Grüneisen parameter

The expression for the Grüneisen constant of a perfect crystal with pair interactions in ${\displaystyle d}$-dimensional space has the form:[2]

$${\displaystyle \Gamma _{0}=-{\frac {1}{2d}}{\frac {\Pi '''(a)a^{2}+(d-1)\left[\Pi ''(a)a-\Pi '(a)\right]}{\Pi ''(a)a+(d-1)\Pi '(a)}},}$$

where ${\displaystyle \Pi }$ is the interatomic potential, ${\displaystyle a}$ is the equilibrium distance, ${\displaystyle d}$ is the space dimensionality. Relations between the Grüneisen constant and parameters of Lennard-Jones, Morse, and Mie[3] potentials are presented in the table below.

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

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

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

$${\displaystyle \gamma _{i}=-{\frac {V}{\omega _{i}}}{\frac {\partial \omega _{i}}{\partial V}}.}$$

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

$${\displaystyle \gamma ={\frac {\alpha K_{T}}{C_{V}\rho }}}$$

if one defines ${\displaystyle \gamma }$ as the weighted average

$${\displaystyle \gamma ={\frac {\sum _{i}\gamma _{i}c_{V,i}}{\sum _{i}c_{V,i}}},}$$

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

Proof

To prove this relation, it is easiest to introduce the heat capacity per particle ${\textstyle {\tilde {C}}_{V}=\sum _{i}c_{V,i}}$; so one can write

$${\displaystyle {\frac {\sum _{i}\gamma _{i}c_{V,i}}{{\tilde {C}}_{V}}}={\frac {\alpha K_{T}}{C_{V}\rho }}={\frac {\alpha VK_{T}}{{\tilde {C}}_{V}}}.}$$

This way, it suffices to prove

$${\displaystyle \sum _{i}\gamma _{i}c_{V,i}=\alpha VK_{T}.}$$

Left-hand side (def):

$${\displaystyle \sum _{i}\gamma _{i}c_{V,i}=\sum _{i}\left[-{\frac {V}{\omega _{i}}}{\frac {\partial \omega _{i}}{\partial V}}\right]\left[k_{\rm {B}}\left({\frac {\hbar \omega _{i}}{k_{\rm {B}}T}}\right)^{2}{\frac {\exp \left({\frac {\hbar \omega _{i}}{k_{\rm {B}}T}}\right)}{\left[\exp \left({\frac {\hbar \omega _{i}}{k_{\rm {B}}T}}\right)-1\right]^{2}}}\right]}$$

Right-hand side (def):

$${\displaystyle \alpha VK_{T}=\left[{\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{P}\right]V\left[-V\left({\frac {\partial P}{\partial V}}\right)_{T}\right]=-V\left({\frac {\partial V}{\partial T}}\right)_{P}\left({\frac {\partial P}{\partial V}}\right)_{T}}$$

Furthermore (Maxwell relations):

$${\displaystyle \left({\frac {\partial V}{\partial T}}\right)_{P}={\frac {\partial }{\partial T}}\left({\frac {\partial G}{\partial P}}\right)_{T}={\frac {\partial }{\partial P}}\left({\frac {\partial G}{\partial T}}\right)_{P}=-\left({\frac {\partial S}{\partial P}}\right)_{T}}$$

Thus

$${\displaystyle \alpha VK_{T}=V\left({\frac {\partial S}{\partial P}}\right)_{T}\left({\frac {\partial P}{\partial V}}\right)_{T}=V\left({\frac {\partial S}{\partial V}}\right)_{T}}$$

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

$${\displaystyle {\frac {\partial S}{\partial V}}={\frac {\partial }{\partial V}}\left\{-\sum _{i}k_{\rm {B}}\ln \left[1-\exp \left(-{\frac {\hbar \omega _{i}(V)}{k_{\rm {B}}T}}\right)\right]+\sum _{i}{\frac {1}{T}}{\frac {\hbar \omega _{i}(V)}{\exp \left({\frac {\hbar \omega _{i}(V)}{k_{\rm {B}}T}}\right)-1}}\right\}}$$

$${\displaystyle V{\frac {\partial S}{\partial V}}=-\sum _{i}{\frac {V}{\omega _{i}}}{\frac {\partial \omega _{i}}{\partial V}}\;\;k_{\rm {B}}\left({\frac {\hbar \omega _{i}}{k_{\rm {B}}T}}\right)^{2}{\frac {\exp \left({\frac {\hbar \omega _{i}}{k_{\rm {B}}T}}\right)}{\left[\exp \left({\frac {\hbar \omega _{i}}{k_{\rm {B}}T}}\right)-1\right]^{2}}}=\sum _{i}\gamma _{i}c_{V,i}}$$

This yields

$${\displaystyle \gamma ={\dfrac {\sum _{i}\gamma _{i}c_{V,i}}{\sum _{i}c_{V,i}}}={\dfrac {\alpha VK_{T}}{{\tilde {C}}_{V}}}.}$$

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

• Definition from Eric Weisstein's World of Physics