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

Thermodynamic definitions[edit]

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[edit]

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.[nb 1] 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[edit]

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. γi'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[edit]

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_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 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 ωi 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_BT} \right) \right] + \sum_i \frac{1}{T} \frac{\hbar\omega_i (V)}{\exp\left(\frac{\hbar\omega_i (V)}{k_BT}\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_BT}\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 V K_T}{\tilde{C}_V}

Notes[edit]

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

See also[edit]

External links[edit]

Gruneisen parameter has no units

References[edit]

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