金属铅的宽区多相物态方程

方俊; 赵艳红; 高兴誉; 张其黎; 王越超; 孙博; 刘海风; 宋海峰

doi:10.7498/aps.74.20250569

摘要

本文从Helmholtz自由能出发构建了铅的宽区多相物态方程, 覆盖从常温到10 MK、从常压到10⁷ GPa的温压范围, 计算了冲击雨贡纽线、300 K等温线、熔化线及温稠密过渡区热力学物性, 并与实验值、铅已有的宽区物态方程数据库SESAME-3200以及第一性原理模拟结果进行了对比分析. 一方面, 本文的模型能较好地再现各类实验数据; 另一方面, 在温稠密过渡区, 本文的模型获得了扩展的第一性原理分子动力学模拟结果的验证, 相比SESAME-3200更符合第一性原理的模拟结果. 本文数据集可在https://www.doi.org/10.57760/sciencedb.j00213.00166中访问获取.

关键词:

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:

作者及机构信息

北京应用物理与计算数学研究所, 计算物理全国重点实验室, 北京　100094

通信作者: 赵艳红, zhao_yanhong@iapcm.ac.cn

基金项目: 计算物理全国重点实验室基金(批准号: 6142A05230302)、国家自然科学基金(批准号: U23A20537, U2230401)和科学挑战专题(批准号: TZ2025013)资助的课题.

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).

文章全文

参考文献

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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 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

施引文献

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

Fig. 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.

下载: 全尺寸图片幻灯片

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

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

下载: 全尺寸图片幻灯片

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

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

下载: 全尺寸图片幻灯片

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

Fig. 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.

下载: 全尺寸图片幻灯片

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

Fig. 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.

下载: 全尺寸图片幻灯片

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

下载: 导出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

下载: 导出CSV

map

[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 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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