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In the energy levels of some atoms, like Be, you sometimes see $S(F=0)$, $S(F=1)$ and these are further divided into sublevels $m_F = -1, 0, +1$. I know this $F$ does not stand for the orbital angular momentum, as in $S, P , D, F$. What is it exactly, and why is it also called $F$?

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The quantum number $F$ is due to the coupling of the net orbital angular momentum and the net nuclear spin.

The hyperfine interaction is due to an interaction between the nuclear magnetic dipole moment and the magnetic field due to the electron spin.

$$H_{HF} = - \vec{\mu_N} . \vec{B_J}$$

since $\vec B_J \propto \vec J$ and depends only on J, we have $H_{HF} = A(J) \vec I . \vec J$

The quantum number $\vec F = \vec I + \vec J$ represents the total angular momentum of the atom, and obeys the same eigenvalue equations as you might expect.

$$\vec F^2 =F(F + 1)\hbar^2$$

$$F_z = m_F \hbar$$

with $$|I - J| \leq F \leq I + J$$ and $$m_F = -F, -F+1, ..., F$$

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Found an answer:

Hyperfine structure arises from the coupling between the magnetic moment of the electron and the nuclear magnetic moment. For the quantum number I defining the net nuclear spin (analagous to the net electron spin $S$), you can construct another quantum number $F=I+J$, which takes on values $J-I,J-I+1, ... J+I-1, J+I$.

Source: A Primer on Quantum Numbers and Spectroscopic Notation, Frederick M. Walter, lecture notes at SUNY SB.

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