Is it possible to use this formula $E = En{n}*(Z\alpha)^2*(\frac{1}{j+1/2}-\frac{3}{4n})$ to determine the spin orbit coupling of an atom with atomic number Z at energy level n ? $\alpha=1/137$ is the fine structure constant.

The formula comes from the fine structure article in wikipedia and takes into account the relativistic correction : https://en.wikipedia.org/wiki/Fine_structure?

I would like to find the energy splitting between the levels $3p^{1/2}$ and $3p^{3/2}$ of Na which experimentally is found to be 0.0021 eV. However when I used the formula $E_n = -E_0*Z^2/n^2$ with $E_0 = 13.6$ eV and $Z = 11$, $j=3/2$ and $n=3$ for Na $3p^{1/2}$ I don't the right order of magnitude : $\Delta E_{SO} \approx -2$ eV.

I am not even sure if the formula $E_n = -E_0*Z^2/n^2$ gives realistic values for the energy levels because they are far away from the binding energies measured with XPS or XAS. Example $E_1(Na) = 1645$ eV but $Na1s = 1070.8$ eV and $E_2(Na) = 411$ eV where $Na2s = 63$ eV.

Thanks,