A wide-range multiphase equation of state for lead

FANG Jun; ZHAO Yanhong; GAO Xingyu; ZHANG Qili; WANG Yuechao; SUN Bo; LIU Haifeng; SONG Haifeng

doi:10.7498/aps.74.20250569

Abstract

We present a multi-phase equation of state (EOS) for lead (Pb, Z = 82) in wide ranges of densities and temperatures: $ {11}{.34}\;{\text{g}}/{\text{c}}{{\text{m}}^3} < \rho < 80\; {\text{g}}/{\text{c}}{{\text{m}}^3}{,} $ $ 300\;{\mathrm{K}} < T < 10\;{\mathrm{MK}}. $ The EOS model is based on a standard decomposition of the Helmholtz free energy that is regarded as a function of the specific volume and the temperature into cold term, ion-thermal term, and electronic excitation term. The cold term models both the compression and the expansion states; the ion-thermal term introduces the Debye approximation and the melting entropy; the electronic excitation term employs the Thomas-Fermi-Kirzhnits (TFK) model. The thermodynamic properties of the warm-dense lead are calculated using the extended first-principles molecular dynamics (ext-FPMD) method, with the density reaching five times that of ambient density and the temperature up to 0.4 MK. Our EOS model is used to predict the principle Hugoniot, the room-temperature isotherm, the melting curve, and the thermodynamic properties in the warm-dense region. A systematic comparison with the experimental data, the SESAME-3200 table, and the ext-FPMD calculations is made and shows that our EOS model is consistent with not only the various experimental data, but also the ext-FPMD calculations, indicating some superiority over the SESAME-3200 table in the warm-dense region. The datasets presented in this paper, including the tabular EOS consisting of internal energy and pressure at the different densities and temperatures, are openly available at https://www.doi.org/10.57760/sciencedb.j00213.00166.

Keywords:

Authors and contacts

National Key Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100094, China

Corresponding author: ZHAO Yanhong, zhao_yanhong@iapcm.ac.cn

Funds: Project supported by the Foundation of the National Key Laboratory of Computational Physics (Grant No. 6142A05230302), the National Natural Science Foundation of China (Grant Nos. U23A20537, U2230401), and the Science Challenge Project, China (Grant No. TZ2025013).

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References

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[2]	Holzapfel W B, Hartwig M, Sievers W 2001 J. Phys. Chem. Ref. Data 30 515 Google Scholar
[3]	Lyon S P, Johnson J D 1992 Los Alamos Technical Report No. LA-UR-92-3407
[4]	More R M, Warren K H, Young D A, Zimmerman G B 1988 Phys. Fluids 31 3059 Google Scholar
[5]	Liu H F, Song H F, Zhang Q L, Zhang G M, Zhao Y H 2016 Matter Radiat. Extrem. 1 123 Google Scholar
[6]	Zhao Y H, Wang L F, Zhang Q L, Zhang L, Song H Z, Gao X Y, Sun B, Liu H F, Song H F 2025 Chin. Phys. B 34 036401 Google Scholar
[7]	汤文辉, 徐彬彬, 冉宪文, 徐志宏 2017 物理学报 66 030505 Google Scholar Tang W H, Xu B B, Ran X W, Xu Z H 2017 Acta Phys. Sin. 66 030505 Google Scholar
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[9]	刘千锐 2023 博士学位论文(北京: 北京大学) Liu Q R 2023 Ph. D. Dissertation (Beijing: Peking University
[10]	Hohenberg P, Kohn W 1964 Phys. Rev. B 136 B864 Google Scholar
[11]	Kohn W, Sham L J 1965 Phys. Rev. A 140 A1133 Google Scholar
[12]	Martin R M 2004 Electronic Structure: Basic Theory and Practical Methods (Cambridge: Cambridge University Press) p119
[13]	Cytter Y, Rabani E, Neuhauser D, Baer R 2018 Phys. Rev. B 97 115207 Google Scholar
[14]	Militzer B, González-Cataldo F, Zhang S, Driver K P, Soubiran F 2021 Phys. Rev. E 103 013203 Google Scholar
[15]	Zhang S, Wang H W, Kang W, Zhang P, He X T 2016 Phys. Plasmas 23 042707 Google Scholar
[16]	Blanchet A, Clérouin J, Torrent M, Soubiran F 2022 Comput. Phys. Commun. 271 108215 Google Scholar
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[18]	Liu Q R, Chen M H 2022 Phys. Rev. B 106 125132 Google Scholar
[19]	Wilson B G, Johnson D D, Alam A 2011 High Energy Density Phys. 7 61 Google Scholar
[20]	Starrett C E 2018 Phys. Rev. E 97 053205 Google Scholar
[21]	Walsh J M, Rice M H, Mcqueen R G, Yarger F L 1957 Phys. Rev. 108 196 Google Scholar
[22]	Al'tshuler L V, Krupnikov K K, Brazhnik M I 1958 Zh. Eksp. Teor. Fiz. 34 886 (in Russian
[23]	Al'tshuler L V, Kormer S B, Bakanova A A, Trunin R F 1960 Zh. Eksp. Teor. Fiz. 38 790 (in Russian
[24]	McQueen R G, Marsh S P 1960 J. Appl. Phys. 31 1253 Google Scholar
[25]	Al'tshuler L V, Bakanova A A, Bushman A V, Dudoladov I P, Zubarev V N 1977 Zh. Eksp. Teor. Fiz. 73 1866 (in Russian
[26]	Marsh S P (Ed.) 1980 LASL Shock Hugoniot Data (Berkeley: Univ. California Press) p100
[27]	Avrorin E N, Vodolaga B K, Voloshin N P, Kuropatenko V F, Kovalenko G V, Simonenko V A, Chernodolyuk B T 1986 Pis'ma Zh. Eksp. Teor. Fiz. 43 241 (in Russian
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[34]	Smirnov N A 2021 J. Phys.: Condens. Matter 33 035402 Google Scholar
[35]	Zhang S, Morales M A 2020 AIP Conf. Proc. 2272 090004
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[37]	Strässle Th, Klotz S, Kunc K, Pomjakushin V, White J S 2014 Phys. Rev. B 90 014101
[38]	Kozyrev N V, Gordeev V V 2022 Metals 12 16
[39]	Schulte O, Holzapfel W B 1995 Phys. Rev. B 52 12636 Google Scholar
[40]	Morita K, Sobolev V, Flad M 2007 J. Nucl. Mater. 362 227 Google Scholar
[41]	Sobolev V P, Schuurmans P, Benamati G 2008 J. Nucl. Mater. 376 358 Google Scholar
[42]	Song P, Cai L C 2010 Physica B 405 1509 Google Scholar
[43]	Gao X Y, Mo Z Y, Fang J, Song H F, Wang H 2017 Comput. Phys. Commun. 211 54 Google Scholar
[44]	Zhou Y Z, Wang H, Liu Y, Gao X Y, Song H F 2018 Phys. Rev. E 97 033305 Google Scholar
[45]	Fang J, Gao X Y, Song H F 2019 Commun. Comput. Phys. 26 1196 Google Scholar
[46]	Blanchet A, Torrent M, Clérouin J 2020 Phys. Plasmas 27 122706 Google Scholar
[47]	Sjostrom T, Crockett S, Rudin S 2016 Phys. Rev. B 94 144101 Google Scholar
[48]	Kadatskiy M A 2019 High Energy Density Phys. 33 100700 Google Scholar
[49]	Liu X, Zhang X H, Gao C, Zhang S, Wang C, Li D F, Zhang P, Kang W, Zhang W Y, He X T 2021 Phys. Rev. B 103 174111 Google Scholar
[50]	Schwarz K 2003 J. Solid State Chem. 176 319 Google Scholar
[51]	Benedict L X, Driver K P, Hamel S, Militzer B, Qi T, Correa A A, Saul A, Schwegler E 2014 Phys. Rev. B 89 224109 Google Scholar
[52]	Mattsson A E 2012 Sandia Technical Report No. SAND2012-7389
[53]	Eliezer S, Ricci R A (Eds.) 1991 High Pressure Equations of State: Theory and Applications (Amsterdam: North Holland) p249
[54]	Johnson J D 1991 Int. J. High Press. Res. 6 277 Google Scholar
[55]	Benedict L X, Ogitsu T, Trave A, Wu C J, Sterne P A, Schwegler E 2009 Phys. Rev. B 79 064106 Google Scholar
[56]	Wilson B, Sonnad V, Sterne P, Isaace W 2006 J. Quant. Spectrosc. Ra. 99 658 Google Scholar

Cited By

图 1 面心立方铅的冷压线(压强-体积) 22个价电子的PAW势计算结果与WIEN2k全势结果的对比

Figure 1. Cold pressure results of face-centered-cubic lead calculated by CESSP with the 22 electron PAW potential, compared with full-potential results from WIEN2k.

DownLoad: Full-Size Img PowerPoint

图 2 冲击雨贡纽线(压强-压缩比)理论计算值, 以及与SESAME-3200和实验数据的对比

Figure 2. Principle Hugoniot predicted by the theoretical model in this work, compared with the SESAME-3200 table and the experimental data.

DownLoad: Full-Size Img PowerPoint

图 3 300 K等温压缩线(体积比-压强)理论计算值与SESAME-3200^[3]和实验数据^[31]的对比

Figure 3. Room-temperature isotherm predicted by the theoretical model in this work, compared with the SESAME-3200 table^[3] and the experimental data^[31].

DownLoad: Full-Size Img PowerPoint

图 4 熔化线及雨贡纽线(温度-压强)理论计算值与SESAME-3200和实验数据的对比

Figure 4. Melting curve predicted by the theoretical model in this work, compared with the experimental data. The T-P Hugoniot curves from our model and the SESAME-3200 table are also given to illustrate the difference.

DownLoad: Full-Size Img PowerPoint

图 5 温稠密过渡区物性(压强-压缩比)理论模型与SESAME-3200^[3]和第一性原理ext-FPMD模拟的对比

Figure 5. Thermodynamic properties of the warm-dense lead predicted by the theoretical model in this work, compared with the SESAME-3200 table^[3] and the ext-FPMD calculations.

DownLoad: Full-Size Img PowerPoint

表 1 Ext-FP方法与标准DFT方法下铅的计算结果比较

Table 1. Comparison of results from the ext-FP method and the DFT method.

电子温度/MK	Ext-FP方法			标准DFT方法
电子温度/MK	N_cut	P/GPa	ΔP/%	N_occ	P/GPa
0.1	40	134.654	–0.01	100	134.670
0.2	60	414.783	–0.03	200	414.890
0.3	80	791.144	–0.23	350	792.977

DownLoad: CSV

表 2 不同温度密度条件下最低价态(铅的5s态)的占据数

Table 2. Occupation numbers of the lowest energy valence state (5s state of lead) at the different temperatures and densities.

	ρ₀	2ρ₀	3ρ₀	4ρ₀	5ρ₀
T = 0.05 MK	1.00000	1.00000	1.00000	1.00000	1.00000
T = 0.1 MK	1.00000	1.00000	1.00000	1.00000	1.00000
T = 0.2 MK	0.99973	0.99977	0.99981	0.99984	0.99988
T = 0.3 MK	0.99492	0.99576	0.99640	0.99701	0.99746
T = 0.4 MK	0.97704	0.98074	0.98364	0.98612	0.98821
T = 0.5 MK	0.94243	0.95155	0.95881	0.96504	0.96917

DownLoad: CSV

[1]	徐锡申, 张万箱 1986 实用物态方程理论导引(北京: 科学出版社)第1, 191页 Xu X S, Zhang W X 1986 Introduction to Practical Equation of States (Beijing: Scientific Press) pp1, 191
[2]	Holzapfel W B, Hartwig M, Sievers W 2001 J. Phys. Chem. Ref. Data 30 515 Google Scholar
[3]	Lyon S P, Johnson J D 1992 Los Alamos Technical Report No. LA-UR-92-3407
[4]	More R M, Warren K H, Young D A, Zimmerman G B 1988 Phys. Fluids 31 3059 Google Scholar
[5]	Liu H F, Song H F, Zhang Q L, Zhang G M, Zhao Y H 2016 Matter Radiat. Extrem. 1 123 Google Scholar
[6]	Zhao Y H, Wang L F, Zhang Q L, Zhang L, Song H Z, Gao X Y, Sun B, Liu H F, Song H F 2025 Chin. Phys. B 34 036401 Google Scholar
[7]	汤文辉, 徐彬彬, 冉宪文, 徐志宏 2017 物理学报 66 030505 Google Scholar Tang W H, Xu B B, Ran X W, Xu Z H 2017 Acta Phys. Sin. 66 030505 Google Scholar
[8]	Graziani F, Desjarlais M P, Redmer R, Trickey S B (Eds.) 2014 Frontiers and Challenges in Warm Dense Matter (Cham: Springer International Publishing) p123
[9]	刘千锐 2023 博士学位论文(北京: 北京大学) Liu Q R 2023 Ph. D. Dissertation (Beijing: Peking University
[10]	Hohenberg P, Kohn W 1964 Phys. Rev. B 136 B864 Google Scholar
[11]	Kohn W, Sham L J 1965 Phys. Rev. A 140 A1133 Google Scholar
[12]	Martin R M 2004 Electronic Structure: Basic Theory and Practical Methods (Cambridge: Cambridge University Press) p119
[13]	Cytter Y, Rabani E, Neuhauser D, Baer R 2018 Phys. Rev. B 97 115207 Google Scholar
[14]	Militzer B, González-Cataldo F, Zhang S, Driver K P, Soubiran F 2021 Phys. Rev. E 103 013203 Google Scholar
[15]	Zhang S, Wang H W, Kang W, Zhang P, He X T 2016 Phys. Plasmas 23 042707 Google Scholar
[16]	Blanchet A, Clérouin J, Torrent M, Soubiran F 2022 Comput. Phys. Commun. 271 108215 Google Scholar
[17]	White A J, Collins L A 2020 Phys. Rev. Lett. 125 055002 Google Scholar
[18]	Liu Q R, Chen M H 2022 Phys. Rev. B 106 125132 Google Scholar
[19]	Wilson B G, Johnson D D, Alam A 2011 High Energy Density Phys. 7 61 Google Scholar
[20]	Starrett C E 2018 Phys. Rev. E 97 053205 Google Scholar
[21]	Walsh J M, Rice M H, Mcqueen R G, Yarger F L 1957 Phys. Rev. 108 196 Google Scholar
[22]	Al'tshuler L V, Krupnikov K K, Brazhnik M I 1958 Zh. Eksp. Teor. Fiz. 34 886 (in Russian
[23]	Al'tshuler L V, Kormer S B, Bakanova A A, Trunin R F 1960 Zh. Eksp. Teor. Fiz. 38 790 (in Russian
[24]	McQueen R G, Marsh S P 1960 J. Appl. Phys. 31 1253 Google Scholar
[25]	Al'tshuler L V, Bakanova A A, Bushman A V, Dudoladov I P, Zubarev V N 1977 Zh. Eksp. Teor. Fiz. 73 1866 (in Russian
[26]	Marsh S P (Ed.) 1980 LASL Shock Hugoniot Data (Berkeley: Univ. California Press) p100
[27]	Avrorin E N, Vodolaga B K, Voloshin N P, Kuropatenko V F, Kovalenko G V, Simonenko V A, Chernodolyuk B T 1986 Pis'ma Zh. Eksp. Teor. Fiz. 43 241 (in Russian
[28]	Mitchell A C, Nellis W J, Moriarty J A, Heinle R A, Holmes N C, Tipton R E, Repp G W 1991 J. Appl. Phys. 69 2981 Google Scholar
[29]	Trunin R F, Il'kaeva L A, Podurets M A, Popov L V, Pechenkin B V, Prokhorov L V, Sevast'yanov A G, Khrustalev V V 1994 Teplofiz. Vys. Temp. 32 692 (in Russian
[30]	Trunin R F 1994 Usp. Fiz. Nauk 164 1215 (in Russian
[31]	Mao H K, Wu Y, Shu J F 1990 Solid State Commun. 74 1027 Google Scholar
[32]	Partouche-Sebban D, Pélissier J L 2005 J. Appl. Phys. 97 043521 Google Scholar
[33]	Dewaele A, Mezouar M, Guignot N, Loubeyre P 2007 Phys. Rev. B 76 144106 Google Scholar
[34]	Smirnov N A 2021 J. Phys.: Condens. Matter 33 035402 Google Scholar
[35]	Zhang S, Morales M A 2020 AIP Conf. Proc. 2272 090004
[36]	Yang X, Zeng X G, Chen H Y, Wang Y T, He L, Wang F 2019 J. Alloy. Comp. 808 151702 Google Scholar
[37]	Strässle Th, Klotz S, Kunc K, Pomjakushin V, White J S 2014 Phys. Rev. B 90 014101
[38]	Kozyrev N V, Gordeev V V 2022 Metals 12 16
[39]	Schulte O, Holzapfel W B 1995 Phys. Rev. B 52 12636 Google Scholar
[40]	Morita K, Sobolev V, Flad M 2007 J. Nucl. Mater. 362 227 Google Scholar
[41]	Sobolev V P, Schuurmans P, Benamati G 2008 J. Nucl. Mater. 376 358 Google Scholar
[42]	Song P, Cai L C 2010 Physica B 405 1509 Google Scholar
[43]	Gao X Y, Mo Z Y, Fang J, Song H F, Wang H 2017 Comput. Phys. Commun. 211 54 Google Scholar
[44]	Zhou Y Z, Wang H, Liu Y, Gao X Y, Song H F 2018 Phys. Rev. E 97 033305 Google Scholar
[45]	Fang J, Gao X Y, Song H F 2019 Commun. Comput. Phys. 26 1196 Google Scholar
[46]	Blanchet A, Torrent M, Clérouin J 2020 Phys. Plasmas 27 122706 Google Scholar
[47]	Sjostrom T, Crockett S, Rudin S 2016 Phys. Rev. B 94 144101 Google Scholar
[48]	Kadatskiy M A 2019 High Energy Density Phys. 33 100700 Google Scholar
[49]	Liu X, Zhang X H, Gao C, Zhang S, Wang C, Li D F, Zhang P, Kang W, Zhang W Y, He X T 2021 Phys. Rev. B 103 174111 Google Scholar
[50]	Schwarz K 2003 J. Solid State Chem. 176 319 Google Scholar
[51]	Benedict L X, Driver K P, Hamel S, Militzer B, Qi T, Correa A A, Saul A, Schwegler E 2014 Phys. Rev. B 89 224109 Google Scholar
[52]	Mattsson A E 2012 Sandia Technical Report No. SAND2012-7389
[53]	Eliezer S, Ricci R A (Eds.) 1991 High Pressure Equations of State: Theory and Applications (Amsterdam: North Holland) p249
[54]	Johnson J D 1991 Int. J. High Press. Res. 6 277 Google Scholar
[55]	Benedict L X, Ogitsu T, Trave A, Wu C J, Sterne P A, Schwegler E 2009 Phys. Rev. B 79 064106 Google Scholar
[56]	Wilson B, Sonnad V, Sterne P, Isaace W 2006 J. Quant. Spectrosc. Ra. 99 658 Google Scholar

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