A New Spin on Photoemission Spectroscopy Page: 34 of 259
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and inserted into equation 1.20 to give
1 FdV x 1
HLS = 2mec2 [ dr r J
1 1 dV
(L S), (1.22)
2mec2 r dr'
where we have used the definition of orbital angular momentum, L = x x p. This last
expression shows the origin of the name 'spin-orbit coupling' as the interaction depends on
the term L - S.
To proceed further, we define the total angular momentum of the electron to be J with
J = L + S (1.23)
Taking the dot product of equation 1.23 with itself gives
J2 = L2 + S2 + 2 (L - S). (1.24)
With this simple trick, equation 1.22 can be rewritten as
HLS = 2 7 Jr (J2 - L2 _ 52) . (1.25)
2mec2 r dr 2
The spin-orbit Hamiltonian, HLS, then splits atomic energy levels according to the total
and orbital quantum numbers. Perturbation theory gives the first order energy shift of the
levels with j, 1, and the standard quantum number n as
1 ldVc h2 l j=l+1/2
On = - - -mc x .r/i (1.26)
= 2m~c2 'r dr 2 -(l + 1) j = 1 - 1/2
This is Lande's interval rule.
It is important to remember that the spin-orbit interaction is a relativistic effect. As
such, it is a large effect only when electron velocities are near the speed of light, clearly
reflected in the (v/c) term in equation 1.17. The spin-orbit interaction is thus important
when dealing with free electrons with many keV of energy, or when dealing with electrons
near high Z nuclei with large electric fields (large dV/dr in equation 1.26). It should also
be noted that the r -v term in equation 1.22 and on show that in Coulomb potentials, the
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Jozwiak, Chris. A New Spin on Photoemission Spectroscopy, thesis or dissertation, December 1, 2008; United States. (https://digital.library.unt.edu/ark:/67531/metadc1014237/m1/34/: accessed April 23, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.