CHEM 322 – 01
Chapter 13
Spin-Orbit Coupling
I. Spin-Orbit Coupling in Hydrogenic Atoms
a. Because an electron has spin angular momentum, and because a moving charge generates a magnetic field, an electron has a magnetic moment that arises from its spin. Also, that same electron has an orbital angular momentum and possesses a magnetic moment that arises from its orbital angular momentum. The interaction of the spin magnetic moment and the magnetic field arising from the orbital angular momentum is known as spin-orbit coupling.
b. The extent of the spin-orbit coupling depends on the relative orientation of the spin and orbital magnetic moments, and therefore on the relative orientation of the spin and orbital angular momenta.
c. The total angular momentum of an electron is described by the quantum numbers l and . s
jls=+
When the two angular momenta are in the same direction, 12jl=+, and when they are in the opposite direction, 12=−jl. The different values of that can arise for a given value of l label levels of a term. j
d. The dependence of the spin-orbit interaction on the value of is expressed in terms of the spin-orbit coupling constant, . The energies of the levels with quantum numbers l, , and jAsj are given by the equation:
The Essay on Magnetic Field Angular Momentum Direction
Some materials have a feature known as ferromagnetism. The prefix 'ferro' refers to Iron, which is one such material. Ferromagnetic materials have the ability to 'remember' the magnetic fields they have been subjected to. An atom consists of a number of negatively charged electrons, orbiting around a positively charged nucleus. These electrons also possess a quantity known as spin, which is ...
()()(),,1112lsjEhcAjjllss=+−+−−1
e. The strength of the spin-orbit coupling increases sharply with atomic number ( as 4Z).
So there is less spin-orbit coupling in hydrogen than in lead.
f. Observed transitions in the electronic spectrum of atoms arising from levels of different values of j are known as fine structure of the spectrum.
II. Term Symbols of Multi-Electron Atoms
a. A term symbol is a label used to indicate the values of the orbital, spin, and total angular momentum of an electronic state, and since the energy of that state depends on the values of these angular momenta, the term symbol becomes a useful way to label an electronic state. The general form of a term symbol is:
21SJL+
is the quantum number indicating the total orbital angular momentum of an electronic state. is the quantum number representing the total spin angular momentum of the electronic state. is the quantum number representing the total angular momentum of the electronic state. The quantity is called the multiplicity of the state. L2SSJ
1 +
b. The value of is obtained by coupling the individual orbital angular momentum of two electrons. This is done using the Clebsch-Gordan series: L
12121212, 1, 2, …, Lllllllll=++−+−−
If more than two electrons are present in an atom, the orbital angular momentum of each additional electron is coupled to previous value of , using the same formula. The Clebsch-Gordan series terminates at L12ll− because the total orbital angular momentum , like each individual orbital momentum l cannot be less than zero. A closed shell has zero orbital angular momentum because all of the individual orbital angular momenta add up to be zero. Thus, we need to only consider the electrons of the unfilled shell. L
c. The value of is obtained by coupling the individual spin angular momentum of two electrons using the Clebsch-Gordan series: S
12121212, 1, 2, …, sssssss=++−+−−Ss
If more than two electrons are present in an atom, the spin angular momentum of each additional electron is coupled to the previous value of , using the same formula. For an electron, recall that S12s=.
The Term Paper on The Atom Ground State
... orbital angular momentum, and projection of spin angular momentum.) It is also assumed -- as had first been suggested by Wolfgang PAULI in 1924 -- that not wo electrons ... in certain 'states'. The states in which they can exist are those in which the ANGULAR MOMENTUM of their orbits ... invisible substructure of objects with unfamiliar properties. Democritus stated this especially clearly in one of the few ...
d. The total angular momentum quantum number gives the relative orientations of the spin and orbital angular momenta for several electrons. The strength of spin-orbit coupling affects the value of . If the spin-orbit coupling is weak (as in atoms of low atomic number
