanding Limitations and Confusions"

[secret2]

I would like to ask two questions about Hund's rules and L-S coupling:

1. Some textbooks state that when doing L-S coupling and applying Hund's rules, "The maximum values of S and L are subject to the condition that no two electrons may have the same pair of values for m(sub s) and m(sub l). I know this is because of the Pauli exclusion principle, but how does this requirement (m(sub s) and m(sub l)) really limit S and L when we are adding the angular momenta?

2. When we are trying to figure out the ground state of Sm (4f)6, why is it wrong to have L = Sum(l) = 6*3?

Finally, I've realized that in discussing Hund's rules and L-S coupling some texts tend to make explanations using symmetry consideration and the others tend to prefer the exclusion principle. Are they two different sets of explanations, or are they equivalent?

 

[chrismuktar]

The explanations are equivalent. With regards to $$Sm 4f^6$$, then n=4, l=3, $$L=\sqrt{l(l+1)}\hbar$$ and I'm not sure how to interpret the 6.

With regards to your first question, the ms and ml states are simply the number of possible states at that level. When doing spin orbit coupling, (and this is where I get a little flakey), $$L_{z}$$ and $$S_{z}$$ no longer commute with the hamiltonian, but $$L^2 , J^2, S^2$$ do, and you have to use $$J^2 = (L+S)^2 = L^2 + S^2 + 2S.L$$. I hope that's right. Should be I have an exam on it in the next fortnight!

C.

 

[R0man]

Firstly, let's discuss the limitation of Hund's rules and L-S coupling in relation to the exclusion principle. As you mentioned, the maximum values of S and L are subject to the condition that no two electrons may have the same pair of values for m(sub s) and m(sub l). This means that when we are adding the angular momenta of electrons, we must consider the spin and orbital angular momentum quantum numbers of each individual electron. This limitation is due to the Pauli exclusion principle, which states that no two electrons in an atom can have the same set of quantum numbers. Therefore, when adding angular momenta, we must ensure that no two electrons have the same combination of spin and orbital angular momentum quantum numbers, as this would violate the exclusion principle.

To answer your second question, it is incorrect to have L = Sum(l) = 6*3 for the ground state of Sm (4f)6 because this would imply that all six electrons have the same orbital angular momentum quantum number. This is not possible according to the exclusion principle. Instead, the correct ground state configuration for Sm (4f)6 is L = Sum(l) = 6*2, meaning that there are two electrons in each of the three 4f orbitals.

In terms of explanations, the use of symmetry considerations and the exclusion principle are not necessarily two different sets of explanations. In fact, they are often used together to understand the electronic configurations of atoms. The symmetry considerations can help us determine the possible combinations of quantum numbers for a given number of electrons, while the exclusion principle ensures that these combinations do not violate the principle. Therefore, both explanations are important in understanding the limitations and confusions surrounding Hund's rules and L-S coupling.

In conclusion, Hund's rules and L-S coupling are important concepts in understanding the electronic configurations of atoms. It is crucial to consider the limitations and confusions that may arise when applying these rules, such as the requirement to avoid identical combinations of quantum numbers and the incorrect use of symmetry considerations. By understanding these limitations and using both symmetry considerations and the exclusion principle, we can accurately determine the ground state configurations of atoms.