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 物質の温度・圧力・体積の関係を状態方式 (Equation Of State, EOS) といいます。最も簡単な状態方程式は 高校物理でおなじみの \(P V = n R T\) という式ですが、これが成り立つのは残念ながら理想気体の場合のみです。現実の結晶性物質に適用できる状態方程式はもっと複雑であり、さまざまなモデルに基づく表式があります。このページでは結晶に適用できる主要な状態方程式のいくつかを紹介します。なお、以降は次のように記号を定義します。

$$\begin{array}{rcl}
V_0 &=& V_{(T = T_0, P=0)} \\
V &=& V_{(T = T_0, P)} \\
K_0 &=& K_{(T=T_0, P=0)}\\
K’_0 &=& (\partial K_0 / \partial P)_T\\
K”_0 &=& (\partial^2 K_0 / \partial P^2)_T \\
a&=&\left(\frac{V_0}{V}\right)^{1/3}
\end{array}$$


Birch-Murnaghan

Birch, F. Finite elastic strain of cubic crystals. Phys. Rev. 71, 809-824 (1947).

2nd-order

\(V_0/V\)と体積弾性率 \(K_0\)によって圧力を求めます。

$$\begin{array}{rcl}
P_{(T=T_0, V)}^{2nd} &=& \frac{3}{2}K_0 \left[ \left( \frac{V_0}{V} \right)^{\frac{7}{3}} – \left( \frac{V_0}{V} \right)^{\frac{5}{3}} \right]\\
&=& \frac{3}{2}K_0 ( a^7 – a^5)
\end{array}$$


3rd-order

\(V_0/V\)と体積弾性率 \(K_0\)およびその圧力による1次の微分値 \(K’_0\)によって圧力を求めます。

$$\begin{array}{rcl}
P_{(T=T_0, V)}^{3rd} &=& \frac{3}{2}K_0 \left[ \left( \frac{V_0}{V} \right)^{\frac{7}{3}} – \left( \frac{V_0}{V} \right)^{\frac{5}{3}} \right] \left[ 1 + \frac{3}{4} (K’_0-4) \left\{ \left( \frac{V_0}{V} \right)^{\frac{2}{3}} -1 \right\} \right] \\
&=& \frac{3}{2}K_0 (a^7 – a^5) \left\{ 1 + \frac{3}{4} (K’_0-4) (a^2 -1) \right\} \\
&=& P_{(T=T_0, V)}^{2nd} \left\{ 1 + \frac{3}{4} (K’_0-4) (a^2 -1) \right\}
\end{array}$$


4th-order

\(V_0/V\)と体積弾性率 \(K_0\)およびその圧力による1次と2次の微分値 \(K’_0, K”_0\)によって圧力を求めます。

$$\begin{array}{rcl}
P_{(T=T_0, V)}^{4th} &=& \frac{3}{2}K_0
\left[ \left( \frac{V_0}{V} \right)^{\frac{7}{3}} – \left( \frac{V_0}{V} \right)^{\frac{5}{3}} \right]
\left[
1 + \frac{3}{4} (K’_0-4) \left\{ \left( \frac{V_0}{V} \right)^{\frac{2}{3}} -1 \right\} +
\frac{9K_0 K”_0 + 9{K’_0}^2 – 63K’_0 + 143}{24} \left\{ \left(\frac{V_0}{V} \right)^{\frac{2}{3}} -1 \right\}^2
\right]\\
&=& \frac{3}{2}K_0 (a^7- a^5) \left\{ 1 + \frac{3}{4} (K’_0-4) (a^2 -1) +
\frac{9K_0 K”_0 + 9{K’_0}^2 – 63K’_0 + 143}{24} (a^2 -1)^2 \right\}\\
&=& P_{(T=T_0, V)}^{3rd} + P_{(T=T_0, V)}^{2nd} \left\{ \frac{9K_0 K”_0 + 9{K’_0}^2 – 63K’_0 + 143}{24} (a^2 -1)^2 \right\}
\end{array}$$


T-dependence Birch-Murnaghan

$$\begin{array}{rcl}
V_{(T,P=0)} &=&\displaystyle V_{(T_0, P=0)} \exp\left\{\int_{T_0}^T a + b T^2+c/T^2 dT \right\}\\
K_{(T,P=0)} &=&\displaystyle K_{(T_0, P=0)} + (\partial K_{(T,P=0)}/ \partial T) (T-T_0)
\end{array}$$


Vinet (Morse Rydverg)

Normal Vinet

Vinet, P., Ferrante, J., Rose, J. H., & Smith, J. R. Compressibility of Solids. J. Geophys. Res. 92, 9319-9325 (1987).

$$\begin{array}{rcl}
P_{(T=T_0, V)} &=& \displaystyle 3K_0
\left(\frac{V_0}{V}\right)^{\frac{2}{3}}
\left\{ 1- \left(\frac{V_0}{V}\right)^{-\frac{1}{3}} \right\}
\exp \left[ \frac{3}{2} (K’_0-1) \left\{ 1- \left(\frac{V_0}{V}\right)^{-\frac{1}{3}} \right\} \right] \\
&=& \displaystyle 3K_0 ( a^2- a ) \exp \left[ \frac{3}{2} (K’_0-1) ( 1- a^{-1} ) \right]
\end{array}$$

3rd order, 4th order, …

Chijioke A.D., Nellis W.J., and SilveraI. F., High-pressure equations of state of Al, Cu, Ta, and W, J. Appl. Phys. 98,
073526 (2005).

Kraus R.G., Davis J.-P., Steagle C.T., Fratanduono D.E., Swift D. C., Brown J.L. and Eggert J.H. Dynamic compression of copper to over 450 GPa: A high-pressure standard. PHYSICAL REVIEW B 93, 134105 (2016).

$$\begin{array}{rcl}
P_{(T=T_0, V)} &=& 3K_0
\left(\frac{V_0}{V}\right)^{\frac{2}{3}}
\left\{ 1- \left(\frac{V_0}{V}\right)^{ -\frac{1}{3}} \right\}
\exp \left[ \eta \left\{ 1- \left(\frac{V_0}{V}\right)^{-\frac{1}{3}} \right\} +
\beta \left\{ 1- \left(\frac{V_0}{V}\right)^{-\frac{1}{3}} \right\} ^2 +
\psi \left\{ 1- \left(\frac{V_0}{V}\right)^{-\frac{1}{3}} \right\} ^3 + \cdots \right] \\
&=& \displaystyle 3K_0 ( a^2- a ) \exp \left[
\eta (1- a^{-1}) + \beta (1- a^{-1})^2 + \psi (1- a^{-1})^3 + \cdots \right]
\end{array}$$


Adapted Polynomial expansion of 2nd order (AP2)

Holzapfel, W. B. Equations of state for solids under strong compression. High Press. Res. 16, 81–126 (1998).

$$\begin{array}{rcl}
P&=&\displaystyle 3K_0 \frac{1-\left(\frac{V}{V_0}\right)^{1/3}}{\left(\frac{V}{V_0}\right)^{5/3}}
\left[ 1+aC_2 \left\{ 1-\left(\frac{V}{V_0}\right)^{1/3} \right\} \right] \exp\left[C_0 \left\{1-\left(\frac{V}{V_0}\right)^{1/3} \right\} \right]\\
&=&\displaystyle 3K_0 \frac{1-a^{-1}}{a^{-5}} [1+a^{-1}C_2(1-a^{-1}) ] \exp\left[C_0 (1-a^{-1}) \right]
\end{array}$$ただし、$$
\begin{array}{lll}
a = \left(\frac{V_0}{V}\right)^{1/3} &
C_0 = -\ln (3K_0 / p_{FG_0}) &
C_2 = \frac{3}{2} (K’_0 -3) -C_0\\
p_{FG_0}=a_{FG} ( Z_e/V_0) ^{5/3}&
a_{FG} = 2336.965\, [\mathrm{GPa}\, \textbf{Å}^5 ] &
\end{array}
$$\(p_{FG_0}\) : Fermi gas pressure
\(a_{FG}\) : Universal Fermi gas parameter
\(Zܼ_e\): Total number of electrons in the unit cell with \(V_0\)


Keane

Keane, A. An investigation of finite strain in an isotropic material subjected to hydrostatic pressure and its seismological applications. Aust. J. Phys. 7, 322–333 (1954).

Stacey, F. D. & Davis, P. M. High pressure equations of state with applications to the lower mantle and core. Phys. Earth Planet. Inter. 142, 137–184, doi: 10.1016/j.pepi.2004.02.003 (2004).

$$ \frac{P}{K_0} = \frac{K’_0}{{K’_{\infty}}^2} \left[ \left( \frac{V_0}{V} \right)^{K’_{\infty}} -1 \right]
– \left( \frac{K’_0}{K’_{\infty}} -1 \right) \ln\left( \frac{V_0}{V} \right)
$$ただし、\(K’_0-1 > K’_{\infty} >K’_0 /2 \)


Mie-Grüneisen(-Debye)

Mie-Grüneisen(-Debye)モデルによる熱圧力効果 (\(\Delta P_{th}\)) の導出は、以下の通りです。
$$ \Delta P_{th} = (e-e_0) \frac{\gamma}{v}
$$ただし、$$
\begin{array}{lll}
e=9 R n T \left( \frac{T}{\theta} \right)^3 \displaystyle\int_0^{\frac{\theta}{T}} \frac{z^3}{e^z-1} dz \,\,\,&
e_0= 9 R n T_0 \left( \frac{T_0}{\theta_0} \right)^3 \displaystyle\int_0^{\frac{\theta_0}{T_0}} \frac{z^3}{e^z-1} dz \,\,\,&
\theta = \theta_0 \exp\left[(\gamma_0-\gamma)/q\right] \,\,\,\\
\gamma = \gamma_0 (v/v_0)^q &
v= \displaystyle \frac{V}{z} N_A \times 10^{-30}\,\,\,[\mathrm{m^3/mol}] &
v_0= \displaystyle \frac{V_0}{z} N_A \times 10^{-30} \,\,\,[\mathrm{m^3/mol}]
\end{array}
$$\(N_A\) : Avogadro constant, 6.02214129(27) × 1023 [mol-1];

\(R\) : Gas constant, 8.3144621(75) [J K-1 mol-1];
\(z\): Number of formula in unit cell;  \(n\): Atoms per formula;  \(\theta_0\): Debye temperature at standard volume;
\(\gamma_0\): Grüneisen parameter at standard volume;  \(q\): Volume dependence of Grüneisen parameter
\(t_0\): Standard temperature;   \(t_0\): Target temperature;  \(V_0\): Standard volume (Å3);   \(V\): Target volume (Å3)   


代表的な物質のEOSパラメータ

Au

Author(s)\(K_0\)\(K’_0\)\(z\)\(n\)\(\partial K_{(T,P=0)}/ \partial T\)\(T_0\) [K]\(V_0\) [Å3]\(\theta_0\) [K]\(\gamma_0\)\(q\)Note
Anderson (1989)*166.655.4823-0.01153003BM+*
Sim (2002)1675.0413001702.971.03BM+MG
Fratanduono et al., 2021170.095.880V
*Thermal pressure \(P_{th}\) は次のように計算: \(P_{th}=0.00714+\frac{\partial K}{\partial T} (T-300) \ln(V_0/V) \)
Author(s)\(K_0\)\(\eta\), 1st Vinet
parameter
\(\beta\), 2nd Vinet
parameter
3rd Vinet
parameter
4th Vinet
parameter
Note
Chijioke et al.
(2005)*
177.266.380001.9334−1.029233.9414Vinet
Pressure limit: 513 GPa
*Reproduced by Chijioke et al. (2005) from Wang et al (2002) J. Appl. Phys. 92, 6616.


Pt

Author(s)\(K_0\)\(K’_0\)\(z\)\(n\)\(T_0\) [K]\(V_0\) [Å3]\(\theta_0\) [K]\(\gamma_0\)\(q\)Note
Holmes (1989)*2665.81V+*
Matsui et al. (2009)2735.204130060.382302.701.10V+MG
Fei et al. (2007)2775.08(2)4130060.382302.72(3)0.5V+MG
Zha et al. (2008)273.5(10)4.70(6)4130060.382302.75(3)0.25 (\(V/V_0\))V+MG
Fratanduono et al. (2021)259.75.839V
*Thermal pressure \(P_{th}\) は次のように計算: \(P_{th}=\alpha_T K_0 (T-300) \times 10^-4, \,\, \alpha_T=0.261\)
Author(s)\(K_0\)\(\eta\), 1st Vinet
parameter
\(\beta\), 2nd Vinet
parameter
3rd Vinet
parameter
4th Vinet
parameter
Note
Chijioke et al.
(2005)*
280.036.3289−1.381161.492−156.484Vinet
Pressure limit: 660 GPa
*Reproduced by Chijioke et al. (2005) from Wang et al (2002) J. Appl. Phys. 92, 6616.

MgO

Author(s)\(K_0\)\(K’_0\)\(z\)\(n\)\(T_0\) [K]\(V_0\) [Å3]\(\theta_0\) [K]\(\gamma_0\)\(q\)\(a\)\(b\)Note
Jacson (1998)162.54.13423006731.411.33BM+MG
Dewaele (2000)1613.94423008001.450.8
Aizawa (2006)1604.15423007731.410.7
Tange et al. (2009)*160.634.367423007611.4420.1385.4V+MG*
Tange et al. (2009)*160.644.221423007611.4310.293.5BM+MG*
*Thermal pressure \(P_{th}\) については原著論文を見て下さい。

NaCl B2

Author(s)\(K_0\)\(K’_0\)\(z\)\(n\)\(T_0\) [K]\(V_0\) [Å3]\(\theta_0\) [K]\(\gamma_0\)\(q\)Note
Sakai et al. (2009)47.00(46)4.10(2)37.73 (4.05)3BM
Sakai et al. (2009)40.40(54)5.04(4)37.73 (4.05)V
Ueda et al. (2008)*28.45(31)5.16(4)30041.115V+*
*Thermal pressure \(P_{th}\) は次のように計算: \(P_th= \frac{\partial P}{\partial T}(T−300), \,\, \frac{\partial P}{\partial T}=0.00468(4)\)

NaCl B1

Author(s)\(K_0\)\(K’_0\)\(K”_0\)\(z\)\(n\)\(T_0\) [K]\(V_0\) [Å3]\(\theta_0\) [K]\(\gamma_0\)\(q\)Note
Matsui et al. (2012)23.75.14(5)-0.392(21)423002791.560.964BM + MG
Decker (1971)*23.70(1)4.91(1)-0.267(2)423002791.590.934BM + MG
Sata et al. (2002)
based on Pt
\(31.14 \left(V / 27.17\right)^{-2.0 / 3.0} \exp\left[-(3 \times 143.5 / 31.14 – 2) \left\{ \left(V / 27.17\right)^{1.0 / 3.0} – 1 \right\} \right] \)
Sata et al. (2002)
based on MgO
\(32.15 \left(V / 27.17\right)^{-2.0 / 3.0} \exp\left[-(3 \times 141.0 / 32.15 – 2) \left\{ \left(V / 27.17\right)^{1.0 / 3.0} – 1 \right\} \right] \)
*Recalculated by Matsui et al. (2012)

Al2O3

Author(s)\(K_0\)\(K’_0\)\(\partial K_{(T,P=0)}/ \partial T\)\(a\)\(b\)\(c\)\(T_0\) [K]\(V_0\)
[cm3/mol]
Note
Dubrovinsky
 et al. (1998)
258(2)4.88(4)-0.0202.6×10-51.81(9)×10-9-0.6730025.59(2)T-dependence
BM

Diamond

Author(s)\(K_0\)\(K’_0\)\(K”_0\)\(z\)\(n\)\(T_0\) [K]\(V_0\) [cm3/mol]\(\theta_0\) [K]\(\gamma_0\)\(q\)Note
Occelli et al. (2003)446(1)3.0(1)3.4170(8)3BM

Al

Author(s)\(K_0\)\(\eta\), 1st Vinet
parameter
\(\beta\), 2nd Vinet
parameter
3rd Vinet
parameter
4th Vinet
parameter
Note
Chijioke et al.
(2005)
72.64.126726.1269−154.33326.694Vinet
Pressure limit: 200 GPa

Cu

Author(s)\(K_0\)\(\eta\), 1st Vinet
parameter
\(\beta\), 2nd Vinet
parameter
3rd Vinet
parameter
4th Vinet
parameter
Note
Chijioke et al.
(2005)
133.35.416715.921−90.223235.814Vinet
Pressure limit: 200 GPa

Ta

Author(s)\(K_0\)\(\eta\), 1st Vinet
parameter
\(\beta\), 2nd Vinet
parameter
3rd Vinet
parameter
4th Vinet
parameter
Note
Chijioke et al.
(2005)
194.11.926552.348−402.721031.54Vinet
Pressure limit: 200 GPa

W

Author(s)\(K_0\)\(\eta\), 1st Vinet
parameter
\(\beta\), 2nd Vinet
parameter
3rd Vinet
parameter
4th Vinet
parameter
Note
Chijioke et al.
(2005)
308.62.791445.694−409.971290.24Vinet
Pressure limit: 200 GPa

Mo

Author(s)\(K_0\)\(K’_0\)\(K”_0\)\(\partial K_{(T,P=0)}/ \partial T\)\(a\)\(b\)\(c\)\(z\)\(n\)\(T_0\) [K]\(V_0\)
[cm3/mol]
\(\theta_0\)\(\gamma_9\)\(q\)Note
Huang (2016)
MGD
2554.25213004702.010.63BM+MG
Zhao
(2000)
2683.81-0.0141-0.02131.3111.23004BM + T-dependence
BM
Author(s)\(K_0\)\(\eta\), 1st Vinet
parameter
\(\beta\), 2nd Vinet
parameter
3rd Vinet
parameter
4th Vinet
parameter
Note
Chijioke et al.
(2005)*
264.874.7127−8.179583.532−189.674Vinet
Pressure limit: 1020 GPa
*Reproduced by Chijioke et al. (2005) from Wang et al (2002) J. Appl. Phys. 92, 6616.

Re

Author(s)\(K_0\)\(K’_0\)\(K”_0\)\(z\)\(n\)\(T_0\) [K]\(V_0\) [Å3]\(\theta_0\) [K]\(\gamma_0\)\(q\)Note
Anz. (####)352.64.5629.467V
Sakai (####)3584.829.47V
Dub3426.15-0.02929.464BM

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