锡的多相物态方程的第一性原理计算研究

陈凯乐; 王越超; 徐远骥; 刘瑜; 咸家伟; 王丽芳; 简单; 刘海风; 宋海峰

doi:10.11858/gywlxb.20251054

锡的多相物态方程的第一性原理计算研究

doi: 10.11858/gywlxb.20251054

陈凯乐^1,,
王越超¹,
徐远骥^2,*, ,,
刘瑜^1,*, ,,
咸家伟¹,
王丽芳¹,
简单³,
刘海风¹,
宋海峰¹

1.
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　100088
2.
北京科技大学应用物理研究所, 北京　100083
3.
中国工程物理研究院材料研究所表面物理与化学重点实验室, 四川绵阳　621908

基金项目: 国家重点研发计划（2021YFB3501503）；国家自然科学基金（U23A20537）；国家重大科研仪器研制专项（62327804）；计算物理全国重点实验室基金

详细信息

作者简介:
陈凯乐（2000－），男，硕士，主要从事材料物态方程研究. E-mail：chenkaile22@gscaep.ac.cn

通讯作者:
徐远骥（1990－），男，博士，讲师，主要从事高压强关联模拟研究. E-mail：yuanjixu@ustb.edu.cn

刘　瑜（1985－），男，博士，副研究员，主要从事强关联材料平衡态和动力学物性的理论计算与数值模拟研究. E-mail：liu_yu@iapcm.ac.cn

中图分类号: O521.2
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出版历程
- 收稿日期: 2025-03-20
- 修回日期: 2025-04-08
- 录用日期: 2025-09-08
- 网络出版日期: 2025-04-23

First-Principles Study on the Multiphase Equation of State of Tin

CHEN Kaile^1
,,
WANG Yuechao¹,
XU Yuanji^{2,*
, ,},
LIU Yu^{1,*
, ,},
XIAN Jiawei¹,
WANG Lifang¹,
JIAN Dan³,
LIU Haifeng¹,
SONG Haifeng¹

1.
National Key Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
2.
Institute for Applied Physics, University of Science and Technology Beijing, Beijing 100083, China
3.
Science and Technology on Surface Physics and Chemistry Laboratory, Institute of Materials, China Academy of Engineering Physics, Mianyang 621908, Sichuan, China

摘要

摘要: 金属锡是高压物理领域研究的热点，也是国防科技领域关注的重要材料。锡具有丰富的物相，无论是基础研究，还是工业应用，锡的多相物态方程和相界都至关重要。采用密度泛函理论结合平均场势方法，系统研究了锡的高温高压多相物态方程、相界、弹性模量、声速和Hugoniot线等，获得了高温高压下锡的多相物态方程，计算得到的β-γ相界、β-Sn的常压声速与实验结果吻合较好。此外，进一步研究了不同密度泛函对锡的高温高压物态方程的影响。结果表明：通过局域密度梯度近似（local density approximation，LDA）和PBEsol泛函得到的主Hugoniot线及常压弹性模量与实验结果具有较好的一致性；与其他泛函相比，通过SCAN（strongly constrained and appropriately normed）泛函描述的相界的偏差较大，但描述的β-Sn的常压声速与实验结果更接近。
- 多相物态方程 /
- 弹性模量 /
- 第一性原理计算 /
- 平均场势
Abstract: Metallic tin is a focal point in high-pressure physics research and a critical material of strategic importance in defense-related technologies. Due to the rich physical phases of tin, it is crucial to study the multiphase equation of state and phase boundaries of tin, whether in basic research or industrial applications. This work systematically investigates the high-temperature and high-pressure multiphase equation of state (EOS), phase boundaries, elastic modulus, sound velocities, and Hugoniot curves of tin using density functional theory (DFT) combined with the mean-field potential (MFP) method. The results not only provide the multiphase EOS of tin under extreme conditions but also demonstrate good agreement with experimental data for the β-γ phase boundary and ambient-pressure sound velocities of β-Sn. Furthermore, this study evaluates the effects of different density functionals (LDA, PBEsol, and SCAN) on the high-pressure EOS. The LDA and PBEsol functionals show superior consistency with experimental Hugoniot curves and ambient-pressure elastic moduli, while the SCAN functional exhibits larger deviations in phase boundary predictions but achieves closer agreement with experimental ambient-pressure sound velocities for β-Sn.
- multiphase equation of state /
- elastic modulus /
- first-principles calculations /
- mean-field potential

HTML全文

图 1 采用4种泛函计算得到的锡的4个相的E-V曲线

Figure 1. Calculated E-V curves of four phases of Sn using four exchange-correlation functionals

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图 2 采用PBE泛函分别计算4个固相得到的不同压力下的零温态密度

Figure 2. Calculated density of states at 0 K of four ground state at different pressures using PBE exchange-correlation functional

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图 3 采用4种交换-关联泛函计算得到的锡的不同相相对γ相的焓差随压力的变化曲线

Figure 3. Calculated ΔH-p curves of different phases (relative to γ phase) of Sn using four exchange-correlation functionals

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图 4 采用4种泛函计算得到零温下锡的4个相的p-V/V₀曲线（V₀为0 GPa下α-Sn的平衡体积）

Figure 4. Calculated p-V/V₀ curves of four different phases of Sn using four exchange-correlation functionals at 0 K(V₀ means equilibrium volume of α-Sn at 0 GPa)

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图 5 采用PBE泛函计算得到的锡的4个相的c/a随压强的变化

Figure 5. Calculated c/a-p curves of four structure of Sn calculated with PBE exchange-correlation functional

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图 6 采用4种泛函计算的300 K下Sn的4个相的p-V曲线与实验数据^{[3, 6–8, 67–68]}的对比

Figure 6. Comparision of the calculated p-V curves at 300 K of of four structure of Sn using four exchange-correlation functionals with experimental data^{[3, 6–8, 67–68]}

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图 7 采用4种泛函计算得到的Sn的β-γ-液相相图和Hugoniot线与实验及理论数据^{[3, 65–66, 69]}的对比

Figure 7. Comparision of calculated β-γ-liquid phase diagram and primary Hugoniot curves of tin using four exchange-correlation functionals with experimental and theoretical data^{[3, 65–66, 69]}

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图 8 采用4种泛函计算得到的β-Sn和γ-Sn在主Hugoniot线上的粒子速度（u）与冲击波速度（D）的关系与实验数据^[71]的对比

Figure 8. Comparision of calculated D-u curves of β-Sn and γ-Sn on the primary Hugoniot curve using four exchange-correlation functionals with the experimental data^[71]

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图 9 采用4种泛函计算的0 GPa下α-Sn的体弹模量与实验数据^{[18, 75–76]}的对比

Figure 9. Comparision of bulk modulus of α-Sn at 0 GPa calculated by four exchange-correlation functionals with the experimental data^{[18, 75–76]}

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图 10 采用4种泛函计算的0 GPa下β-Sn的体弹模量与实验数据^[77–85]的对比

Figure 10. Comparision of calculated bulk modulus of β-Sn at 0 GPa using four exchange-correlation functionals with the experimental data^[77–85]

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图 11 采用4种泛函计算的1 000 K下γ-Sn和δ-Sn的弹性模量

Figure 11. Calculated bulk modulus of γ-Sn and δ-Sn at 1 000 K using four exchange-correlation functionals

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图 12 采用4种泛函计算的0 GPa下β-Sn的剪切模量（对比参考数据为Rehn等^[66]根据Kammer等^[86]得到的弹性常数计算得到）

Figure 12. Calculated shear modulus of β-Sn at 0 GPa using four exchange-correlation functionals (The data used for comparison were from Rehn, et al.^[66], calculated based on the elastic constants obtained from Kammer, et al.^[86])

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图 13 采用4种泛函计算得到的0 GPa下α-Sn的杨氏模量和泊松比

Figure 13. Calculated Young’s modulus and Possion’s ratio of α-Sn using four exchange-correlation functionals

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图 14 采用4种泛函结合能量-应变法计算的0 GPa下β-Sn的声速（对比实验数据来自文献[14]）

Figure 14. Calculated sound velocity of β-Sn at 0 GPa using four exchange-correlation functionals combined with energy-strain method (The experimental data used for comparison were from the literature [14])

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图 15 采用4种泛函结合能量-应变法计算的沿主Hugoniot线的β-Sn和γ-Sn的声速（对比实验数据来自文献[13, 71, 87]）

Figure 15. Calculated sound velocity of β-Sn and γ-Sn along primary Hugoniot curves using four exchange-correlation functionals combined with energy-strain method (The experimental data used for comparison were from the literature [13, 71, 87])

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表 1 采用不同交换-关联泛函计算得到的零温相变压力

Table 1. Pressures of zero temperature phase transition calculated by different exchange-correlation functional

Method	Phase transition pressure/GPa
Method	α-β	β-γ	γ-δ	δ-ε
LDA^[24]	0.80	23.5		>90
LDA^[25]	0.95	15.9	42.9	163
GGA^[26]	0.70	12.0	45.0	160
SCAN^[27]		16.2

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表 2 基于冷能计算BM方程拟合的结构参数、平衡体积和体积模量

Table 2. Lattice parameters, equilibrium volumes, and bulk modulus from BM equation fitting

Phase	Method	a₀/Å	c₀/a₀	V₀/(Å³/atom)	E₀/(eV/atom)	B₀/GPa	$B'_0 $
	Exp.	6.483^[60]	1.000^[60]	34.06^[60]	3.140^[61]	42.6^[62]
	LDA	6.476	1.000	33.94	3.958	45.0	4.9
α-Sn	PBE	6.654	1.000	36.83	3.152	35.9	4.8
	PBEsol	6.539	1.000	34.95	3.509	41.7	4.8
	SCAN	6.549	1.000	35.11	3.414	41.5	4.6
	Exp.	5.831^[60]	0.546^[60]	27.07^[60]	3.100^[16]	57.0^[62]
	LDA	5.778	0.543	26.19	3.977	60.1	4.8
β-Sn	PBE	5.929	0.544	28.40	3.112	47.6	4.8
	PBEsol	5.827	0.543	26.87	3.536	56.1	4.8
	SCAN	5.868	0.544	27.45	3.335	52.7	4.8
	LDA	3.920	0.854	25.63	3.938	57.0	4.9
	PBE	4.041	0.846	27.80	3.091	45.5	4.9
γ-Sn	PBEsol	3.945	0.859	26.26	3.517	54.3	4.8
	SCAN	3.956	0.866	26.99	3.226	47.3	5.2
	Exp.					76.4^[9]	4.04^[9]
	LDA	3.708	1.000	25.48	3.928	59.4	4.8
δ-Sn	PBE	3.809	1.000	27.63	3.081	47.3	4.8
	PBEsol	3.738	1.000	26.12	3.507	55.8	4.8
	SCAN	3.770	1.000	26.78	3.215	51.4	4.8

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表 3 采用4种交换-关联泛函计算得到的Sn的4个相分别稳定存在的压强范围

Table 3. Calculated stable pressure ranges of the four phases of tin using four different exchange-correlation functionals

Method	Pressure/GPa
Method	α	β	γ	δ
Exp.	0−0.6^[65]	0.6^[65]−9.8^[64]	9.8^[64]−19.0^[63]、50.0^[13]	19^[63]、50^[13]−146^[66]
LDA	Negative	0−10.7	10.7−31.3	31.3−50.0
PBE	0−0.8	0.8−5.3	5.3−21.1	21.1−50.0
PBEsol	Negative	0−5.2	5.2−50.0
SCAN	0−7.1	7.1−30.9		30.9−50.0

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