The Spin-Orbit Interaction in Sodium

The Spin-Orbit Coupling Energy

In the classical description of an electron orbiting a proton, we have a picture of two charges moving with relative velocities with respect to each other. And, any moving charge produces a magnetic field. So an electron orbiting a nucleus creates a magnetic moment (mu) U that points in the direction opposite of L, the orbital angular momentum.

Generally R X V gives the direction of the magnetic moment by the right hand rule, but since the charge is negative for the electron, (mu)u points in the direction opposite L. The classical magnetic moment relation is (mu)u=IA for a loop of current I, enclosed by area A. And, the orientation of the magnetic moment vector is perpindicular to the loop. For a circulting charge: I=q/T Where the revolution period can be linked to the orbital angular momentum. So for an electron, (mu)u=-eL/2m.

And not only does the electron have orbital angular momentum, but it also has intrinsic angular momentum. The quantum number for the intrinsic angular momentum of an electron is S=1/2, and the electron has an intrinsic angular moment (mu)us=-2g*S/2m. Where g is a factor that accounts for the intrinsic charge to mass ratio of the electron. The g factor has been measured to be nearly equal to 2. So, the total magnetic moment of an electron is just the addition of the two vectors us+u=-e(L+gS)/2m. The total angular momentum J, is just the vector sum of the intrinsic and orbital parts. J=L+S.

The total angular momentum is conserved in all interaction and not necessairly L or S individually. The total angular momentum has the value J=(j(j+1)^1/2 h bar, where j can equal |l-s|, |l-s|+1...|l+s|. The z component of angular momentum has the value Jz=mjh bar where mj can be -j,-j+1,..j. It is common to label the value of the total momentum quantum number (j) with a subscript following the letter code for the orbital angualr momentum. Thus the 2p states of hydrogen are called the 2p(1/2) and 2p(3/2) states for j=(1/2) and j=(3/2) respectfully.

The spin-orbit ineteraction and the fine structure

If the orbital angular momentum is not zero, then the orbital motion of the electron creates an internal magnetic field (Bint). If we look at an atom in the rest frame of an electron, the fuzzy electron cloud and nucleus appear to revolve around the electron with orbital angular momentum L. (This time it is a positive charge doing the orbiting.) The intrinsic magnetic moment of the electron then interacts with the B-field so a tourqe force is felt on the electron which tries to align (mu)u with B. The magnetic moment is a generalized force that can do work, thus able to change the energy of the electrn. The size of the resulting energy shift is Delta E=-u dot B and proportional to S dot L. So the amount of energy shift depends on the relative orientations of the electron's intrinsic angular momentum vector S with respect to L. If the two vectors are aligned, then there will be a greater positiveshift in the energy. If the two are anti-aligned, and since the magnitude of L is larger than the magnitude of S, the addition of the two will yield a negative shift in energy, that is the state will have less energy than before.

Since the groundstate electron configuration fills up the first two principle quantum levels, it seems very likely tht the excited states will be transtioning between the levels in the third and fourth principle shells.

The total angular momentum vector for the 3s state has the following configuration. LZ=0 and S=1/2

The total angular momentum vector J for the 3p state has the following possible configurations:

--------Lz----------S---------J  
        1          1/2       3/2
        1         -1/2       1/2
       -1         -1/2      -3/2
       -1          1/2      -1/2
	0          1/2       1/2
	0         -1/2      -1/2

So the 3p(3/2) state electrons will have a total energy greater than the 3p state by a small amount since the momentum vectors S and L add to give a greater total angular momentum vector J. The 3p(1/2) state electrons on the other hand will be shifted to an energy level less than that of 3p since the angular momentum vectors more or less substract to give a smaller magnitude for the total angular momentum.

The 3d electron states also have a fine structure energy shift. The possible values for J are:

-------Lz-----------S--------J  
        2          1/2       5/2
        2         -1/2       3/2
       -2         -1/2      -5/2
       -2          1/2      -3/2
        1          1/2       3/2
        1         -1/2       1/2
       -1         -1/2      -3/2
       -1          1/2      -1/2
	0          1/2       1/2
	0         -1/2      -1/2

So again the 3d(5/2) state will have energy slightly higher than the 3d state due to the alinged momenta vectors S and L. Likewise the 3d(3/2) state will have slightly less energy.

Finally, forConclusions and Results