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:
Grüneisen constant for perfect crystals with pair interactions edit
The expression for the Grüneisen constant of a perfect crystal with pair interactions in -dimensional space has the form:[2]
Lattice | Dimensionality ( ) | Lennard-Jones potential | Mie Potential | Morse potential |
---|---|---|---|---|
Chain | ||||
Triangular lattice | ||||
FCC, BCC | ||||
"Hyperlattice" | ||||
General formula |
The expression for the Grüneisen constant of a 1D chain with Mie potential exactly coincides with the results of MacDonald and Roy.[4] Using the relation between the Grüneisen parameter and interatomic potential one can derive the simple necessary and sufficient condition for Negative Thermal Expansion in perfect crystals with pair interactions A proper description of the Grüneisen parameter represents a stringent test for any type of interatomic potential.
Microscopic definition via the phonon frequencies edit
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 . The Grüneisen parameter of an individual vibrational mode can then be defined as (the negative of) the logarithmic derivative of the corresponding frequency :
Relationship between microscopic and thermodynamic models edit
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
Proof edit
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
See also edit
External links edit
References edit
- ^ Grüneisen, E. (1912), "Theorie des festen Zustandes einatomiger Elemente", Annalen der Physik, 344 (12): 257–306, Bibcode:1912AnP...344..257G, doi:10.1002/andp.19123441202
- ^ Krivtsov, A.M.; Kuzkin, V.A. (2011), "Derivation of Equations of State for Ideal Crystals of Simple Structure", Mechanics of Solids, 46 (3): 387–399, Bibcode:2011MeSol..46..387K, doi:10.3103/S002565441103006X, S2CID 51837957
- ^ "Mie potential page on SklogWiki - a wiki for statistical mechanics and thermodynamics". www.sklogwiki.org. Retrieved 2019-11-19.
- ^ MacDonald, D. K. C.; Roy, S.K. (1955), "Vibrational Anharmonicity and Lattice Thermal Properties. II", Phys. Rev., 97 (3): 673–676, Bibcode:1955PhRv...97..673M, doi:10.1103/PhysRev.97.673