Spin orbit interaction energy

AiiEohc ntgcao 137.036 Thus, the spin-orbit interaction energy is about 5 X 10 times smaller than 1 11-... 

Since the spin-orbit interaction energy is small, the solution of equations (7.43) to obtain E is most easily accomplished by means of perturbation theory, a technique which is presented in Chapter 9. The evaluation of E is left as a problem at the end of Chapter 9. 

Using first-order perturbation theory, show that the spin-orbit interaction energy for a hydrogen atom is given by... 

Calculating the spin-orbit interaction energy 8.0 by finding the eigenfunctions and eigenvalues of the operator + Hs.o. where (u.i) is the Hamiltonian of Eq. (11.1), is difficult. One therefore usually estimates 8.0. by using perturbation theory. Except for heavy atoms, the effect of s.o. is small compared with the effect of (u.i), and first-order perturbation theory can be used to estimate 80.-... 

For a raany-electron atom, it can be shown (Bethe and Jackiw, p. 164) that the spin-orbit interaction energy is... 

We have based the discussion on a scheme in which we first added the individual electronic orbital angular momenta to form a total-orbital-angular-momentum vector and did the same for the spins L = S,- L, and S = 2i S,. We then combined L and S to get J. This scheme is called Russell-Saunders couplit (or L-S coupling) and is appropriate where the spin-orbit interaction energy is small compared with the interelec-tronic repulsion energy. The operators L and S commute with + W,ep, but when is included in the Hamiltonian, L and no longer commute with H. (J does commute with + //rep + Q ) If the spin-orbit interaction is small, then L and S almost commute with (t, and L-S coupling is valid. 

Hie spin-orbit interaction energy in diatomic molecules can be shown to be well approximated by A A2, where A depends on A and on the intemuclear distance R but not on 2. The spacing between levels of the multiplet is thus constant. When A is positive, the level with the lowest value of A + 2 lies lowest, and the multiplet is regular. When A is negative, the multiplet is inverted. Note that for A 0 the spin multiplicity 25+1 always equals the number of multiplet components. This is not always true for atoms. 

The nature of the spin-orbit part of the Hamiltonian operator can be obtained from the classical spin-orbit interaction energy by replacing the classical dynamical variables by their quantum mechanical operators. The magnetic dipole moment of a spinning electron is given by... 

This value gives a typical order of magnitude for the spin-orbit interaction energy, in hydrogen almost 500,000 times smaller than the ionization energy Jbut still a large effect in certain kinds of spectroscopy. 

The term symbols encode the values of the quantum numbers L, S, and /, and an expression for the spin-orbit interaction energy of a many-electron atom follows directly. With the Hamiltonian in Equation 10.21 and with a phenomenological constant, y, specific to the atomic system, the energy associated with spin-orbit interaction in the absence of an exfernal magnetic field is... 

Compute the spin-orbit interaction energy in terms of the parameter y in Equation 10.24 for the all states of an atom with occupancy Is 2s 2p 3s 3d. ... 

Among these reports, there exists a discrepancy in the quoted values for the spin-orbit interaction energy and the exciton binding energies, which most likely may have its genesis in the difficulty of interpreting the complex spectra near the band edge. 

When the individual electron spin-orbit interaction energy prevails over the spin-orbit interactions of different electrons, we first have to sum the orbital and spin moments of every electron (to obtain all /), then sum these / and further obtain the total atomic angular momentum quantum numbers and finally the overall quantum number 7, characterizing the total angular momentum of the atom. This type of binding is called a j-j bond. Since the spin-orbit interaction is proportional to Z, this type of binding occurs mainly in heavy atoms. We will not consider further this type of coupling. 

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