The Grüneisen parameter, γ, named after Eduard Grüneisen, describes the effect that changing the volume of a crystal lattice has on its vibrational properties, and, as a consequence, the effect that changing temperature has on the size or dynamics of the lattice. The term is usually reserved to describe the single thermodynamic property γ, which is a weighted average of the many separate parameters γi entering the original Grüneisen's formulation in terms of the phonon nonlinearities.
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Thermodynamic definitions
Because of the equivalences between many properties and derivatives within thermodynamics (e.g. see Maxwell Relations), there are many formulations of the Grüneisen parameter which are equally valid, leading to numerous distinct yet correct interpretations of its meaning.
Some formulations for the Grüneisen parameter include:
where V is volume,
Grüneisen constant for perfect crystals with pair interactions
The expression for the Grüneisen constant of a perfect crystal with pair interactions in
where
The expression for the Grüneisen constant of a 1D chain with Mie potential exactly coincides with the results of MacDonald and Roy. Using the relation between Grüneisen parameter and interatomic potential one can derive the simple necessary and sufficient condition for Negative Thermal Expansion in perfect crystals with pair interactions
Microscopic definition via the phonon frequencies
The physical meaning of the parameter can also be extended by combining thermodynamics with a reasonable microphysics model for the vibrating atoms within a crystal. When the restoring force acting on an atom displaced from its equilibrium position is linear in the atom's displacement, the frequencies ωi of individual phonons do not depend on the volume of the crystal or on the presence of other phonons, and the thermal expansion (and thus γ) is zero. When the restoring force is non-linear in the displacement, the phonon frequencies ωi change with the volume
Relationship between microscopic and thermodynamic models
Using the quasi-harmonic approximation for atomic vibrations, the macroscopic Grüneisen parameter (γ) can be related to the description of how the vibration frequencies (phonons) within a crystal are altered with changing volume (i.e. γi's). For example, one can show that
if one defines
where
Proof
To prove this relation, it is easiest to introduce the heat capacity per particle
This way, it suffices to prove
Left-hand side (def):
Right-hand side (def):
Furthermore (Maxwell relations):
Thus
This derivative is straightforward to determine in the quasi-harmonic approximation, as only the ωi are V-dependent.
This yields