# A Difference between russel-saunders-coupling and spin-orbit-coupling

#### JanSpintronics

Hello everybody,

Im confused with the difference of the both coupling phenomena, is it just the difference in the amount of electron?
So spin-orbit-coupling is just the coupling between orbit and spin of one electron and the russel-saunders is the coupling of a spin of many electrons and the orbit momentum of many electrons?
So if we just consider 1electron the both couplings are the same?

Thank you for any help

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#### TeethWhitener

Gold Member
Russell-Saunders is spin-orbit coupling, where spin-orbit coupling is small (light atoms where the electrons aren't particularly relativistic). In this case, the total orbital angular momentum $\mathbf{L}$ and the total spin angular momentum $\mathbf{S}$ are both good quantum numbers (meaning their operators commute with the Hamiltonian to a good approximation). So we obtain total angular momentum $\mathbf{J}$ simply by adding total spin and total orbital angular momentum: $\mathbf{J} = \mathbf{L} + \mathbf{S}$.

In heavier atoms, where the electrons are relativistic, the spin orbit coupling is quite large, and total spin/orbital angular momentum are no longer good quantum numbers. In this case, we have to add the spin and orbital angular momenta for each electron separately ($\mathbf{\ell}_i$ and $\mathbf{s}_i$, respectively) to get total angular momentum:
$$\mathbf{J} = \sum_i {\left(\mathbf{\ell}_i + \mathbf{s}_i\right)}$$

• vanhees71

#### JanSpintronics

Hello

Well that sentense:
Russell-Saunders is spin-orbit coupling, where spin-orbit coupling is small
That doesnt make sense for me, but obviously you talk about 2 different spin orbit coupling but i dont know what you meant with the "small" spin orbit coupling and the other which you identify with Russel saunders coupling. Can you explain that please?

Do you meant with the "small" spin orbit coupling the coupling between the electron spin and orbit momentum, therefore with the russel saunders the coupling of the collective spin and orbit momentum?

And to understand what you wrote, you mean that this is true:

$$J_1=\sum_{i}(s_i + l_i) \neq \sum_i s_i + \sum_i l_i= J_2$$ ??

And just quick to come back to the question if we just consider 1 electron, are they both the same, identical?

#### TeethWhitener

Gold Member
That doesnt make sense for me, but obviously you talk about 2 different spin orbit coupling but i dont know what you meant with the "small" spin orbit coupling and the other which you identify with Russel saunders coupling. Can you explain that please?
I don’t know if I can be clearer than the part you quoted. This link:
http://users.aber.ac.uk/ruw/teach/327/lsjj.php
might be of more help. It goes through Russell-Saunders (LS) coupling and jj coupling with examples.

• vanhees71