Grüneisen parameter

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 γ_i entering the original Grüneisen's formulation in terms of the phonon nonlinearities.^[1]

1 Thermodynamic definitions
2 Microscopic definition via the phonon frequencies
3 Relationship between microscopic and thermodynamic models
- 3.1 Derivation
4 See also
5 External links
6 References

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:

$\gamma = V \left(\frac{dP}{dE}\right)_V = \frac{\alpha K_S}{C_P \rho} = \frac{\alpha K_T}{C_V \rho}$

where V is volume, $C_P$ and $C_V$ are the principal (i.e. per-mass) heat capacities at constant pressure and volume, E is energy, α is the volume thermal expansion coefficient, $K_S$ and $K_T$ are the adiabatic and isothermal bulk moduli, and ρ is density.

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 $\omega_i$ :

$\gamma_i= - \frac{V}{\omega_i} \frac{\partial \omega_i}{\partial V}.$

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. γ_i's). For example, one can show that

$\gamma = \frac{\alpha K_T}{C_V \rho}$

if one defines $\gamma$ as the weighted average

$\gamma = \frac{\sum_i \gamma_i c_{V,i} }{ \sum_i c_{V,i} },$

where $c_{V,i}$ 's are the partial vibrational mode contributions to the heat capacity, such that $C_{V} = \frac{1}{\rho} \sum_i c_{V,i} .$

Derivation

To arrive at this result, one can start from the definitions of the two key quantities,

$\alpha = \frac{1}{V} (\partial V/\partial T)_P$ and $K_T = - V (\partial P/\partial V)_T$ ,

by noting that $(\partial P/\partial V)_T (\partial V/\partial T)_P = - (\partial T/\partial P)_V$ and thus

$\alpha = \frac{1}{K_T (\partial P/\partial T)_V}.$

To calculate the partial derivative in this equation, let us first relate $P$ to the internal energy $E$ . Writing the thermodynamic relationship $dQ = dE %2B P dV = TdS$ as

$(\partial E /\partial T)_V dT %2B (\partial E/\partial V)_T dV %2B P dV = T(\partial S/\partial T)_V dT %2B T(\partial S/\partial V)_T dV$ ,

we conclude that

$(\partial S/\partial T)_V = \frac{1}{T} (\partial E /\partial T)_V$ , and $(\partial S/\partial V)_T = \frac{1}{T} [ P %2B (\partial E/\partial V)_T ]$ .

Using $(\partial S/\partial V)_T = \int \frac{\partial^2 S}{\partial V \partial T }dT$ ,

we find the relationship between $P$ and $E$ :

$P(T) = - \frac({\partial E}{\partial V})_T %2B T \int_0^T \frac{1}{T^\prime} \frac{\partial^2 E }{\partial V \partial T^\prime} d T^\prime = - \frac{\partial}{\partial V}[ E(T) - T \int_0^T \frac{1}{T^\prime} (\frac{\partial E(T^\prime) }{\partial T^\prime})_V d T^\prime ]_T.$

The derivative $(\partial P/\partial T)_V$ can then be calculated, resulting in

$\alpha = \frac{1}{ K_T \frac{\partial}{\partial V} [ \int_0^T \frac{1}{T^\prime} (\frac{\partial E(T^\prime) }{\partial T^\prime})_V d T^\prime ]_T }.$

We can now express the latter partial derivatine via the parameters of the quasi-harmonic model. The internal energy is related to the vibrational mode frequencies $\omega_i(V)$ via

$E = E_{static} %2B \sum_i \hbar \omega_i (n_i %2B1/2),$

where $E_{static}$ is the equilibrium energy of the lattice, and $n_i$ is the statistical occupation of the $i$ 's mode, given by the Bose-Einstein distribution, $n_i=\frac{1}{\exp[-\omega_i/T]-1}$ . Since the only value that depends on the volume are the frequencies $\omega_i(V)$ , the heat capacity per unit volume is