In condensed matter, Grüneisen parameter γ is a dimensionless thermodynamic parameter named after German physicist Eduard Grüneisen, whose original definition was formulated in terms of the phonon nonlinearities.[1]
Because of the equivalences of 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 interpretations of its meaning. Some formulations for the Grüneisen parameter include:
where V is volume, and are the principal (i.e. per-mass) heat capacities at constant pressure and volume, E is energy, S is entropy, α is the volume thermal expansion coefficient, and are the adiabatic and isothermal bulk moduli, is the speed of sound in the medium, and ρ is density. The Grüneisen parameter is dimensionless.
Using the quasi-harmonic approximation for atomic vibrations, the macroscopic Grüneisen parameter (γ) can be related to the description of how the vibrational frequencies (phonons) within a crystal are altered with changing volume (i.e. γi's).
For example, one can show that
if one defines as the weighted average
where 's are the partial vibrational mode contributions to the heat capacity, such that
Proof
To prove this relation, it is easiest to introduce the heat capacity per particle ; so one can write
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