Chapter 8, Problem 60E

The well-known sodium doublet is two yellow spectral lines of very close wavelength. 589.0 nm and 589.6 nm. It is caused by splitting of the 3p energy level, due to the spin-orbit interaction. In its ground state, sodium’s single valence electron is in the 3s level, it may be excited to the next higher level, the
3p, then emit a photon as it drops back to the 3s. However, the 3p is actually two levels, in which L and S are aligned and antialigned. (In the notation of Section 8.7 these are, respectively, the $3p_{3/2}$ and the
$3p_{1/2}$ .) Because the transitions start from slightly different initial energies yet have identical final energies

(the 3s having no orbital angular momentum to lead to spin−orbit interaction), there are two different wavelengths possible for the emitted photon. Calculate the difference in energy between the two photons. From this, obtain a rough value of the average strength of the internal magnetic field experienced by sodium’s valence electron.