Although nominally outside the terms of reference of this thesis, the following example serves to illustrate the application of the preceding theory, to clear up an historical misunderstanding and to provide continuity with recent studies of muonium diffusion.

Experiments performed by Kiefl *et.al.* [66]
and MacFarlane *et.al.* [67]
measured the muonium hop rate in KCl
over a wide temperature range, clearly showing the
position of the minimum hop rate, characteristic of the
crossover from two-phonon limited diffusion to one-phonon
activated diffusion, at K.
The measured hop rate below was fitted to a
power law relation , yielding
a value .It was noted at the time that the low temperature limiting value of
predicted by theory was 7 (or 9 in a perfect fcc lattice
if the sites are symmetric with respect to the phonon modes);
the discrepancy with the measured result remained unexplained.

To answer the question of why the hop rate in KCl
did not follow a *T ^{-7}* dependence,
Kagan and Prokof'ev argued in their 1990 paper [54]
that if one used the real
phonon spectrum in Eq. (6.8) one could obtain agreement
with the measured

The total phonon density of states of KCl,
measured by inelastic neutron scattering [55],
is shown in Fig. 6.15.
The damping rate can then be calculated
using this spectrum for in a numerical integration over phonon frequency .(The Debye temperature of KCl is about 230 K, but the real phonon
density of states has its upper cut-off at
.)
In this calculation, is normalized
so that its integral is unity,
which does not affect its *T*-dependence, only an overall factor.
The correct normalization of the phonon density of states can be
obtained by calculating the lattice specific heat

(10) |

Figure 6.16 shows the temperature dependence of , along with the result obtained if it is assumed that .Figure 6.17 shows the temperature dependence of the power law exponent

for the same cases.
The principal cause of the weak temperature dependence of
the muonium hop rate in KCl is that the temperature where the
hop rate minimum occurs is already a sufficiently large fraction of
that the entire phonon spectrum contributes to
.
The population of low frequency phonons
for which increases only linearly with *T*;
the temperature dependence of drops off.
Since the temperature at which the minimum hop rate occurs is
70 K, a substantial fraction of the Debye temperature,
we are far from the low temperature limit
where one obtains .It can be seen from the graph that this is the case whether the real
spectrum or a Debye-like is used. It is an intrinsic
property of two-phonon diffusion, *not* the structure in the spectrum,
that causes to be characterized by a lower at temperatures greater than about or so.
The structure in the real has a small effect on , as can be seen,
but the overall shape of the function is hardly affected.
A weak temperature dependence of the 2-phonon diffusion rate
will occur in any crystal for which (the temperature
where the crossover between two-phonon and one-phonon
regimes occurs) happens to be more than
about , so that two-phonon diffusion
is important at relatively high temperatures.
Related to this, it must be stressed that the
dependence is expected
*only* in the low temperature limit .It is a simple exercise to show that in the high temperature limit
(several times ), .At all intermediate temperatures, the characteristic exponent is a function of temperature, and it is not very meaningful
to apply a single value to over a range of temperatures.
It is also possible that the coupling to
high-frequency modes doesn't
follow the simple low frequency limiting
behavior.
If this is the case, the model will fail completely at higher
temperatures.