Ph.D. thesis (pdf) - dirac

4.3. Inelastic Scattering Experiments 63
Combining this observation with equation 4.3.28 it follows that the temperature
dependence of the non-ergodicity factor is given by
f(Q, t) =
exp(−2W)S is (Q)
exp(−2W)S is (Q) + exp(−2W)aT = 1
1 + a ′ T
(4.3.34)
where the factor a encompasses all the Q-dependent prefactors in equation 4.3.31
and a ′ = a/S is (Q). This temperature dependence is the starting point for the
definition of the parameter α which is studied in section 6.4.
The one-phonon contribution to the incoherent structure factor is not dependent on
the wave vector of the modes but only on their frequency. The dynamical structure
factor is given by an expression similar to equation 4.3.31
S inc,inel (Q, ω) =
exp(−2W)
2MN
∑ (Q · e s ) 2
s
ω s
[〈n s + 1〉δ(ω − ω s ) + 〈n s 〉δ(ω + ω s )]
where the sum over s is to be taken over all modes independent of their wave vector.
It is conventional (and convenient) to introduce the vibrational density of states,
g(ω), and to replace the sum over the modes by an integral:
S inel,inc (Q, ω) = 1
2M exp(−2W)Q2n(ω) g(ω). (4.3.35)
ω
Here (Q · e s ) 2 is replaced by its averaged over all modes with frequency ω s , which
in a cubic Bravais crystal or an isotropic system is given by Q 2 /3. This expression
is used for analyzing the data presented in chapter 8.
4.3.8 Simple models for liquids
In this work we do not study “normal” non-viscous liquids, but a notion of the
dynamics in this high temperature limiting case is useful in the interpretation of the
dynamics in highly viscous systems. Liquids are traditionally modeled as classical
systems. The comparison between the result of models and experimental data is
made by adjusting the result of the model to obey the principle of detailed balance
[Squires, 1978];
( ) ω
S(ω) = exp S classic (ω). (4.3.36)
2k B T
Liquids are disordered and therefore isotropic on average. This means that the structure
factor and the intermediate scattering function depend only on the magnitude
of the Q-vector, not on its direction.