Page 282 - Tunable Lasers Handbook

242 Norman P. Barnes

where A, is a rate constant. E is a coupling constant, p is the number of phonons
required to span the gap between the manifolds. and

is the thermal occupation factor for the phonons, vp being the phonon frequency.
In this formulation, the first two factors are nominally temperature independent
so that the temperature dependence is carried by the thermal occupation factor
for the phonons.
To reconcile these two theories. Struck and Fonger relied on a single config-
urational coordinate model. In the simplest application of the single configura-
tional coordinate model. the interaction of the active atom and its nearest neigh-
bors is considered to be described by a single configurational parameter. A
configurational parameter can describe one aspect of the geometrical configura-
tion of the active atom with its nearest neighbors. As an example, a configura-
tional parameter for an active atom in a position of octahedral symmetry could
be the average distance between the active atom and its six nearest neighbors. As
the single configurational coordinate changes, the average distance between the
active atom and its six nearest neighbors expands or contracts. In this case, the
expansion and contraction is reminiscent of the breathing motion; consequently,
it is often referred to as the breathing mode.
Energies associated with different manifolds are dependent on this single
configurational coordinate. Typically. energy as a function of the configuration
coordinate appears as a parabola as shown in Fig. 8. Equilibrium positions are
found near the lowest point in the parabola. That is, it would require energy to
either expand or contract the configurational coordinate. For example, as the
length between the active atom and its nearest neighbors contracts, the mutual
repulsion of like charges would tend to dominate and push the nearest neighbors
away. The strength of the interaction can be gauged from the shape of the para-
bolic curves. If the energy depends strongly on the configurational coordinate,
the parabola will be more strongly curved. Conversely, if the parabola is weakly
curved, the energy depends only weakly on the configurational coordinate.
Although the curvature of the parabolas for different manifolds can be different,
a case can be made for them being roughly equal.
The curvature of the parabolas describing the energy versus configurational
coordinate determines the energy spacing between adjacent energy levels within
the manifold. If a particle is trapped in a potential well described by a parabolic
form. the particle will undergo simple harmonic motion. For the atoms involved
in the configurational coordinate model, the harmonic motion must be described
using quantum mechanics. For this reason, Struck and Fonger refer to a quantum
mechanical single configurational coordinate. Quantizing the simple harmonic
motion introduces two effects not found in classical simple harmonic oscillators,