This is to answer the question from Alexandros:
The simplest approximation to the thermal expansion coefficient uses the
Debye or Einstein specific heat through the so-called Gruneisen formula
alpha=gamma*cv/(3*B*Vm)
where alpha is the linear expansion coefficient (1/3 of the volume TEC), B
is the Bulk modulus, cv is the lattice specific heat, Vm is the molar volume
and gamma is the Gruneisen coefficient. Note that the high-temperature
limit for both the Debye and Einstein formulas (cfr. Ashcroft & Mermin,
chapter 23) is cv->3*n*Kb, where n is the number of atoms in a molecule and
Kb is the Boltzmann constant (8.31 J/mol/K). For oxides with the perovskite
structure, B~150-180 GPa, Vm~3x10^-5 m^3/mol and n=5, so
alpha->gamma*7x10^-6 at high temperatures. Typical values of gamma are of
the order of 2.
The problem with using this formula to extract the Debye temperature
(thetad) from thermal expansion data is that it invariably yields
irrealistically low values of Thetad. The reason of this is that gamma is
an anharmonicity coefficient, which is not the same for different phonons.
So, strictly speaking, one gets an anharmonicity-weighted Debye temperature.
This problem has been addressed for alkali halides and perovskites in a
useful paper by H. Inaba (J. Cer. Soc. Japan, 106, 1988, p 272), which also
contains references to other materials. Inaba proposes a semi-empirical
model which works well for non-pathological systems, and (in principle)
should allow one to extract the true Debye temperature. In practice, I
think this is more useful when Thetad is known and one wants to fit the TE
data.
By the way, getting Debye tempeatures from Debye-Waller factors is also
risky, because one gets a complicated average over the phonons that project
on that particulat site and direction.
Paolo
