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	~~Hello~~. ~~Let me introduce~~ the ~~writer~~. ~~Her name is Refugia Shryock~~. ~~One~~ of the ~~very best issues in~~ the ~~world~~ for me is ~~to do aerobics~~ and ~~I've been performing it~~ for ~~quite~~ a ~~whilst~~. ~~Managing individuals~~ is ~~his occupation~~. ~~Puerto Rico~~ is ~~exactly~~ where he's ~~usually been residing but she requirements~~ to ~~move simply because of her family~~.<br><br>~~My homepage~~ [http://~~srgame~~.co.kr/~~qna~~/~~21862 home std test kit~~]		The '''Grüneisen parameter''', γ, named after [[Eduard Grüneisen]], describes the effect that changing the volume of a [[crystal lattice]] has on its [[phonon\|vibrational properties]], and, as a consequence, the effect that changing temperature has on the size or dynamics of the [[crystal lattice\|lattice]]. The term is usually reserved to describe the single thermodynamic property γ, which is a weighted average of the many separate parameters γ<sub>i</sub> entering the original Grüneisen's formulation in terms of the [[phonon]] nonlinearities.<ref>E. Grüneisen, The state of a body. Handb. Phys., 10, 1-52. NASA translation RE2-18-59W</ref>
			__TOC__

			== 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:

			<math>\gamma = V \left(\frac{dP}{dE}\right)_V = \frac{\alpha K_S}{C_P \rho} = \frac{\alpha K_T}{C_V \rho}</math>

			where V is volume, <math>C_P</math> and <math>C_V</math> are the principal (i.e. per-mass) heat capacities at constant pressure and volume, E is energy, α is the volume [[thermal expansion\|thermal expansion coefficient]], <math>K_S</math> and <math>K_T</math> are the adiabatic and isothermal [[compressibility\|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 [[Proportionality (mathematics)\|linear]] in the atom's displacement, the frequencies ω<sub>i</sub> of individual [[phonon]]s 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.{{#tag:ref\|Thermal expansion does occur in Harmonic crystals if the force constant is dependent on the lattice parameter, which is the usual case.\|group=nb}} When the restoring force is non-linear in the displacement, the phonon frequencies ω<sub>i</sub> change with the volume <math>V</math>. The Grüneisen parameter of an individual vibrational mode <math>i</math> can then be defined as (the negative of) the logarithmic derivative of the corresponding frequency <math>\omega_i</math>:

			<math>\gamma_i= - \frac{V}{\omega_i} \frac{\partial \omega_i}{\partial V}. </math>

			== 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<sub>i</sub>'s).
			For example,
			one can show that

			<math>\gamma = \frac{\alpha K_T}{C_V \rho}</math>

			if one defines <math>\gamma </math> as the weighted average

			<math>\gamma = \frac{\sum_i \gamma_i c_{V,i} }{ \sum_i c_{V,i} }, </math>

			where <math> c_{V,i}</math>'s are the partial vibrational mode contributions to the heat capacity, such that <math>C_{V} = \frac{1}{\rho} \sum_i c_{V,i} .</math>

			===Proof===
			To prove this relation, it is easiest to introduce the heat capacity per particle <math>\tilde{C}_V = \sum_i c_{V,i}</math>; so one can write

			<math>\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}</math>.

			This way, it suffices to prove

			<math>\sum_i \gamma_i c_{V,i} = \alpha V K_T</math>.

			Left-hand side (def):

			<math>\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]</math>

			Right-hand side (def):

			<math>\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</math>

			Furthermore ([[Maxwell relations]]):

			<math>\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</math>

			Thus

			<math>\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</math>

			This derivative is straightforward to determine in the [[quasi-harmonic approximation]], as only the ω<sub>i</sub> are ''V''-dependent.

			<math>\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\}</math>

			<math>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}</math>

			By which it is proven that

			<math>\gamma = \dfrac{\sum_i \gamma_i c_{V,i}}{\sum_i c_{V,i}} = \dfrac{\alpha V K_T}{\tilde{C}_V}</math>

			==Notes==
			{{Reflist\|group=nb}}

			==See also==
			*[[Debye model]]''

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

			Gruneisen parameter has no units

			==References==
			{{Reflist}}

			{{DEFAULTSORT:Gruneisen Parameter}}
			[[Category:Condensed matter physics]]
			[[Category:Dimensionless numbers of thermodynamics]]

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Hello. Let me introduce the writer. Her name is Refugia Shryock. One of the very best issues in the world for me is to do aerobics and I've been performing it for quite a whilst. Managing individuals is his occupation. Puerto Rico is exactly where he's usually been residing but she requirements to move simply because of her family.<br><br>My homepage [http://srgame.co.kr/qna/21862 home std test kit]

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