Page 273 - Tunable Lasers Handbook

6 Transition Metal Solid-state lasers 33

In octahedrally coordinated crystal fields, the 3d energy levels split into two
levels. One set of these levels, denoted by 3d~, is lower than the initial 3d level
by amount -4Dq. These levels are triply degenerate. The other set of these lev-
els, denoted by 3dT, is higher than the initial 3d levels by amount 6Dq. These
levels are doubly degenerate. The term Dq is referred to as the crystalline field
parameter. It can be regarded as the measure of the overlap of the 3d electron
orbits with the electron orbits of the neighboring atoms comprising the laser
material. Even though Tanabe and Sugano refer to Dq as the crystalline field
parameter, some authors refer to the process of computing the energy levels as
ligand field theory.
In essence. the Tanabe-Sugano theory treats the active atom and the six
nearest neighbors as a molecule. The initial 3d orbits of the active atom are now
combined to form orbits associated with the formation of molecular bonds. That
is, the atomic electron orbits are combined so that the electron can follow com-
plex orbits that can take them in the vicinity of some of the atoms in the mole-
cule. For the 3dT orbits, the departure of the molecular bounding orbits from
the atomic 3d orbits of the active atom can be significant. Energy differ- ,nces
between any of the levels can be determined by calculating all of the various
terms in an energy matrix, Thus. the energies of the various interactions, specifi-
cally the mutual repulsion of the electrons and the crystal field effects, are cal-
culated using all possible combinations of orbits of the electrons and arranged
in a matrix. Energy levels are then computed by diagonalizing the resulting
matrix.
Even though the departure of the orbits from the atomic orbits can be signif-
icant, the orbits can be composed of a sum of atomic orbits, Atomic orbits can be
described as the product of a radial function R(r) and angular function Y,l,(O, $1.
The functions Y,,m(O. @) are referred to as the spherical harmonics and are com-
posed of a product of sine and cosine functions involving 0 and @. Functions
describing the 3dT orbits are the linear combinations

On the other hand. functions describing the 3d& orbits are the linear combi-
nations