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Equation of State (EOS)

　The relationship between temperature, pressure, and volume of a substance is called the Equation Of State (EOS). The simplest equation of state is the $P V = n R T$ equation familiar from high school physics, but unfortunately this only applies to ideal gases. The equation of state applicable to real crystalline materials is more complex, with expressions based on various models. This page introduces several major equations of state applicable to crystals. The symbols are defined as follows.

Unit cell volume at temperature $T_0$, pressure 0 GPa (standard state)	$V_0 = V_{(T = T_0, P=0)}$
Unit cell volume at temperature $T_0$, pressure $P$	$V = V_{(T = T_0, P)}$
Bulk modulus at standard state	$K_0 = K_{(T=T_0, P=0)}$
First derivative of the bulk modulus	$K’_0 = (\partial K_0 / \partial P)_T$
Second derivative of the bulk modulus	$K”_0 = (\partial^2 K_0 / \partial P^2)_T$
Cube root of the ratio of unit cell volume at pressure 0 GPa to unit cell volume at pressure $P$ (For cubic systems, this corresponds to the ratio of the unit cell edge length $a$)	$x=\left(\frac{V_0}{V}\right)^{1/3}$

Birch-Murnaghan

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

2nd-order

$V_0/V$, and bulk modulus $K_0$, the pressure is calculated.

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

3rd-order

The pressure is calculated using $V_0/V$, bulk modulus $K_0$, and its first derivative with respect to pressure $K’_0$.

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

4th-order

The pressure is calculated using $V_0/V$, bulk modulus $K_0$, and its first and second derivatives with respect to pressure $K’_0, K”_0$.

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

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).

$$\begin{array}{rcl}
P&=&\displaystyle 3K_0 \frac{1-\left(\frac{V}{V_0}\right)^{1/3}}{\left(\frac{V}{V_0}\right)^{5/3}}
\left[ 1+aC_2 \left\{ 1-\left(\frac{V}{V_0}\right)^{1/3} \right\} \right] \exp\left[C_0 \left\{1-\left(\frac{V}{V_0}\right)^{1/3} \right\} \right]\\
&=&\displaystyle 3K_0 \frac{1-x^{-1}}{x^{-5}} [1+x^{-1}C_2(1-x^{-1}) ] \exp\left[C_0 (1-x^{-1}) \right]
\end{array}$$where $$
\begin{array}{lll}
x = \left(\frac{V_0}{V}\right)^{1/3} &
C_0 = -\ln (3K_0 / p_{FG_0}) &
C_2 = \frac{3}{2} (K’_0 -3) -C_0\\
p_{FG_0}=a_{FG} ( Z_e/V_0) ^{5/3}&
a_{FG} = 2336.965\, [\mathrm{GPa}\, \textbf{Å}^5 ] &
\end{array}
$$$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).

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

Mie-Grüneisen(-Debye)

The thermal pressure effect ($\Delta P_{th}$) derived from the Mie-Grüneisen(-Debye) model is as follows.
$$ \Delta P_{th} = (e-e_0) \frac{\gamma}{v}
$$where $$
\begin{array}{lll}
e=9 R n T \left( \frac{T}{\theta} \right)^3 \displaystyle\int_0^{\frac{\theta}{T}} \frac{z^3}{e^z-1} dz \,\,\,&
e_0= 9 R n T_0 \left( \frac{T_0}{\theta_0} \right)^3 \displaystyle\int_0^{\frac{\theta_0}{T_0}} \frac{z^3}{e^z-1} dz \,\,\,&
\theta = \theta_0 \exp\left[(\gamma_0-\gamma)/q\right] \,\,\,\\
\gamma = \gamma_0 (v/v_0)^q &
v= \displaystyle \frac{V}{z} N_A \times 10^{-30}\,\,\,[\mathrm{m^3/mol}] &
v_0= \displaystyle \frac{V_0}{z} N_A \times 10^{-30} \,\,\,[\mathrm{m^3/mol}]
\end{array}
$$$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)

EOS Parameters for Representative Materials

Au

Author(s)	$K_0$	$K’_0$	$z$	$n$	$\partial K_{(T,P=0)}/ \partial T$	$T_0$ [K]	$\theta_0$ [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

*Thermal pressure $P_{th}$ is calculated as follows: $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)*	177.26	6.38000	1.9334	−1.0292	33.941	4Vinet 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)*	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

*Thermal pressure $P_{th}$ is calculated as follows: $P_{th}=\alpha_T K_0 (T-300) \times 10^-4, \,\, \alpha_T=0.261$

Author(s)	$K_0$	$\eta$, 1st Vinet parameter	$\beta$, 2nd Vinet parameter	3rd Vinet parameter	4th Vinet parameter	Note
Chijioke et al. (2005)*	280.03	6.3289	−1.3811	61.492	−156.48	4Vinet 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]	$\theta_0$ [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*

*For thermal pressure $P_{th}$, please refer to the original paper.

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+*

*Thermal pressure $P_{th}$ is calculated as follows: $P_th= \frac{\partial P}{\partial T}(T−300), \,\, \frac{\partial P}{\partial T}=0.00468(4)$

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)	23.7	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] $

*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)	-0.020	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

*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. (####)	352.6	4.56	–	–	–	–	29.467	–	–	–	V
Sakai (####)	358	4.8					29.47				V
Dub	342	6.15	-0.029				29.46				4BM

Unit cell volume at temperature \(T_0\), pressure 0 GPa (standard state)	\(V_0 = V_{(T = T_0, P=0)}\)
Unit cell volume at temperature \(T_0\), pressure \(P\)	\(V = V_{(T = T_0, P)}\)
Bulk modulus at standard state	\(K_0 = K_{(T=T_0, P=0)}\)
First derivative of the bulk modulus	\(K’_0 = (\partial K_0 / \partial P)_T\)
Second derivative of the bulk modulus	\(K”_0 = (\partial^2 K_0 / \partial P^2)_T\)
Cube root of the ratio of unit cell volume at pressure 0 GPa to unit cell volume at pressure \(P\) (For cubic systems, this corresponds to the ratio of the unit cell edge length \(a\))	\(x=\left(\frac{V_0}{V}\right)^{1/3}\)