Grüneisen parameter

The expression for the Grüneisen constant of a perfect crystal with pair interactions in $d$-dimensional space has the form:{{Citation

| title = Derivation of Equations of State for Ideal Crystals of Simple Structure

| doi = 10.3103/S002565441103006X

| journal = Mechanics of Solids

| pages = 387–399 | volume = 46 | issue = 3 | year = 2011

| last1 = Krivtsov | first1 = A.M.

| last2 = Kuzkin | first2 = V.A.

| bibcode =2011MeSol..46..387K | s2cid = 51837957

}}

$$\backslash Gamma\_0\; =\; -\backslash frac\{1\}\{2d\}\backslash frac\{\backslash Pi$$'(a)a^2 + (d-1)\left[\Pi(a)a - \Pi'(a)\right]}{\Pi''(a)a + (d-1)\Pi'(a)},

where $\backslash Pi$ 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{{Cite web|url=http://www.sklogwiki.org/SklogWiki/index.php/Mie_potential|title=Mie potential page on SklogWiki - a wiki for statistical mechanics and thermodynamics|website=www.sklogwiki.org|access-date=2019-11-19}} 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.{{Citation

| title = Vibrational Anharmonicity and Lattice Thermal Properties. II

| doi = 10.1103/PhysRev.97.673

| year = 1955

| journal = Phys. Rev.

| pages = 673–676

| volume = 97

| issue = 3

| last1 = MacDonald | first1 = D. K. C.

| last2 = Roy | first2 = S.K.

| bibcode = 1955PhRv...97..673M }}

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 $\backslash Pi$'(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 $V$. The Grüneisen parameter of an individual vibrational mode $i$ can then be defined as (the negative of) the logarithmic derivative of the corresponding frequency $\backslash omega\_i$:

$$\backslash gamma\_i=\; -\; \backslash frac\{V\}\{\backslash omega\_i\}\; \backslash frac\{\backslash partial\; \backslash omega\_i\}\{\backslash partial\; V\}.$$

Using the quasi-harmonic approximation for atomic vibrations, the macroscopic Grüneisen parameter ({{mvar|γ}}) can be related to the description of how the vibrational frequencies (phonons) within a crystal are altered with changing volume (i.e. {{mvar|γ_i}}'s).

For example, one can show that

$$\backslash gamma\; =\; \backslash frac\{\backslash alpha\; K\_T\}\{C\_V\; \backslash rho\}$$

if one defines $\backslash gamma$ as the weighted average

$$\backslash gamma\; =\; \backslash frac\{\backslash sum\_i\; \backslash gamma\_i\; c\_\{V,i\}\; \}\{\; \backslash sum\_i\; c\_\{V,i\}\; \},$$

where $c\_\{V,i\}$'s are the partial vibrational mode contributions to the heat capacity, such that $C\_\{V\}\; =\; \backslash frac\{1\}\{\backslash rho\; V\}\; \backslash sum\_i\; c\_\{V,i\}\; .$

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

$$\backslash frac\{\backslash sum\_i\; \backslash gamma\_i\; c\_\{V,i\}\}\{\backslash tilde\{C\}\_V\}\; =\; \backslash frac\{\backslash alpha\; K\_T\}\{C\_V\; \backslash rho\}\; =\; \backslash frac\{\backslash alpha\; V\; K\_T\}\{\backslash tilde\{C\}\_V\}.$$

This way, it suffices to prove

$$\backslash sum\_i\; \backslash gamma\_i\; c\_\{V,i\}\; =\; \backslash alpha\; V\; K\_T.$$

Left-hand side (def):

$$\backslash sum\_i\; \backslash gamma\_i\; c\_\{V,i\}\; =\; \backslash sum\_i\; \backslash left[-\; \backslash frac\{V\}\{\backslash omega\_i\}\; \backslash frac\{\backslash partial\; \backslash omega\_i\}\{\; \backslash partial\; V\}\; \backslash right]\; \backslash left[\; k\_\{\backslash rm\; B\}\; \backslash left(\backslash frac\{\backslash hbar\; \backslash omega\_i\}\{k\_\{\backslash rm\; B\}\; T\}\backslash right)^2\; \backslash frac\{\backslash exp\backslash left(\; \backslash frac\{\backslash hbar\; \backslash omega\_i\}\{k\_\{\backslash rm\; B\}\; T\}\; \backslash right)\}\{\backslash left[\backslash exp\backslash left(\backslash frac\{\backslash hbar\; \backslash omega\_i\}\{k\_\{\backslash rm\; B\}\; T\}\backslash right)\; -\; 1\backslash right]^2\}\; \backslash right]$$

Right-hand side (def):

$$\backslash alpha\; V\; K\_T\; =\; \backslash left[\; \backslash frac\{1\}\{V\}\; \backslash left(\backslash frac\{\backslash partial\; V\}\{\; \backslash partial\; T\}\backslash right)\_P\; \backslash right]\; V\; \backslash left[-V\; \backslash left(\backslash frac\{\backslash partial\; P\}\{\backslash partial\; V\}\backslash right)\_T\backslash right]\; =\; -\; V\; \backslash left(\; \backslash frac\{\backslash partial\; V\}\{\backslash partial\; T\}\; \backslash right)\_P\; \backslash left(\backslash frac\{\backslash partial\; P\}\{\backslash partial\; V\}\backslash right)\_T$$

Furthermore (Maxwell relations):

$$\backslash left(\; \backslash frac\{\backslash partial\; V\}\{\backslash partial\; T\}\; \backslash right)\_P\; =\; \backslash frac\{\backslash partial\}\{\backslash partial\; T\}\; \backslash left(\backslash frac\{\backslash partial\; G\}\{\backslash partial\; P\}\backslash right)\_T\; =\; \backslash frac\{\backslash partial\}\{\backslash partial\; P\}\; \backslash left(\backslash frac\{\backslash partial\; G\}\{\backslash partial\; T\}\backslash right)\_P\; =\; -\; \backslash left(\; \backslash frac\{\backslash partial\; S\}\{\backslash partial\; P\}\; \backslash right)\_T$$

Thus

$$\backslash alpha\; V\; K\_T\; =\; V\; \backslash left(\; \backslash frac\{\backslash partial\; S\}\{\backslash partial\; P\}\; \backslash right)\_T\; \backslash left(\backslash frac\{\backslash partial\; P\}\{\backslash partial\; V\}\backslash right)\_T\; =\; V\; \backslash left(\; \backslash frac\{\backslash partial\; S\}\{\backslash partial\; V\}\; \backslash right)\_T$$

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

$$\backslash frac\{\backslash partial\; S\}\{\backslash partial\; V\}\; =\; \backslash frac\{\backslash partial\; \}\{\backslash partial\; V\}\; \backslash left\backslash \{\; -\; \backslash sum\_i\; k\_\{\backslash rm\; B\}\; \backslash ln\backslash left[\; 1\; -\; \backslash exp\backslash left(\; -\backslash frac\{\backslash hbar\backslash omega\_i\; (V)\}\{k\_\{\backslash rm\; B\}T\}\; \backslash right)\; \backslash right]\; +\; \backslash sum\_i\; \backslash frac\{1\}\{T\}\; \backslash frac\{\backslash hbar\backslash omega\_i\; (V)\}\{\backslash exp\backslash left(\backslash frac\{\backslash hbar\backslash omega\_i\; (V)\}\{k\_\{\backslash rm\; B\}T\}\backslash right)\; -\; 1\}\; \backslash right\backslash \}$$

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

This yields

$$\backslash gamma\; =\; \backslash dfrac\{\backslash sum\_i\; \backslash gamma\_i\; c\_\{V,i\}\}\{\backslash sum\_i\; c\_\{V,i\}\}\; =\; \backslash dfrac\{\backslash alpha\; V\; K\_T\}\{\backslash tilde\{C\}\_V\}.$$

Regarding Boltzmann-Gibbs (BG) statistical mechanics, it is reported in the literature that the Grüneisen parameter presents an expressive enhancement close to critical points (CPs) and phase transitions. However, for genuine quantum critical phenomena, i.e., in the complete absence of temperature {{mvar|T}}, a thermodynamic definition of the Grüneisen parameter is elusive because it embodies dependences with temperature and exactly at {{mvar|T}} = 0K the Grüneisen parameter is undetermined. Nevertheless, a quantum version $\backslash Gamma^\{0\backslash text\{K\}\}$ was recently proposed.{{Citation | title = Grüneisen parameter as an entanglement compass and the breakdown of the Hellmann-Feynman theorem | doi = 10.1103/PhysRevB.108.L140403 | year = 2023 | journal = Physical Review B | pages = L140403 | volume = 108 | issue = 14 | last1 = Squillante | first1 = L. | last2 = Ricco | first2 = L.S. | last3 = Ukpong | first3 = A.M. | last4 = Lagos-Monaco | first4 = R.E. | last5 = Seridonio | first5 = A.C. | last6 = de Souza | first6 = M.| bibcode = 2023PhRvB.108n0403S | url = https://journals.aps.org/prb/abstract/10.1103/PhysRevB.108.L140403 | arxiv = 2306.00566 }} Using the 1D Ising model under a transverse magnetic field (1DIMTF) the authors have shown that, for the quantum CP of such a model, $\backslash Gamma^\{0\backslash text\{K\}\}$ shows a divergent-like behavior when the magnetic energy {{mvar|B}} is comparable to the exchange coupling energy {{mvar|J}}. Such behavior is associated with the breakdown of the Boltzmann-Gibbs-von Neumann-Shannon entropy extensivity in this regime, which leads to zeros and infinities in physical quantities such as the Grüneisen parameter. However, upon employing the generalized nonadditive entropy $S\_q$, Constantino Tsallis demonstrated that for a unique value of the entropic index {{mvar|q}}, $S\_q$ is extensive right at the CP of the 1DIMTF. Hence, upon making an unprecedented connection of $\backslash Gamma^\{0\backslash text\{K\}\}$ in terms of $S\_q$ and using the 1DIMTF, researchers from Physics Department - Unesp, Rio Claro, have shown that $\backslash Gamma^\{0\backslash text\{K\}\}$ is universally nondivergent when using the appropriate entropy for the critical regime.{{Citation | title = Universally nondiverging Grüneisen parameter at critical points | doi = 10.1103/PhysRevB.111.L060409 | year = 2025 | journal = Physical Review B | pages = L060409 | volume = 111 | issue = 6 | last1 = Soares | first1 = S.M. | last2 = Squillante | first2 = L. | last3 = Lima | first3 = H.S. |last4 = Tsallis | first4 = C. |last5 = de Souza | first5 = M. | bibcode = 2024arXiv240911086S | url = https://journals.aps.org/prb/abstract/10.1103/PhysRevB.111.L060409 | arxiv = 2409.11086 }} Such results suggest that the divergent-like behavior of physical quantities associated with the nonvalidity of BG statistical mechanics should be revisited in terms of $S\_q$.

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

• [http://scienceworld.wolfram.com/physics/GrueneisenParameters.html Definition from Eric Weisstein's World of Physics]

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Category:Condensed matter physics

Category:Dimensionless numbers of thermodynamics