Grüneisen parameter

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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]

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.^{[nb 1]} 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 $C_{{V}}={\frac {1}{\rho }}\sum _{i}c_{{V,i}}.$

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

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

This way, it suffices to prove

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

Left-hand side (def):

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

Right-hand side (def):

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

Furthermore (Maxwell relations):

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

Thus

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

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

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

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

By which it is proven that

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

Notes

↑ Thermal expansion does occur in Harmonic crystals if the force constant is dependent on the lattice parameter, which is the usual case.

External links

Definition from Eric Weisstein's World of Physics

Gruneisen parameter has no units

References

↑ E. Grüneisen, The state of a body. Handb. Phys., 10, 1-52. NASA translation RE2-18-59W

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