Page 275 - Tunable Lasers Handbook

6 Transition Metal Solid-state Lasers 35

S=-K(T)(Y2,(6.$) + &&$))/2': , ;5 j

Electron o'rbits described by these linear combinations of functions are graphed
in Fig. 6. As can be seen, the 3dT orbits are maximized along the .I, y, and I
axes. that is, the orbits are directed ton ard the positions of the nearest neighbors.
On the other hand, the 3d~ orbits are maximized at angles directed between the
nearest neighbors. Because the nearest neighbors usually have a net negative
charge, it is logical that the orbits directed toward the nearest neighbors uould
have a higher energy. In essence, the electrons are being forced to go where they
are being repulsed.
A calculation of the energies of the molecular bonding orbits must include
the effects of the mutual repulsion. Mutual repulsion energy contributions can be
expressed in terms of the Racah parameters, A. B, and C Racah parameters, in
turn. are expressed in term5 of Slater integrals: however, it is beyond the scope
of xhis chapter to delve into the details. Suffice it to say that the 4 term is an
additive term on all of the diagonal elements. When only energy differences are
to be calculated. this term drops out. The B and C energy terms occur on many
off-diagonal elements. However. Tanabe and Sugano observed that the ratio of
C/B is nearly constant and in the range of 4 to 5. A slight increase of this ratio is
noted as the nuclear charge increases while the number of electrons remains
constant. A. ratio of C/B of 3.97 was expected based on Slater integral formalism.
Thus. the mutual repulsion contribution to the energy levels can be approxi-
mated if only a single parameter is known. Usually this parameter is the Racah
parameter B. Hence, many of the Tanabe-Sugano calculations are normalized by
this parameter.
Crystal field contributions to the energy of the molecular orbits can be
described by the parameter Dq. Remember that lODq is the energy difference
between the 3dT and the 3~1e levels for a single 3d electron. Consider the case
where there are N electrons. These electrons can be split between the 3dT and
3d~ orbits. Suppose II of these electrons are in the 3de orbits. leaving N-n of
them in the 3dT orbits. Crystal field effect contributions to the energy can be
approximated as (6N - 1On)Dq. Crystal field energy contributions. in this simpli-
fied approach, occur only for diagonal energy matrix elements.
Energy differences between the various levels have been calculated for all
combinations of electrons in octahedral symmetry and are presented in Tanabe-
Sugano diagrams. Such diagrams often plot the energy difference between vari-
ous energy levels, normalized by the Racah B parameter. as a function of the
crystal field parameter, again normalized by the Racah B parameter. A
Tanabe-Sugano diagram for three electrons in the 3d subshell is presented in