On the creation of thermal equations of state for use in Dioptas

Figures & data

Figure 1. Example of a Version 4 jcpds file. The 6 parameters used to establish the thermal EoS are K0, K0P, DK0DT, DK0PDT, ALPHAT and DALPHADT.

Figure 2. Descriptions of the input parameters of a Version 4 jcpds file [11].

Figure 3. The solid data points show (a) the 0 GPa volumetric thermal expansion, and (b) the 298 K isothermal compressibility of MgO, as calculated from the Sokolova et al. thermal EoS [21]. The dashed line in (a) is the best fit to the thermal expansion using Equation (8(8) $V=V_0(1+{\alpha }_T(T\left)\mathrm{\Delta }T\right)=V_0(1+{\alpha }_{298}\mathrm{\Delta }T+{\frac{\mathrm{\partial }{\alpha }_T}{\mathrm{\partial }T}|}_{298}{\mathrm{\Delta }T}^2)$(8) ), while the dashed line in (b) is the best fit to the compressibility using a 3rd order Birch-Murnaghan EoS. The unfilled data points in (a) show the thermal expansion of MgO at 0.01 GPa, as calculated by Dioptas using the jcpds input file shown in Figure 1. At a given temperature and this very low, but non-zero pressure, most of the MgO unit cell volumes calculated by Dioptas are greater than the associated volumes on the P=0 isobar, particularly at high temperatures. Effectively, Dioptas produces isotherms exhibiting expansion upon compression within a narrow pressure range. The reasons for this non-physical response are described in the main text.

Figure 4. The differences in pressures calculated by the Sokolova et al. thermal EoS for MgO as a function of the unit cell volume and temperature, and the reparameterisation suitable for use in Dioptas via import of a jcpds file. The shown volume scale equates to a pressure range of approximately 0–100 GPa.

Figure 5. The 298, 1000 and 1500 K isotherms for MgO represented by the black dashed lines, as calculated using the thermal EoS of Sokolova et al. [21]. The black square data points show a number of P-V data points along the three isotherms, as calculated by Dioptas using the jcpds file shown in Figure 1.

Figure 6. The 298, 1000 and 2000 K isotherms for Ne, as calculated using the thermal EoS of Fei et al. [23] are shown by the black dashed lines. The filled squares show a number of P-V data points along the 1000 and 2000 K isotherms, as calculated by Dioptas using the jcpds file input parameters for Ne given in Table 1.

Table 1. The optimised thermal EoS parameters required for the jcpds files of various materials. $V_0$ is the unit cell volume for each material at ambient pressure, while the meanings of the other parameters are described in the main text.

Material	$V_0$ (Å${}^3$/cell)	$K_0$ (GPa)	$K^{\mathrm{\prime }}$	$\alpha ({10}^{-5}$ K${}^{-1})$	$\mathrm{\partial }{\alpha }_T/\mathrm{\partial }T$ (${10}^{-8}$ $K^{-2}$)	$\mathrm{\partial }{K}_0/\mathrm{\partial }T$ (GPa $K^{-1}$)	$\mathrm{\partial }{K}^{\mathrm{\prime }}/\mathrm{\partial }T$ ($K^{-1}$)
MgO [21]	74.711${}^{\star }$	161.107	4.010	3.502	0.775	−0.00624	0.000193
Al [21]	66.288${}^{\star }$	76.174	4.136	6.527	4.395	−0.00914	0.000332
Au [21]	67.849${}^{\star }$	169.669	5.662	3.911	1.403	−0.01017	0.000330
Cu [21]	47.239${}^{\star }$	137.606	4.952	4.744	1.735	−0.01078	0.000412
Mo [21]	31.115${}^{\star }$	261.274	4.122	1.359	0.491	−0.00945	0.000139
Nb [21]	35.961${}^{\star }$	169.297	3.711	2.040	0.358	−0.00372	0.000069
Pt [21]	60.384${}^{\star }$	277.936	5.158	2.572	0.578	−0.01312	0.000303
Ta [21]	36.070${}^{\star }$	191.797	3.756	1.942	0.206	−0.00388	0.000077
NaCl (B1) [19]	179.113	25.079	4.569	11.927	7.132	−0.00529	0.001114
NaCl (B2) [23]	41.350	30.690	4.330	10.719	−0.893	0.00096	−0.000047
KCl (B2) [29]	52.953	24.606	4.540	8.944	0.173	−0.00028	0.000036
hcp-Fe [25]	22.642	150.475	5.537	5.051	−0.280	0.00713	−0.000174
Ne [23]	88.967	1.430	8.020	310.340	−7.539	0.00011	−0.000279
Cu [33]	47.219	136.922	4.921	0	0	0	0
Au [34]	67.716	178.548	5.327	0	0	0	0
Pt [34]	60.408	267.319	5.395	0	0	0	0
Mo [32]	31.095	260.607	3.965	0	0	0	0
Ta [35]	36.093	168.015	4.140	1.145	0.675	−0.00079	0.000007

Notes: $V_0$ values denoted ★ were held fixed during the refinement at the values given in [21].

Table 2. The as-published parameters for the room temperature AP2 EoS of Mo [32], and the Vinet EoS of Au and Pt [34].

Metal	$V_0$ (fixed) (Å${}^3$/atom)	EoS	$K_0$ (GPa)	$K^{\mathrm{\prime }}$	EoS	$K_0$ (GPa)	$K^{\mathrm{\prime }}$
Mo	15.547	AP2	260 (fixed)	4.00(1)	B-M	260.607	3.965
Au	16.929	Vinet	170.90(24)	5.880(5)	B-M	178.548	5.327
Pt	15.102	Vinet	259.70(16)	5.839(3)	B-M	267.319	5.395

Notes: $V_0$ is the atomic volume at ambient pressure. For comparison, the values of $K_0$ and $K^{\mathrm{\prime }}$ obtained by refitting these equations of state with a Birch-Murnaghan (B-M) EoS are also shown.

Figure 7. The 298, 1000, 2000 and 3000 K isotherms for hcp-Fe (ϵ-Fe), as calculated using the thermal EoS of Dorogokupets et al. [25]. The filled squares show a number of P-V data points along the 1000, 2000 and 3000 K isotherms, as calculated by using the jcpds file input parameters for hcp-Fe given in Table 1.

Figure 8. The 300, 2000 and 3000 K isotherms for the B2 phase of KCl, as calculated using the thermal EoS of Dewaele et al. [29]. The filled squares show a number of data points along the same isotherms, as calculated by using the jcpds file input parameters for B2-KCl given in Table 1.

Figure 9. The room temperature equations of state of Mo, Cu, Au and Pt to 300 GPa. The data points show the calculated compressions using published AP2 (Mo) [32], 3rd order Vinet (Cu) [33], and 2nd order Vinet (Au and Pt) equations of state [34]. The dashed lines through the data points are the best-fitting B-M equations of state, using the parameters listed in Table 1.

Figure 10. The differences in pressures between those calculated by the Gorman et al. thermal EoS for Ta, and those calculated by Dioptas using the parameters given in Table 1. Despite the very large P-T range (2300 GPa and 5000 K) the maximum difference between the two equations of state is only ∼8 GPa at 240 GPa and 1750 K. No contours are shown in the white area at low pressures (<100 GPa) and high temperatures (3300–5000 K) as Ta is in the liquid state at such conditions.

Merkel S. Example JCPDS file in the Version 4 format; 2006. Available from: https://merkel.texture.rocks/HPDiff/file_jcpds_4.php

 Google Scholar

Sokolova TS, Dorogokupets PI, Dymshits AM, et al. Microsoft excel spreadsheets for calculation of PVT relations and thermodynamic properties from equations of state of MgO, diamond and nine metals as pressure markers in high pressure and high-temperature experiments. Comput Geosci. 2016;94:162–169. DOI: 10.1016/j.cageo.2016.06.002

 Web of Science ®Google Scholar

Fei Y, Ricolleau A, Frank M, et al. Toward an internally consistent pressure scale. Proc Natl Acad Sci USA. 2007;104(22):9182–9186. Available from: https://www.jstor.org/stable/25427827 DOI: 10.1073/pnas.0609013104

 PubMed Web of Science ®Google Scholar

Dorogokupets PI, Dewaele A. Equations of state of MgO, Au, Pt, NaCl-B1, and NaCl-B2: internally consistent high-temperature pressure scales. High Press Res. 2007;27(4):431–446. DOI: 10.1080/08957950701659700

 Web of Science ®Google Scholar

Dewaele A, Belonoshko AB, Garbarino G, et al. High pressure–high-temperature equation of state of KCl and KBr. Phys Rev B. 2012;85(21):Article ID 214105. DOI: 10.1103/PhysRevB.85.214105

 Web of Science ®Google Scholar

Dorogokupets PI, Dymshits AM, Litasov KD, et al. Thermodynamics and equations of state of iron to 350 GPa and 6000 K. Sci Rep. 2017;7(1):Article ID 41863. DOI:10.1038/srep41863

 PubMedGoogle Scholar

Fratanduono DE, Smith RF, Ali SJ, et al. Probing the solid phase of noble metal copper at terapascal conditions. Phys Rev Lett. 2020;124:1Article ID 015701. DOI:10.1103/PhysRevLett.124.015701

 PubMed Web of Science ®Google Scholar

Fratanduono DE, Millot M, Braun DG, et al. Establishing gold and platinum standards to 1 terapascal using shockless compression. Science. 2021;372(6546):1063–1068. DOI:10.1126/science.abh0364

 PubMed Web of Science ®Google Scholar

Shen G, Wang Y, Dewaele A, et al. Toward an international practical pressure scale: a proposal for an IPPS ruby gauge (IPPS-Ruby2020). High Press Res. 2020;40(3):299–314. DOI:10.1080/08957959.2020.1791107

 Web of Science ®Google Scholar

Gorman MG, Wu CJ, Smith RF, et al. Ramp compression of tantalum to multiterapascal pressures: constraints of the thermal equation of state to 2.3 TPa and 5000 K. Phys Rev B. 2023;107(1):Article ID 014109. DOI: 10.1103/PhysRevB.107.014109

 Web of Science ®Google Scholar