Elastic properties and their pressure dependence of rare earth metals

HUANG Chengning; LIU Beilei; WANG Yuechao; GAO Xingyu; XIAN Jiawei; LIU Haifeng; SONG Haifeng

doi:10.7498/aps.74.20250574

Abstract

Rare earth metals are of significant importance in engineering and technological applications, and their unique f-electron-related behaviors have attracted widespread interest in condensed matter physics. In this work, we investigate the elastic properties of rare earth metals ranging from Ce to Yb by combining first-principles calculations with systematic data compilation. Taking Ce and Yb as representative cases, we investigate the evolution of their elastic properties under high-pressure conditions (0–15 GPa), and we systematically compare the simulation performances of different f-electron treatment approaches. The results indicate a significant difference in ductility between light and heavy rare earth metals under ambient pressure. Under pressure, the elastic properties of Ce and Yb undergo marked changes in phase transitions. Specifically, the B/G ratio, a key indicator of ductility, decreases from about 2.0 in light lanthanides to around 1.5 in heavy lanthanides, crossing the critical threshold of 1.75. Notably, during the fcc iso-structural phase transition in Ce and the fcc-bcc phase transition in Yb, a significant brittle-ductile transition is observed. These transitions are closely related to the bonding characteristics modulated by atomic number or pressure condition. For instance, as the atomic number increases, the Cauchy pressure (C₁₂–C₄₄) decreases with the variation of s and d valence electrons, indicating an enhanced covalent bonding tendency. In addition, this study reveals that simulating f-electrons as core electrons can adequately describe the elastic properties and trends of rare earth metals under ambient pressure. However, when modeling high-pressure structural phase transitions and their related elastic evolution, the method of treating f-electrons as valence electrons and performing electron correlation correction shows better accuracy. The datasets presented in this paper are openly available at https://doi.org/10.57760/sciencedb.j00213.00150.

Keywords:

Authors and contacts

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

Corresponding author: WANG Yuechao, yuechao_wang@126.com ; SONG Haifeng, song_haifeng@iapcm.ac.cn

Funds: Project supported by the National Key Research and Development Program of China (Grant No. 2021YFB3501503) and the National Natural Science Foundation of China (Grant No. U2230401).

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References

[1]	McMahan A K, Huscroft C, Scalettar R T, Pollock E L 1998 J. Comput. Aid Mol. Des. 5 131 Google Scholar
[2]	Grimvall G 1999 Thermophysical Properties of Materials (Amsterdam: Elsevier
[3]	Jia T T, Chen G, Zhang Y S 2017 Phys. Rev. B 95 155206 Google Scholar
[4]	Li W G, Kou H B, Zhang X Y, Ma J Z, Li Y, Geng P J, Wu X Z, Chen L M, Fang D N 2019 Mech. Mater. 139 103194 Google Scholar
[5]	Pugh S 1954 Philos. Mag. 45 823 Google Scholar
[6]	Xu L, Li X, He Q, Yang J, Sun S L, Li J, Hu J B, Wu Q 2025 J. Appl. Phys. 137 015902 Google Scholar
[7]	谢明宇, 李法新 2022 力学进展 52 33 Google Scholar Xie M Y, Li F X 2022 Adv. Mech. 52 33 Google Scholar
[8]	Kittel C 1996 Introduction to Solid State Physics (New York: Wiley
[9]	Boguslavskii Y Y, Goncharov V A, Il'ina G G 1989 J. Less-Common Met. 147 249 Google Scholar
[10]	Jeong I K, Darling T W, Graf M J, Proffen T, Heffner R H, Lee Y, Vogt T, Jorgensen J D 2004 Phys. Rev. Lett. 92 105702 Google Scholar
[11]	Decremps F, Antonangeli D, Amadon B, Schmerber G 2009 Phys. Rev. B 80 132103 Google Scholar
[12]	Wang Z G, Bi Y, Xu L, Liu L 2014 Mater. Res. 1 026501 Google Scholar
[13]	Lipp M J, Jackson D, Cynn H, Aracne C, Evans W J, Mcmahan A K 2008 Phys. Rev. Lett. 101 165703 Google Scholar
[14]	Lipp M J, Jenei Z, Cynn H, Kono Y, Park C, Kenney-Benson C, Evans W J 2017 Nat. Commun. 8 1198 Google Scholar
[15]	Xu L, Wang Z G, Li Z G, Li X H, Yao S L, Li J, Zhou X M, Yu Y Y, Hu J B, Wu Q 2021 Appl. Phys. Lett. 118 074102 Google Scholar
[16]	Söderlind P, Turchi P E A, Landa A, Lordi V 2014 J. Phys. : Condens. Matter. 26 416001 Google Scholar
[17]	Hill R 1952 Proc. Phys. Soc. A 65 349 Google Scholar
[18]	Yang Z, Xian J W, Gao X Y, Tian F Y, Song H F 2024 J. Chem. Phys. 161 194101 Google Scholar
[19]	Blöchl P E 1994 Phys. Rev. B Condens Matter. 50 17953 Google Scholar
[20]	Perdew J P, Burke K, Ernzerhof M 1996 Phys. Rev. Lett. 77 3865 Google Scholar
[21]	Delin A, Fast L, Johansson B, Eriksson O, Wills J M 1998 Phys. Rev. B 58 4345 Google Scholar
[22]	Ouyang Y F, Tao X M, Zeng F J, Chen H M, Du Y, Feng Y P, He Y 2009 Physica B 404 2299 Google Scholar
[23]	Seitz F, Turnbull D 1964 Solid State Physics: Advance in Research and Applications (New York: Academic press
[24]	Material Properties https://material-properties.org [2025-03-19]
[25]	Stassis C, Gould T, McMasters O D, Gschneidner K A, Nicklow R M 1979 Phys. Rev. B 19 5746 Google Scholar
[26]	Pettifor D G 1992 Mater. Sci. Tech. 8 345 Google Scholar
[27]	Eberhart M E, Jones T E 2012 Phys. Rev. B 86 134106 Google Scholar
[28]	Kim S, Chen J, Cheng T, et al. PubChem 2025 update. 2025 Nucleic Acids Res. 53 D1516. https://pubchem.ncbi.nlm.nih.gov/periodic-table/melting-point/ [2025-03-19]
[29]	Nikolaev A V, Tsvyashchenko A V 2012 Phys. -Usp. 55 657 Google Scholar
[30]	Voronov F F, Goncharova V A, Stal'gorova O V 1979 J. Exp. Theor. Phys. 49 687
[31]	Chen J L, Kaltsoyannis N 2022 J. Phys. Chem. C 126 11426 Google Scholar
[32]	Eryigit S, Parlak C, Eryigit R 2022 J. Phys. : Condens. Matter 34 295402 Google Scholar
[33]	Tran F, Karsai F, Blaha P 2014 Phys. Rev. B 89 155106 Google Scholar
[34]	Chesnut G N, Vohra Y K 1999 Phys. Rev. Lett. 82 1712 Google Scholar
[35]	Morée J B, Amadon B 2018 Phys. Rev. B 98 205101 Google Scholar
[36]	Satikunvar D D, Bhatt N K, Thakore B Y 2021 J. Appl. Phys. 129 035107 Google Scholar

Cited By

图 1 稀土金属Ce -Yb的模量(实验数据取自文献[23,24], 除本文计算外, 理论数据取自文献[16,21,22])　(a) 体模量; (b) 剪切模量

Figure 1. Elastic modulus of rare earth metals (Ce -Yb) (Experimental data are taken from Refs. [23,24], and theoretical data are taken from Refs. [16,21,22], except for the computational results from this work): (a) Bulk modulus; (b) shear modulus.

DownLoad: Full-Size Img PowerPoint

图 2 (a) 稀土金属体模量和剪切模量的比值B/G, 并与已有的实验^[23,24]进行比较; (b) f-core方法计算获得s, d价电子填充数; (c) 柯西压力C₁₂ – C₄₄随原子序数变化; (d) 稀土金属的熔点^[28]

Figure 2. (a) Ratio of bulk modulus to shear modulus (B/G ) for rare earth metals. Comparisons are made with the experimental results^[23,24]. (b) The s- and d-valence electron occupation numbers calculated using the f-core method. (c) The variation of Cauchy pressure (C₁₂ – C₄₄) with the atomic number. (d) Melting points for rare-earth metals^[28].

DownLoad: Full-Size Img PowerPoint

图 3 (a) Ce体模量随压强的变化; (b) 剪切模量随压强的变化; (c) 纵波声速随压强的变化; (d) 横波声速随压强的变化. 并与已有的实验结果^[11–13,30]进行比较, 图中虚线是实验给出的Ce的γ-α相变压力点

Figure 3. (a) Bulk modulus B for Ce as a function of pressure. (b) Shear modulus G as a function of pressure. (c) Longitudinal wave velocity C_Las a function of pressure. (d) Transverse wave velocity C_T as a function of pressure. Comparisons are made with existing experimental results^[11-13,30]. The dashed line in the figure marks the experimentally reported γ-α phase transition pressure.

DownLoad: Full-Size Img PowerPoint

图 4 (a) B/G随压强的变化关系, 并与已有的实验结果^[11–13,30]进行比较; (b) s价电子数随压强的变化; (c) d, f价电子数随压强的变化; (d) 柯西压力C₁₂ – C₄₄随压强的变化

Figure 4. (a) B/G ratio as a function of pressure. Comparisons with existing experimental results^[11–13,30] are provided. (b) The s-valence electron occupation as a function of pressure. (c) The d, f-valence electron occupation as a function of pressure. (d) Cauchy pressure (C₁₂ – C₄₄) as a function of pressure.

DownLoad: Full-Size Img PowerPoint

图 5 Yb的fcc-bcc相焓差随压强的变化

Figure 5. Enthalpy difference between fcc and bcc phase for Yb as a function of pressure.

DownLoad: Full-Size Img PowerPoint

图 6 (a) Yb的体模量随压强的变化, 小图为相变压力点附近体模量随压强的变化; (b) 剪切模量随压强的变化; (c) 纵波声速随压强的变化; (d) 横波声速随压强的变化. 并与已有的实验结果^[9]进行比较

Figure 6. (a) Bulk modulus for Yb as a function of pressure, with the inset showing the bulk modulus variation near the phase transition pressure. (b) Shear modulus as a function of pressure. (c) Longitudinal wave velocity C_L as a function of pressure. (d) Transverse wave velocity C_T as a function of pressure. Comparisons with existing experimental results^[9] are provided.

DownLoad: Full-Size Img PowerPoint

图 7 (a) 对于Yb, B/G随压强的变化, 并与已有的实验结果^[9]进行比较; (b) s价电子数随压强变化; (c) d价电子数随压强变化; (d) 柯西压力C₁₂–C₄₄随压强的变化

Figure 7. (a) B/G ratio for Yb as a function of pressure. Comparisons with existing experimental results^[9] are provided. (b) The s-valence electron occupation as a function of pressure. (c) The d-valence electron occupation as a function of pressure. (d) Cauchy pressure (C₁₂–C₄₄) as a function of pressure.

DownLoad: Full-Size Img PowerPoint

表 1 实验与理论计算的镧系元素Ce-Yb弹性性质

Table 1. Calculated elastic constants, bulk modulus (B), shear modulus (G) and B/G for rare earth Ce-Yb.

	Method	B/GPa	G/GPa	C₁₁	C₁₂	C₄₄	C₁₃	C₃₃	B/G	Ref.
Ce	f-band	39.80	33.24	63.20	28.10	50.90	—	—	1.19	This work
	GGA+OP, f-band	40.89	—	—	—	—	—	—	—	[16]
	PBE, f-core	29.23	13.72	39.29	24.21	20.45	—	—	2.13	This work
	PBE, f-core	34.62	—	—	—	—	—	—	—	[21]
	GGA, f-core	30.21	15.86	43.46	23.59	21.71	—	—	1.90	[22]
γ -Ce	PBE+U	27.20	15.76	40.59	20.52	21.33	—	—	1.73	This work
γ -Ce	Expt.	14.83	12.86	24.1	10.2	19.4	—	—	1.15	[25]
Pr	PBE, f-band	20.64	18.75	39.30	11.31	22.80	—	—	1.10	This work
	GGA+OP, f-band	20.88	—	—	—	—	—	—	—	[16]
	PBE, f-core	31.66	16.45	44.80	25.10	23.20	—	—	1.92	This work
	GGA, f-core	36.65	—	—	—	—	—	—	—	[21]
	GGA, f-core	34.57	18.83	60.77	25.36	17.4	17.88	67.34	1.83	[22]
	PBE+U	24.27	11.58	35.20	18.80	14.60	—	—	2.09	This work
	Expt.	28.80	14.80	—	—	—	—	—	1.95	[24]
Nd	PBE, f-band	18.9	14.95	30.90	12.90	21.00	—	—	1.26	This work
	GGA+OP, f-band	20.98	—	—	—	—	—	—	—	[16]
	PBE, f-core	33.9	18.55	49.10	26.29	25.70	—	—	1.83	This work
	GGA, f-core	39.12	—	—	—	—	—	—	—	[21]
	GGA, f-core	36.12	20.77	65.24	25.88	19.11	17.77	71.77	1.74	[22]
	PBE+U	28.57	25.65	46.90	19.41	38.90	—	—	1.11	This work
	Expt.	31.8	16.3	—	—	—	—	—	1.95	[24]
Pm	PBE, f-band	20.67	14.81	31.60	15.20	22.00	—	—	1.2	This work
	GGA+OP, f-band	19.92	—	—	—	—	—	—	—	[16]
	PBE, f-core	35.67	20.37	52.40	27.31	28.20	—	—	1.75	This work
	GGA, f-core	39.21	—	—	—	—	—	—	—	[21]
	GGA, f-core	37.96	23.21	70.36	24.63	21.00	18.62	77.17	1.64	[22]
	PBE+U	16.80	10.94	24.20	13.10	17.20	—	—	1.54	This work
	Expt.	35.37	16.70	—	—	—	—	—	2.12	[23]
Sm	PBE, f-band	18.70	4.87	18.10	19.00	18.50	—	—	3.84	This work
	GGA+OP, f-band	19.91	—	—	—	—	—	—	—	[16]
	PBE, f-core	36.81	21.72	54.60	27.91	30.10	—	—	1.69	This work
	GGA, f-core	38.94		—	—	—	—	—	—	[21]
	GGA, f-core	36.91	19.60	61.81	21.27	18.64	24.56	68.58	1.88	[22]
	PBE+U	12.10	7.39	15.90	10.21	13.70	—	—	1.64	This work
	Expt.	29.46	12.68	—	—	—	—	—	2.32	[23]
Eu	PBE, f-band	14.33	7.51	16.60	13.20	17.70	—	—	2.32	This work
	PBE, f-core	12.93	9.07	17.61	10.60	16.80	—	—	1.43	This work
	GGA, f-core	14.67	—	—	—	—	—	—	—	[21]
	GGA, f-core	12.52	8.40	16.46	10.55	16.34	—	—	1.49	[22]
	PBE+U	12.20	7.56	16.20	10.21	13.80	—	—	1.61	This work
	Expt.	14.75	5.90	—	—	—	—	—	2.5	[23]
Gd	PBE, f-band	30.50	16.31	42.70	24.40	24.00	—	—	1.87	This work
Gd	GGA+OP, f-band	28.99	—	—	—	—	—	—	—	[16]
Gd	PBE, f-core	39.54	24.12	59.60	29.50	33.10	—	—	1.64	This work
	GGA, f-core	36.74	—	—	—	—	—	—	—	[21]
	GGA, f-core	41.73	22.11	68.26	21.00	21.01	30.04	80.3	1.89	[22]
	PBE+U	31.64	19.06	47.10	23.91	26.60	—	—	1.66	This work
	Expt.	38.40	22.31	—	—	—	—	—	1.72	[23]
Tb	PBE, f-band	24.37	16.76	36.50	18.30	25.20	—	—	1.45	This work
	GGA+OP, f-band	30.15	—	—	—	—	—	—	—	[16]
	PBE, f-core	41.37	25.17	62.70	30.70	34.10	—	—	1.64	This work
	GGA, f-core	36.28	—	—	—	—	—	—	—	[21]
	GGA, f-core	40.87	22.77	68.43	20.07	21.85	28.59	79.25	1.79	[22]
	PBE+U	32.90	15.70	46.10	26.31	21.40	—	—	2.09	This work
	Expt.	39.99	22.90	—	—	—	—	—	1.75	[23]
Dy	GGA+OP, f-band	29.08	—	—	—	—	—	—	—	[16]
	PBE, f-core	41.27	25.48	62.80	30.50	34.60	—	—	1.62	This work
	GGA, f-core	36.74	—	—	—	—	—	—	—	[21]
	GGA, f-core	42.14	24.48	70.93	20.53	23.97	20.53	28.75	1.72	[22]
	Expt.	38.50	25.45	—	—	—	—	—	1.51	[23]
Ho	PBE, f-band	29.09	14.26	35.90	25.69	27.50	—	—	2.04	This work
	GGA+OP, f-band	29.88	—	—	—	—	—	—	—	[16]
	PBE, f-core	42.14	26.15	64.80	30.80	34.90	—	—	1.61	This work
	GGA, f-core	38.20	—	—	—	—	—	—	—	[21]
	GGA, f-core	44.12	26.26	75.40	22.30	26.74	29.63	85.06	1.68	[22]
	PBE+U	14.63	7.03	15.50	14.19	20.40	—	—	2.08	This work
	Expt.	39.75	26.73	—	—	—	—	—	1.49	[23]
Er	PBE, f-band	32.73	9.51	34.40	31.90	26.00	—	—	3.46	This work
	GGA+OP, f-band	29.95	—	—	—	—	—	—	—	[16]
	PBE, f-core	42.60	27.78	65.60	31.10	34.80	—	—	1.53	This work
	GGA, f-core	40.12	—	—	—	—	—	—	—	[21]
	GGA, f-core	45.82	28.60	81.54	24.27	28.85	28.34	88.05	1.60	[22]
	PBE+U	29.43	17.13	35.70	26.30	37.50	—	—	1.72	This work
	Expt.	41.15	29.68	—	—	—	—	—	1.38	[23]
Tm	PBE, f-band	20.75	15.21	47.70	8.86	9.02	3.86	58.40	1.36	This work
	GGA+OP, f-band	27.93	—	—	—	—	—	—	—	[16]
	PBE, f-core	42.93	26.46	67.00	30.90	34.20	—	—	1.62	This work
	GGA, f-core	42.41	—	—	—	—	—	—	—	[21]
	GGA, f-core	48.23	31.02	88.44	25.58	30.28	28.04	94.21	1.58	[22]
	PBE+U	21.81	10.97	25.60	19.92	24.59	—	—	1.99	This work
	Expt.	44.5	30.5	—	—	—	—	—	1.45	[24]
Yb	PBE, f-band	16.63	10.55	20.10	14.89	24.10	—	—	1.58	This work
	PBE, f-core	15.87	9.35	18.60	14.50	22.30	—	—	1.69	This work
	GGA, f-core	15.58	—	—	—	—	—	—	—	[21]
	GGA, f-core	16.34	10.72	23.21	12.91	17.44	—	—	1.52	[22]
	PBE+U	10.68	3.82	17.66	13.33	20.75	—	—	2.79	This work
	Expt.	13.13	9.9	—	—	—	—	—	1.33	[24]

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[1]	McMahan A K, Huscroft C, Scalettar R T, Pollock E L 1998 J. Comput. Aid Mol. Des. 5 131 Google Scholar
[2]	Grimvall G 1999 Thermophysical Properties of Materials (Amsterdam: Elsevier
[3]	Jia T T, Chen G, Zhang Y S 2017 Phys. Rev. B 95 155206 Google Scholar
[4]	Li W G, Kou H B, Zhang X Y, Ma J Z, Li Y, Geng P J, Wu X Z, Chen L M, Fang D N 2019 Mech. Mater. 139 103194 Google Scholar
[5]	Pugh S 1954 Philos. Mag. 45 823 Google Scholar
[6]	Xu L, Li X, He Q, Yang J, Sun S L, Li J, Hu J B, Wu Q 2025 J. Appl. Phys. 137 015902 Google Scholar
[7]	谢明宇, 李法新 2022 力学进展 52 33 Google Scholar Xie M Y, Li F X 2022 Adv. Mech. 52 33 Google Scholar
[8]	Kittel C 1996 Introduction to Solid State Physics (New York: Wiley
[9]	Boguslavskii Y Y, Goncharov V A, Il'ina G G 1989 J. Less-Common Met. 147 249 Google Scholar
[10]	Jeong I K, Darling T W, Graf M J, Proffen T, Heffner R H, Lee Y, Vogt T, Jorgensen J D 2004 Phys. Rev. Lett. 92 105702 Google Scholar
[11]	Decremps F, Antonangeli D, Amadon B, Schmerber G 2009 Phys. Rev. B 80 132103 Google Scholar
[12]	Wang Z G, Bi Y, Xu L, Liu L 2014 Mater. Res. 1 026501 Google Scholar
[13]	Lipp M J, Jackson D, Cynn H, Aracne C, Evans W J, Mcmahan A K 2008 Phys. Rev. Lett. 101 165703 Google Scholar
[14]	Lipp M J, Jenei Z, Cynn H, Kono Y, Park C, Kenney-Benson C, Evans W J 2017 Nat. Commun. 8 1198 Google Scholar
[15]	Xu L, Wang Z G, Li Z G, Li X H, Yao S L, Li J, Zhou X M, Yu Y Y, Hu J B, Wu Q 2021 Appl. Phys. Lett. 118 074102 Google Scholar
[16]	Söderlind P, Turchi P E A, Landa A, Lordi V 2014 J. Phys. : Condens. Matter. 26 416001 Google Scholar
[17]	Hill R 1952 Proc. Phys. Soc. A 65 349 Google Scholar
[18]	Yang Z, Xian J W, Gao X Y, Tian F Y, Song H F 2024 J. Chem. Phys. 161 194101 Google Scholar
[19]	Blöchl P E 1994 Phys. Rev. B Condens Matter. 50 17953 Google Scholar
[20]	Perdew J P, Burke K, Ernzerhof M 1996 Phys. Rev. Lett. 77 3865 Google Scholar
[21]	Delin A, Fast L, Johansson B, Eriksson O, Wills J M 1998 Phys. Rev. B 58 4345 Google Scholar
[22]	Ouyang Y F, Tao X M, Zeng F J, Chen H M, Du Y, Feng Y P, He Y 2009 Physica B 404 2299 Google Scholar
[23]	Seitz F, Turnbull D 1964 Solid State Physics: Advance in Research and Applications (New York: Academic press
[24]	Material Properties https://material-properties.org [2025-03-19]
[25]	Stassis C, Gould T, McMasters O D, Gschneidner K A, Nicklow R M 1979 Phys. Rev. B 19 5746 Google Scholar
[26]	Pettifor D G 1992 Mater. Sci. Tech. 8 345 Google Scholar
[27]	Eberhart M E, Jones T E 2012 Phys. Rev. B 86 134106 Google Scholar
[28]	Kim S, Chen J, Cheng T, et al. PubChem 2025 update. 2025 Nucleic Acids Res. 53 D1516. https://pubchem.ncbi.nlm.nih.gov/periodic-table/melting-point/ [2025-03-19]
[29]	Nikolaev A V, Tsvyashchenko A V 2012 Phys. -Usp. 55 657 Google Scholar
[30]	Voronov F F, Goncharova V A, Stal'gorova O V 1979 J. Exp. Theor. Phys. 49 687
[31]	Chen J L, Kaltsoyannis N 2022 J. Phys. Chem. C 126 11426 Google Scholar
[32]	Eryigit S, Parlak C, Eryigit R 2022 J. Phys. : Condens. Matter 34 295402 Google Scholar
[33]	Tran F, Karsai F, Blaha P 2014 Phys. Rev. B 89 155106 Google Scholar
[34]	Chesnut G N, Vohra Y K 1999 Phys. Rev. Lett. 82 1712 Google Scholar
[35]	Morée J B, Amadon B 2018 Phys. Rev. B 98 205101 Google Scholar
[36]	Satikunvar D D, Bhatt N K, Thakore B Y 2021 J. Appl. Phys. 129 035107 Google Scholar

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Vol. 74, Issue 15 - 2025-08-05

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