Equations of state of iron and nickel to the pressure at the center of the Earth

Naohisa Hirao; Yuichi Akahama; Yasuo Ohishi

doi:10.1063/5.0074340

Matter and Radiation at Extremes, Volume. 7, Issue 3, 038403(2022)

Equations of state of iron and nickel to the pressure at the center of the Earth

Naohisa Hirao1... Yuichi Akahama2,a) and Yasuo Ohishi1 |Show fewer author(s)

Author Affiliations

¹Japan Synchrotron Radiation Research Institute (JASRI), Sayo, Sayo-gun, Hyogo 679-5198, Japan

²Graduate School of Material Science, University of Hyogo, Kouto, Kamigohri, Hyogo 678-1297, Japan

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Figures & Tables(11)

$Representative x-ray diffraction pattern (red crosses) and Rietveld simulation (green line) for hcp-Fe at 354 GPa. The blue line shows a residual. Lattice constants are calculated to be a = 2.1294(17) Å and c = 3.390(5) Å for hcp-Fe and a = 3.4049(11) Å for fcc-Pt.$

Fig. 1. Representative x-ray diffraction pattern (red crosses) and Rietveld simulation (green line) for hcp-Fe at 354 GPa. The blue line shows a residual. Lattice constants are calculated to be a = 2.1294(17) Å and c = 3.390(5) Å for hcp-Fe and a = 3.4049(11) Å for fcc-Pt.

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Fig. 2. Compression curve of Fe up to a pressure of 354 GPa, together with curves from previous reports. Solid black, green, blue, and red curves correspond to the experimental results of the present study and those conducted by Mao et al.,³ Dubrovinsky et al.,⁴ and Dewaele et al.,⁵ respectively. ΔV_a denotes the deviations of the atomic volume for each previous curve from the present one.

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$Representative x-ray diffraction pattern (red crosses) and Rietveld simulation (blue line) for Ni at 368 GPa. The green line shows a residual. The estimated lattice constants a for fcc-Ni and fcc-Pt are 2.9521(10) and 3.3977(6) Å, respectively.$

Fig. 3. Representative x-ray diffraction pattern (red crosses) and Rietveld simulation (blue line) for Ni at 368 GPa. The green line shows a residual. The estimated lattice constants a for fcc-Ni and fcc-Pt are 2.9521(10) and 3.3977(6) Å, respectively.

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Fig. 4. Compression curve of Ni up to a pressure of 368 GPa, together with curves from previous reports. Solid black, green, and red curves are Vinet EOS fits to the experimental data from the present study, Kennedy and Keeler,¹⁴ and Dewaele et al.,⁶ respectively.

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Fig. 5. Gamma plots constructed using (a) the fcc-Pt data at 354 GPa from Table II, (b) the fcc-Pt data at 368 GPa from Table IV, and (c) the fcc-Ni data at 368 GPa from Table IV.

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Fig. 6. Densities of Fe, Ni, and Fe_0.91Ni_0.09 at 298 K as functions of pressure. The Preliminary Reference Earth Model (PREM)³⁷ is also shown for comparison.

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Table 1. Conditions of each experimental run.

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Table 1. Conditions of each experimental run.

Run No.	Diamond anvil geometry^a (μm)	PTM^b	Scale	P range (GPa)
Fe 1	500	He	Pt + ruby	5–20
Fe 2	350	He	Ruby	0–54
Fe 3	35/250/350	None	Pt	0–296
Fe 4	28/250/350	None	Pt	219–354
Ni 1	500	MEW^c	Ruby	0–10
Ni 2	500	He	Pt	0–21
Ni 3	100/300	None	Pt	2–179
Ni 4	27/250/450	None	Pt	257–368
Ni 5	50/230/450	None	Pt	163–202

Table 2. 2θ and d-values (d_o) of the observed diffraction lines with λ = 0.4176 Å for the pattern at 354 GPa, indices for these lines, and calculated d-values (d_c) for the hcp-Fe and fcc-Pt phases. Calculated lattice constants are also listed.

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Table 2. 2θ and d-values (d_o) of the observed diffraction lines with λ = 0.4176 Å for the pattern at 354 GPa, indices for these lines, and calculated d-values (d_c) for the hcp-Fe and fcc-Pt phases. Calculated lattice constants are also listed.

2θ (deg)	d_o (Å)	hcp-Fe			fcc-Pt
		a = 2.1294(17) (Å)			a = 3.4049(11) (Å)
		c = 3.390(5) (Å)
		Index	d_c (Å)	d_o − d_c (Å)	Index	d_c (Å)	d_o − d_c (Å)
12.193	1.9661				111	1.9658	0.0003
12.970	1.8487	010	1.8441	0.0046
14.090	1.7024	(002)			200	1.7025	−0.0001
14.833	1.6176	011	1.6200	−0.0024
19.290	1.2463	012	1.2480	−0.0017
20.004	1.2022				220	1.2038	−0.0016
22.639	1.0638	110	1.0647	−0.0009
23.451	1.0275				311	1.0266	0.0008
24.518	0.9834				222	0.9829	0.0005
25.003	0.9646	013	0.9636	0.0010
26.808	0.9007	112	0.9016	−0.0009
27.113	0.8908	021	0.8897	0.0010
28.406	0.8510				400	0.8512	−0.0002

Table 3. Parameters of the Vinet EOS obtained by a least-squares fit of the experimental compression data for hcp-Fe, together with those from previous reports. The parameters of these EOSs are the atomic volume V₀, bulk modulus K₀, and its pressure derivative $K_{0}^{'}$ under ambient conditions.

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Table 3. Parameters of the Vinet EOS obtained by a least-squares fit of the experimental compression data for hcp-Fe, together with those from previous reports. The parameters of these EOSs are the atomic volume V₀, bulk modulus K₀, and its pressure derivative $K_{0}^{'}$ under ambient conditions.

V₀ (Å³)	K₀ (GPa)	$K_{0}^{'}$	P Range (GPa)	P Gauge	Formula	PTM	References
11.215(29)	159.27(99)	5.86(4)	14–354	Ruby or Pt	Vinet	He or none	Present
11.176(17)	164.8(3.6)	5.33(9)	35–300	Pt	B-M^a	Ar or none	Mao et al.³
11.1780(2)	160.67(5)	5.36(1)			B-M		Corrected EOS
11.1989	156	5.81	20–200	Pt	B-M	None (heating)	Dubrovinsky et al.⁴
11.2022(1)	151.65(2)	5.854(3)			B-M		Corrected EOS
11.214(49)	163.4(7.9)	5.38(16)	17–197	Ruby or W	Vinet	He	Dewaele et al.⁵
11.214	156.7(9)	5.8(1.0)			Vinet		Corrected EOS

Table 4. 2θ and d-values (d_o) of the observed diffraction lines with λ = 0.4134 Å for the pattern at 368 GPa and indices for these lines and calculated d-values (d_c) for the fcc-Ni and fcc-Pt phases. Calculated lattice constants are also listed.

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Table 4. 2θ and d-values (d_o) of the observed diffraction lines with λ = 0.4134 Å for the pattern at 368 GPa and indices for these lines and calculated d-values (d_c) for the fcc-Ni and fcc-Pt phases. Calculated lattice constants are also listed.

2θ (deg)	d_o (Å)	fcc-Ni			fcc-Pt
		a = 2.9521(10) (Å)			a = 3.3977(6) (Å)
		Index	d_c (Å)	d_o − d_c (Å)	Index	d_c (Å)	d_o − d_c (Å)
12.048	1.9614				111	1.9616	−0.0001
13.856	1.7065	111	1.7053	0.0011	(200)
16.008	1.4783	200	1.4769	0.0015
19.722	1.2020				220	1.2012	0.0008
22.757	1.0434	220	1.0443	−0.0008
23.186	1.0243				311	1.0244	−0.0001
24.244	0.9803				222	0.9808	−0.0005
26.746	0.8900	311	0.8906	−0.0005
27.969	0.8518	222	0.8527	−0.0004	(400)
30.613	0.7798				331	0.7794	0.0003
31.446	0.7596				420	0.7597	−0.0001
32.359	0.7387	400	0.7384	0.0001

Table 5. Parameters of the Vinet EOS obtained by a least-squares fit of the experimental compression data for Ni, together with those from previous reports. The parameters of these EOSs are the atomic volume V₀, bulk modulus K₀, and its pressure derivative $K_{0}^{'}$ , under ambient conditions.

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Table 5. Parameters of the Vinet EOS obtained by a least-squares fit of the experimental compression data for Ni, together with those from previous reports. The parameters of these EOSs are the atomic volume V₀, bulk modulus K₀, and its pressure derivative $K_{0}^{'}$ , under ambient conditions.

V₀ (Å³)	K₀ (GPa)	$K_{0}^{'}$	Method	P Range (GPa)	Formula	References
10.9376(4)	173.5(1.4)	5.55(5)	DAC	0–368	Vinet	Present
⋯	183(3)	⋯	Ultrasonic	⋯	⋯	Yamamoto¹⁵
10.940	176.7(2.5)	5.23(9)	DAC	0–156	Vinet	Dewaele et al.⁶
10.940	169.3(8)	5.67(18)			Vinet	Corrected EOS
10.9376	187.8(5)	4.89(4)	Shock wave	0–120	Vinet	Kennedy and Keeler¹⁴

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Naohisa Hirao, Yuichi Akahama, Yasuo Ohishi. Equations of state of iron and nickel to the pressure at the center of the Earth[J]. Matter and Radiation at Extremes, 2022, 7(3): 038403

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Paper Information

Category:

Received: Oct. 7, 2021

Accepted: Mar. 8, 2022

Published Online: Jan. 11, 2023

The Author Email: Akahama Yuichi (akahama@sci.u-hyogo.ac.jp)

DOI:10.1063/5.0074340

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Table 1. Conditions of each experimental run.

Table 1. Conditions of each experimental run.

Table 2. 2θ and d-values (do) of the observed diffraction lines with λ = 0.4176 Å for the pattern at 354 GPa, indices for these lines, and calculated d-values (dc) for the hcp-Fe and fcc-Pt phases. Calculated lattice constants are also listed.

Table 2. 2θ and d-values (do) of the observed diffraction lines with λ = 0.4176 Å for the pattern at 354 GPa, indices for these lines, and calculated d-values (dc) for the hcp-Fe and fcc-Pt phases. Calculated lattice constants are also listed.

Table 3. Parameters of the Vinet EOS obtained by a least-squares fit of the experimental compression data for hcp-Fe, together with those from previous reports. The parameters of these EOSs are the atomic volume V0, bulk modulus K0, and its pressure derivative K0′ under ambient conditions.

Table 3. Parameters of the Vinet EOS obtained by a least-squares fit of the experimental compression data for hcp-Fe, together with those from previous reports. The parameters of these EOSs are the atomic volume V0, bulk modulus K0, and its pressure derivative K0′ under ambient conditions.

Table 4. 2θ and d-values (do) of the observed diffraction lines with λ = 0.4134 Å for the pattern at 368 GPa and indices for these lines and calculated d-values (dc) for the fcc-Ni and fcc-Pt phases. Calculated lattice constants are also listed.

Table 4. 2θ and d-values (do) of the observed diffraction lines with λ = 0.4134 Å for the pattern at 368 GPa and indices for these lines and calculated d-values (dc) for the fcc-Ni and fcc-Pt phases. Calculated lattice constants are also listed.

Table 5. Parameters of the Vinet EOS obtained by a least-squares fit of the experimental compression data for Ni, together with those from previous reports. The parameters of these EOSs are the atomic volume V0, bulk modulus K0, and its pressure derivative K0′, under ambient conditions.

Table 5. Parameters of the Vinet EOS obtained by a least-squares fit of the experimental compression data for Ni, together with those from previous reports. The parameters of these EOSs are the atomic volume V0, bulk modulus K0, and its pressure derivative K0′, under ambient conditions.

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Table 2. 2θ and d-values (d_o) of the observed diffraction lines with λ = 0.4176 Å for the pattern at 354 GPa, indices for these lines, and calculated d-values (d_c) for the hcp-Fe and fcc-Pt phases. Calculated lattice constants are also listed.

Table 2. 2θ and d-values (d_o) of the observed diffraction lines with λ = 0.4176 Å for the pattern at 354 GPa, indices for these lines, and calculated d-values (d_c) for the hcp-Fe and fcc-Pt phases. Calculated lattice constants are also listed.

Table 4. 2θ and d-values (d_o) of the observed diffraction lines with λ = 0.4134 Å for the pattern at 368 GPa and indices for these lines and calculated d-values (d_c) for the fcc-Ni and fcc-Pt phases. Calculated lattice constants are also listed.

Table 4. 2θ and d-values (d_o) of the observed diffraction lines with λ = 0.4134 Å for the pattern at 368 GPa and indices for these lines and calculated d-values (d_c) for the fcc-Ni and fcc-Pt phases. Calculated lattice constants are also listed.