Grüneisen parameter

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.

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: $$\gamma = V \left(\frac{dP}{dE}\right)_V = \frac{\alpha K_T}{C_V \rho} = \frac{\alpha K_S}{C_P \rho} = \frac{\alpha v_s^2}{C_P} = -\left(\frac{\partial \ln T}{\partial \ln V}\right)_S$$ 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, $S$ is entropy, $α$ 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 $ρ$ 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: $$ \Gamma_0 = -\frac{1}{2d}\frac{\Pi'(a)a^2 + (d-1)\left[\Pi(a)a - \Pi'(a)\right]}{\Pi''(a)a + (d-1)\Pi'(a)}, $$ 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 potentials are presented in the table below.

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 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 $$V$$. 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 vibrational 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 $C_{V} = \frac{1}{\rho V} \sum_i c_{V,i} .$

Proof
To prove this relation, it is easiest to introduce the heat capacity per particle $\tilde{C}_V = \sum_i c_{V,i}$ ; so one can write $$\frac{\sum_i \gamma_i c_{V,i}}{\tilde{C}_V} = \frac{\alpha K_T}{C_V \rho} = \frac{\alpha V K_T}{\tilde{C}_V}.$$

This way, it suffices to prove $$\sum_i \gamma_i c_{V,i} = \alpha V K_T.$$

Left-hand side (def): $$\sum_i \gamma_i c_{V,i} = \sum_i \left[- \frac{V}{\omega_i} \frac{\partial \omega_i}{ \partial V} \right] \left[ k_{\rm B} \left(\frac{\hbar \omega_i}{k_{\rm B} T}\right)^2 \frac{\exp\left( \frac{\hbar \omega_i}{k_{\rm B} T} \right)}{\left[\exp\left(\frac{\hbar \omega_i}{k_{\rm B} T}\right) - 1\right]^2} \right]$$

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

Furthermore (Maxwell relations): $$\left( \frac{\partial V}{\partial T} \right)_P = \frac{\partial}{\partial T} \left(\frac{\partial G}{\partial P}\right)_T = \frac{\partial}{\partial P} \left(\frac{\partial G}{\partial T}\right)_P = - \left( \frac{\partial S}{\partial P} \right)_T$$

Thus $$\alpha V K_T = V \left( \frac{\partial S}{\partial P} \right)_T \left(\frac{\partial P}{\partial V}\right)_T = V \left( \frac{\partial S}{\partial V} \right)_T$$

This derivative is straightforward to determine in the quasi-harmonic approximation, as only the $ω_{i}$ are V-dependent.

$$\frac{\partial S}{\partial V} = \frac{\partial }{\partial V} \left\{ - \sum_i k_{\rm B} \ln\left[ 1 - \exp\left( -\frac{\hbar\omega_i (V)}{k_{\rm B}T} \right) \right] + \sum_i \frac{1}{T} \frac{\hbar\omega_i (V)}{\exp\left(\frac{\hbar\omega_i (V)}{k_{\rm B}T}\right) - 1} \right\}$$

$$V \frac{\partial S}{\partial V} = - \sum_i \frac{V}{\omega_i} \frac{\partial \omega_i}{\partial V} \;\; k_{\rm B} \left(\frac{\hbar \omega_i}{k_{\rm B}T}\right)^2 \frac{\exp\left( \frac{\hbar \omega_i}{k_{\rm B} T} \right)}{\left[\exp\left(\frac{\hbar \omega_i}{k_{\rm B} T}\right) - 1\right]^2} = \sum_i \gamma_i c_{V,i}$$

This yields $$\gamma = \dfrac{\sum_i \gamma_i c_{V,i}}{\sum_i c_{V,i}} = \dfrac{\alpha V K_T}{\tilde{C}_V}.$$