Abstract
The tensor-operator and group-theoretical methods of Racah are used to examine the following contributions to the Hamiltonian of a rare-earth ion: (
a
) the Coulomb interaction between the electrons, (
b
) the spin-orbit coupling of the electrons, and (
c
) the term arising from the influence of an external electrostatic field such as occurs when the rare-earth ion is situated in a crystal lattice. With regard to (
a
), the positions of all the terms of the configurations
f
2
,
f
3
and
f
4
, as well as all the terms of other configurations
f
n
whose multiplicities are the highest or the highest but one, are tabulated on the basis of a hydrogenic 4
f
wave function. The theory of (
b
) is developed, and the parameter λ defining the spin-orbit splitting of a term is given for all terms whose energies have been found, (
c
) is treated by similar techniques and the theory is related to the one at present in use. All the general methods are illustrated by examples.
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