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、次のように記号を定義します。
| 温度 \(T_0\), 圧力 0 GPa (標準状態)における単位格子体積 | \(V_0 = V_{(T = T_0, P=0)}\) |
| 温度 \(T_0\), 圧力 \(P\) における単位格子体積 | \(V = V_{(T = T_0, P)}\) |
| 標準状態における体積弾性率 | \(K_0 = K_{(T=T_0, P=0)}\) |
| 体積弾性率の1階微分 | \(K’_0 = (\partial K_0 / \partial P)_T\) |
| 体積弾性率の2階微分 | \(K”_0 = (\partial^2 K_0 / \partial P^2)_T\) |
| 圧力 0 Gpa の単位格子体積と 圧力 \(P\) の単位格子体積の比率の1/3乗 (立方晶系の場合は単位格子の一辺の長さ \(a\) の比率に相当) |
\(x=\left(\frac{V_0}{V}\right)^{1/3}\) |
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。
$$\begin{array}{rcl}
P_{(T=T_0, V)}^{2nd} &=& \frac{3}{2}K_0 \left[ \left( \frac{V_0}{V} \right)^{\frac{7}{3}} – \left( \frac{V_0}{V} \right)^{\frac{5}{3}} \right]\\
&=& \frac{3}{2}K_0 ( x^7 – x^5)
\end{array}$$
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。
$$\begin{array}{rcl}
P_{(T=T_0, V)}^{3rd} &=& \frac{3}{2}K_0 \left[ \left( \frac{V_0}{V} \right)^{\frac{7}{3}} – \left( \frac{V_0}{V} \right)^{\frac{5}{3}} \right] \left[ 1 + \frac{3}{4} (K’_0-4) \left\{ \left( \frac{V_0}{V} \right)^{\frac{2}{3}} -1 \right\} \right] \\
&=& \frac{3}{2}K_0 (x^7 – x^5) \left\{ 1 + \frac{3}{4} (K’_0-4) (x^2 -1) \right\} \\
&=& P_{(T=T_0, V)}^{2nd} \left\{ 1 + \frac{3}{4} (K’_0-4) (x^2 -1) \right\}
\end{array}$$
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。
$$\begin{array}{rcl}
P_{(T=T_0, V)}^{4th} &=& \frac{3}{2}K_0
\left[ \left( \frac{V_0}{V} \right)^{\frac{7}{3}} – \left( \frac{V_0}{V} \right)^{\frac{5}{3}} \right]
\left[
1 + \frac{3}{4} (K’_0-4) \left\{ \left( \frac{V_0}{V} \right)^{\frac{2}{3}} -1 \right\} +
\frac{9K_0 K”_0 + 9{K’_0}^2 – 63K’_0 + 143}{24} \left\{ \left(\frac{V_0}{V} \right)^{\frac{2}{3}} -1 \right\}^2
\right]\\
&=& \frac{3}{2}K_0 (x^7- x^5) \left\{ 1 + \frac{3}{4} (K’_0-4) (x^2 -1) +
\frac{9K_0 K”_0 + 9{K’_0}^2 – 63K’_0 + 143}{24} (x^2 -1)^2 \right\}\\
&=& P_{(T=T_0, V)}^{3rd} + P_{(T=T_0, V)}^{2nd} \left\{ \frac{9K_0 K”_0 + 9{K’_0}^2 – 63K’_0 + 143}{24} (x^2 -1)^2 \right\}
\end{array}$$
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).
$$\begin{array}{rcl}
P_{(T=T_0, V)} &=& \displaystyle 3K_0
\left(\frac{V_0}{V}\right)^{\frac{2}{3}}
\left\{ 1- \left(\frac{V_0}{V}\right)^{-\frac{1}{3}} \right\}
\exp \left[ \frac{3}{2} (K’_0-1) \left\{ 1- \left(\frac{V_0}{V}\right)^{-\frac{1}{3}} \right\} \right] \\
&=& \displaystyle 3K_0 ( x^2- x ) \exp \left[ \frac{3}{2} (K’_0-1) ( 1- x^{-1} ) \right]
\end{array}$$
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).
$$\begin{array}{rcl}
P_{(T=T_0, V)} &=& 3K_0
\left(\frac{V_0}{V}\right)^{\frac{2}{3}}
\left\{ 1- \left(\frac{V_0}{V}\right)^{ -\frac{1}{3}} \right\}
\exp \left[ \eta \left\{ 1- \left(\frac{V_0}{V}\right)^{-\frac{1}{3}} \right\} +
\beta \left\{ 1- \left(\frac{V_0}{V}\right)^{-\frac{1}{3}} \right\} ^2 +
\psi \left\{ 1- \left(\frac{V_0}{V}\right)^{-\frac{1}{3}} \right\} ^3 + \cdots \right] \\
&=& \displaystyle 3K_0 ( x^2- x ) \exp \left[
\eta (1- x^{-1}) + \beta (1- x^{-1})^2 + \psi (1- x^{-1})^3 + \cdots \right]
\end{array}$$
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).
$$\begin{array}{rcl}
P&=&\displaystyle 3K_0 \frac{1-\left(\frac{V}{V_0}\right)^{1/3}}{\left(\frac{V}{V_0}\right)^{5/3}}
\left[ 1+aC_2 \left\{ 1-\left(\frac{V}{V_0}\right)^{1/3} \right\} \right] \exp\left[C_0 \left\{1-\left(\frac{V}{V_0}\right)^{1/3} \right\} \right]\\
&=&\displaystyle 3K_0 \frac{1-x^{-1}}{x^{-5}} [1+x^{-1}C_2(1-x^{-1}) ] \exp\left[C_0 (1-x^{-1}) \right]
\end{array}$$ただし、$$
\begin{array}{lll}
x = \left(\frac{V_0}{V}\right)^{1/3} &
C_0 = -\ln (3K_0 / p_{FG_0}) &
C_2 = \frac{3}{2} (K’_0 -3) -C_0\\
p_{FG_0}=a_{FG} ( Z_e/V_0) ^{5/3}&
a_{FG} = 2336.965\, [\mathrm{GPa}\, \textbf{Å}^5 ] &
\end{array}
$$\(p_{FG_0}\) : 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)* | 166.65 | 5.4823 | -0.0115 | 300 | 3BM+* | ||||||
| Sim (2002) | 167 | 5.0 | 4 | 1 | 300 | 170 | 2.97 | 1.0 | 3BM+MG | ||
| Fratanduono et al., 2021 | 170.09 | 5.880 | V |
| Author(s) | \(K_0\) | \(\eta\), 1st Vinet parameter |
\(\beta\), 2nd Vinet parameter |
3rd Vinet parameter |
4th Vinet parameter |
Note |
|---|---|---|---|---|---|---|
| Chijioke et al. (2005)* |
177.26 | 6.38000 | 1.9334 | −1.0292 | 33.941 | 4Vinet Pressure limit: 513 GPa |
Pt
| Author(s) | \(K_0\) | \(K’_0\) | \(z\) | \(n\) | \(T_0\) [K] | \(V_0\) [Å3] | \(\theta_0\) [K] | \(\gamma_0\) | \(q\) | Note |
|---|---|---|---|---|---|---|---|---|---|---|
| Holmes (1989)* | 266 | 5.81 | V+* | |||||||
| Matsui et al. (2009) | 273 | 5.20 | 4 | 1 | 300 | 60.38 | 230 | 2.70 | 1.10 | V+MG |
| Fei et al. (2007) | 277 | 5.08(2) | 4 | 1 | 300 | 60.38 | 230 | 2.72(3) | 0.5 | V+MG |
| Zha et al. (2008) | 273.5(10) | 4.70(6) | 4 | 1 | 300 | 60.38 | 230 | 2.75(3) | 0.25 (\(V/V_0\)) | V+MG |
| Fratanduono et al. (2021) | 259.7 | 5.839 | V |
| Author(s) | \(K_0\) | \(\eta\), 1st Vinet parameter |
\(\beta\), 2nd Vinet parameter |
3rd Vinet parameter |
4th Vinet parameter |
Note |
|---|---|---|---|---|---|---|
| Chijioke et al. (2005)* |
280.03 | 6.3289 | −1.3811 | 61.492 | −156.48 | 4Vinet Pressure limit: 660 GPa |
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) | 162.5 | 4.13 | 4 | 2 | 300 | 673 | 1.41 | 1.3 | 3BM+MG | |||
| Dewaele (2000) | 161 | 3.94 | 4 | 2 | 300 | 800 | 1.45 | 0.8 | ||||
| Aizawa (2006) | 160 | 4.15 | 4 | 2 | 300 | 773 | 1.41 | 0.7 | ||||
| Tange et al. (2009)* | 160.63 | 4.367 | 4 | 2 | 300 | 761 | 1.442 | 0.138 | 5.4 | V+MG* | ||
| Tange et al. (2009)* | 160.64 | 4.221 | 4 | 2 | 300 | 761 | 1.431 | 0.29 | 3.5 | BM+MG* |
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) | – | – | 300 | 41.115 | – | – | – | V+* |
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) | 23.7 | 5.14(5) | -0.392(21) | 4 | 2 | 300 | – | 279 | 1.56 | 0.96 | 4BM + MG |
| Decker (1971)* | 23.70(1) | 4.91(1) | -0.267(2) | 4 | 2 | 300 | – | 279 | 1.59 | 0.93 | 4BM + 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 |
\(32.15 \left(V / 27.17\right)^{-2.0 / 3.0} \exp\left[-(3 \times 141.0 / 32.15 – 2) \left\{ \left(V / 27.17\right)^{1.0 / 3.0} – 1 \right\} \right] \) | ||||||||||
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) | -0.020 | 2.6×10-5 | 1.81(9)×10-9 | -0.67 | 300 | 25.59(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) |
72.6 | 4.1267 | 26.1269 | −154.33 | 326.69 | 4Vinet 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) |
133.3 | 5.4167 | 15.921 | −90.223 | 235.81 | 4Vinet 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) |
194.1 | 1.9265 | 52.348 | −402.72 | 1031.5 | 4Vinet 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) |
308.6 | 2.7914 | 45.694 | −409.97 | 1290.2 | 4Vinet 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 |
255 | 4.25 | – | – | – | – | – | 2 | 1 | 300 | – | 470 | 2.01 | 0.6 | 3BM+MG |
| Zhao (2000) |
268 | 3.81 | -0.0141 | -0.0213 | 1.31 | 11.2 | 300 | 4BM + 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)* |
264.87 | 4.7127 | −8.1795 | 83.532 | −189.67 | 4Vinet Pressure limit: 1020 GPa |
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. (####) | 352.6 | 4.56 | – | – | – | – | 29.467 | – | – | – | V |
| Sakai (####) | 358 | 4.8 | 29.47 | V | |||||||
| Dub | 342 | 6.15 | -0.029 | 29.46 | 4BM |