Grüneisen parameter explained

In condensed matter, Grüneisen parameter is a dimensionless thermodynamic parameter named after German physicist Eduard Grüneisen, whose original definition was formulated in terms of the phonon nonlinearities.

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:$$\backslash gamma\; =\; V\; \backslash left(\backslash frac\backslash right)\_V\; =\; \backslash frac\; =\; \backslash frac\; =\; \backslash frac\; =\; -\backslash left(\backslash frac\backslash right)\_S$$where is volume,

C _{P} and C _{V} are the principal (i.e. per-mass) heat capacities at constant pressure and volume, is energy, is entropy, is the volume thermal expansion coefficient, K _{S} and K _{T} are the adiabatic and isothermal bulk moduli, v _{s} is the speed of sound in the medium, and is density. The Grüneisen parameter is dimensionless.

Grüneisen constant for perfect crystals with pair interactions

The expression for the Grüneisen constant of a perfect crystal with pair interactions in

d -dimensional space has the form:$$\backslash Gamma\_0\; =\; -\backslash frac\backslash frac,$$where \Pi is the interatomic potential, a is the equilibrium distance, d is the space dimensionality. Relations between the Grüneisen constant and parameters of Lennard-Jones, Morse, and Mie^[1] potentials are presented in the table below.

Lattice	Dimensionality (d )	Lennard-Jones potential	Mie Potential	Morse potential
Chain	1	10 (1)/(2)	(m + n +3)/(2)	(3 \alpha a)/(2)
Triangular lattice	2	5	(m + n +2)/(4)	(3 \alpha a -1)/(4)
FCC, BCC	3	(19)/(6)	(n + m +1)/(6)	(3 \alpha a -2)/(6)
"Hyperlattice"	inf ty	- (1)/(2)	- (1)/(2)	- (1)/(2)
General formula		(11)/(- d) (1)/(2)	(m + n +4)/(- 2 d) (1)/(2)	(3 \alpha a +1)/(- 2 d) (1)/(2)

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 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

\Pi'''(a) a > - (d -1 )\Pi''(a). A proper description of the Grüneisen parameter represents a stringent test for any type of interatomic potential.

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 ω_i 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 ω_i change with the volume

V . The Grüneisen parameter of an individual vibrational mode i can then be defined as (the negative of) the logarithmic derivative of the corresponding frequency \omega _{i} :

$$\backslash gamma\_i=\; -\; \backslash frac\; \backslash frac.$$

Relationship between microscopic and thermodynamic models

Using the quasi-harmonic approximation for atomic vibrations, the macroscopic Grüneisen parameter can be related to the description of how the vibrational frequencies (phonons) within a crystal are altered with changing volume (i.e. 's). For example, one can show that$$\backslash gamma\; =\; \backslash frac$$if one defines

\gamma as the weighted average$$\backslash gamma\; =\; \backslash frac,$$where c _{V,i} 's are the partial vibrational mode contributions to the heat capacity, such that $C\_\; =\; \backslash frac\; \backslash sum\_i\; c\_\; .$

Proof

To prove this relation, it is easiest to introduce the heat capacity per particle $\backslash tilde\_V\; =\; \backslash sum\_i\; c\_$; so one can write$$\backslash frac\; =\; \backslash frac\; =\; \backslash frac.$$

This way, it suffices to prove$$\backslash sum\_i\; \backslash gamma\_i\; c\_\; =\; \backslash alpha\; V\; K\_T.$$

Left-hand side (def):$$\backslash sum\_i\; \backslash gamma\_i\; c\_\; =\; \backslash sum\_i\; \backslash left[-\; \backslash frac\{V\}\{\backslash omega\_i\}\; \backslash frac\{\backslash partial\; \backslash omega\_i\}\{\; \backslash partial\; V\}\; \backslash right]\; \backslash left[k\_\{\backslash rm\; B\}\; \backslash left(\backslash frac\{\backslash hbar\; \backslash omega\_i\}\{k\_\{\backslash rm\; B\}\; T\}\backslash right)^2\; \backslash frac\{\backslash exp\backslash left(\backslash frac\{\backslash hbar\; \backslash omega\_i\}\{k\_\{\backslash rm\; B\}\; T\}\; \backslash right)\}\{\backslash left[\backslash exp\backslash left(\backslash frac\{\backslash hbar\; \backslash omega\_i\}\{k\_\{\backslash rm\; B\}\; T\}\backslash right)\; -\; 1\backslash right]^2\}\; \backslash right]$$

Right-hand side (def):$$\backslash alpha\; V\; K\_T\; =\; \backslash left[\backslash frac\{1\}\{V\}\; \backslash left(\backslash frac\{\backslash partial\; V\}\{\; \backslash partial\; T\}\backslash right)\_P\; \backslash right]\; V\; \backslash left[-V\; \backslash left(\backslash frac\{\backslash partial\; P\}\{\backslash partial\; V\}\backslash right)\_T\backslash right]\; =\; -\; V\; \backslash left(\backslash frac\; \backslash right)\_P\; \backslash left(\backslash frac\backslash right)\_T$$

Furthermore (Maxwell relations):$$\backslash left(\backslash frac\; \backslash right)\_P\; =\; \backslash frac\; \backslash left(\backslash frac\backslash right)\_T\; =\; \backslash frac\; \backslash left(\backslash frac\backslash right)\_P\; =\; -\; \backslash left(\backslash frac\; \backslash right)\_T$$

Thus$$\backslash alpha\; V\; K\_T\; =\; V\; \backslash left(\backslash frac\; \backslash right)\_T\; \backslash left(\backslash frac\backslash right)\_T\; =\; V\; \backslash left(\backslash frac\; \backslash right)\_T$$

This derivative is straightforward to determine in the quasi-harmonic approximation, as only the are V-dependent.

$$\backslash frac\; =\; \backslash frac\; \backslash left\backslash$$

$$V\; \backslash frac\; =\; -\; \backslash sum\_i\; \backslash frac\; \backslash frac\; \backslash ;\backslash ;\; k\_\; \backslash left(\backslash frac\backslash right)^2\; \backslash frac\; =\; \backslash sum\_i\; \backslash gamma\_i\; c\_$$

This yields$$\backslash gamma\; =\; \backslash dfrac\; =\; \backslash dfrac.$$

See also

• Debye model
• Negative thermal expansion
• Mie–Grüneisen equation of state

External links

• Definition from Eric Weisstein's World of Physics

Notes and References

1. Web site: Mie potential page on SklogWiki - a wiki for statistical mechanics and thermodynamics. www.sklogwiki.org. 2019-11-19.