Редактирование: Mie–Gruneisen equation of state

[[Уравнение состояния Ми-Грюнайзена | Русская версия]]

== Source ==
This article is based on the paper '''A.M. Krivtsov, V.A. Kuzkin, [[Медиа: Krivtsov_2011_MechSol.pdf | Derivation of Equations of State for Ideal Crystals of Simple Structure]] // Mech. Solids. 46 (3), 387-399 (2011)'''

== Mie-Gruneisen equation of state ==
In high pressure physics it is usual to represent the total pressure  <math>p</math> in condensed matter as a sum of "cold" and  "thermal"   components:

<math>p = p_0 + p_T, ~~~~ p_T = p - p_0</math>

The cold pressure, refereed to as the "cold curve" is caused by deformation of crystal lattice only. The thermal pressure is due to thermal motion of the atoms. In other words, the cold pressure is a function of volume only, while the thermal pressure also depends on thermal energy <math> E_T </math>:

<math>p = p_0(V) + p_T(V,E_T)</math>

The thermal energy is a part of the internal energy caused by the thermal motion of atoms. In the simplest case the thermal energy os equal to <math> c_V T </math>, where <math> c_V </math> is the specific heat. In practice it is usually assumed that the dependence of pressure on thermal energy is linear: 

<math> p = p_0(V) + \frac{\varGamma(V)}{V} E_T</math>

The given equation is refereed to as '''Mie-Gruneisen equation of state (EOS)'''. The function  <math>\varGamma(V)</math> is called '''Gruneisen function'''. The value <math> \varGamma_0 </math> of Gruneisen function in undeformed configuration is called '''Gruneisen coefficient (or Gruneisen constant)'''. 

<math> \varGamma_0 = \varGamma(V_0)</math>

== Equation of state for perfect crystals with simple lattice ==

<math>
    p_0 = \frac{1}{2V_0d\theta^d}\sum_{k=1}^n N_k\varPhi_k A_k^2,~~~~\varGamma = -\frac{\sum_{k=1}^n N_k((d+2)\varPhi'_k A_k^2 + 2\varPhi''_k A_k^4 )}{d\sum_{k=1}^n
    N_k (d\varPhi_k +2\varPhi'_k A_k^2)}
</math>

where <math>k</math> is the number of coordination sphere, <math>n</math> is the number of coordination spheres, <math>N_k</math> is the number of atoms bolonging to the <math>k</math>-th coordination sphere, <math> A_k = \rho_k R \theta</math> is the radius of coordination sphere , <math> \rho_k=A_k/A_1 </math>, <math>R</math> is the radius of the first coordination sphere in undeformed configuration, <math>\varPhi^{(n)}_k = \varPhi^{(n)}(A_k^2)</math>.


== Cold curve for Lennard-Jones, Mie, and Morse potentials ==

In the case of nierest neighbors interactions the cold curve for Lennard-Jones, Mie, and Morse potentials has the following simple form.

* '''Cold curve for Lennard-Jones potential:'''
<math>
 \varPi(r) =D\left[\left(\frac{a}{r}\right)^{12}-2\left(\frac{a}{r}\right)^{6}\right], ~~~~ p_0 = \frac{6MD}{dV_0\theta^{d}}(\theta^{-12}-\theta^{-6})
</math>


* '''Cold curve for Mie potential:'''
<math>
   \varPi(r) =\frac{D}{n-m} \left[m\left(\frac{a}{r}\right)^{n}-n\left(\frac{a}{r}\right)^{m} \right], ~~~~
   p_0 =\frac{m n MD}{2d(n-m)V_0\theta^{d}}\left(\theta^{-n}-\theta^{-m}\right)
</math>

* '''Cold curve for Morse potential:'''
<math>
   \varPi(r) = D\left[e^{2\alpha(a-r)}-2e^{\alpha(a-r)}\right], ~~~~
   p_0 = \frac{\alpha a MD}{d V_0\theta^{d-1}} \left[e^{2\alpha a(1-\theta)}-e^{\alpha a(1-\theta)}\right]
</math>

Here <math>D</math> is the bond energy, <math>a</math> is the bond length, <math>\alpha</math> is the parameter characterising the width of the potential well; <math>m, n</math> are parameters of Mie potential.

== Gruneisen constant for Lennard-Jones, Mie, and Morse potentials == 

The expression for Gruneisen parameter for crystalls with simple lattice and nearest-neighbor interactions in  <math>d</math>-dimmensional space has the form:

<math>
    \varGamma_0 = -\frac{1}{2d}\frac{\varPi'''(a)a^2 + (d-1)\left[\varPi''(a)a - \varPi'(a)\right]}{\varPi''(a)a + (d-1)\varPi'(a)}
</math>

where <math>\Pi</math> is the interatomic potential, <math>a</math> is the equlibrium ditance, <math>d</math> is the space dimmensinality. The relation between Gruneisen constant and parameters of  Lennard-Jones, Mie, and Morse potentials is presented in the table below.

{|class="wikitable"
|-
!Lattice
!Dimmensionality
!Lennard-Jones potential
!Mie potential
!Morse potential
|-
| Chain
! <math> d=1 </math>
! <math>10\frac{1}{2} </math>
! <math>\frac{m+n+3}{2}</math>
! <math>\frac{3\alpha a}{2}</math>
|-
| Triangular lattice
!<math>d=2 </math>
! <math>5</math>
! <math> \frac{m+n+2}{4}</math>
! <math> \frac{3\alpha a - 1}{4}</math>
|-
| ГЦК, ОЦК 
! <math>d=3 </math>
! <math>\frac{19}{6} </math>
! <math>\frac{n+m+1}{6}</math>
! <math>\frac{3\alpha a-2}{6}</math>
|-
| "Hyperlattice"
! <math>d=\infty</math>
! <math>-\frac{1}{2}</math>
! <math>-\frac{1}{2}</math>
! <math>-\frac{1}{2}</math>
|-
| General formula
! <math>d</math>
! <math>\frac{11}{d}-\frac{1}{2}</math>
! <math>\frac{m+n+4}{2d}-\frac{1}{2}</math>
! <math>\frac{3\alpha a + 1}{2d}-\frac{1}{2}</math>
|-
|}

The expression for Gruneisen constant of 1D chain with Mie potential exactly coincides with the results of McDonald and Roy.

== Gruneisen function for Lennard-Jones, Mie, and Morse potentials == 

In the case of nearest-neighbor interactions the Gruneisen function has the following form.
 
* '''Gruneisen function for Lennard-Jones potential:'''
<math>
  \varGamma = \frac{1}{d}\frac{4(8-d)\theta^{6}-7(14-d)}{(8-d)\theta^{6}-(14-d)}.
</math>


* '''Gruneisen function for Mie potential:''' 
<math>
    \varGamma = \frac{1}{2d}\frac{(n+2)(n-d+2)\theta^{m-n}-(m+2)(m-d+2)}{(n-d+2)\theta^{m-n}-(m-d+2)}.
</math>


* '''Gruneisen function for Morse potential:'''
<math>
\varGamma = \frac{1}{2d}\frac{e^{\alpha a(1-\theta)}\left(4\alpha^2a^2\theta^2-2d_1\alpha a
\theta-d_1\right)-\left(\alpha^2 a^2\theta^2-d_1\alpha a\theta-d_1 \right)}{e^{\alpha a(1-\theta)}(2\alpha a\theta-d_1)
-(\alpha a\theta-d_1)},~~
</math>
<math>d_1 = d-1,~~</math> <math>\theta=(V/V_0)^{1/d}</math>




== Papers ==
* A.M. Krivtsov, V.A. Kuzkin, [[Медиа: Krivtsov_2011_MechSol.pdf | Derivation of Equations of State for Ideal Crystals of Simple Structure]] // Mech. Solids. 46 (3), 387-399 (2011).
* MacDonald D. K. C., Roy, S.K. (1955), "Vibrational Anharmonicity and Lattice Thermal Properties. II", Phys. Rev. 97: 673–676, doi:10.1103/PhysRev.97.673

== Links ==

* [http://ru.wikipedia.org/wiki/%D0%A3%D1%80%D0%B0%D0%B2%D0%BD%D0%B5%D0%BD%D0%B8%D0%B5_%D1%81%D0%BE%D1%81%D1%82%D0%BE%D1%8F%D0%BD%D0%B8%D1%8F_%D0%9C%D0%B8_%E2%80%94_%D0%93%D1%80%D1%8E%D0%BD%D0%B0%D0%B9%D0%B7%D0%B5%D0%BD%D0%B0 Статья про уравнение Ми-Грюнайзена в Википедии]

* [http://en.wikipedia.org/wiki/Mie%E2%80%93Gruneisen_equation_of_state  Mie–Gruneisen equation of state (Wikipedia]