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稀土金属在工程技术领域具有重要应用, 同时因其与f电子相关的独特行为受到凝聚态物理的广泛关注. 本文结合第一性原理计算与数据汇编, 对稀土金属的弹性性质随原子序数变化开展分析, 并以Ce和Yb为例, 对高压0—15 GPa范围内的弹性演化进行研究讨论, 对比了不同f电子处理方法的模拟表现. 结果表明, 稀土金属随原子序数变化存在明显的延展性差异, 在压力作用下的相变处弹性性质会发生显著改变. 特别是, 在Ce的fcc同构相变和Yb的fcc-bcc相变中出现脆、延性转变. 这些与随原子序数或压力条件改变发生的成键特性变化密切相关. 此外, 研究发现, 将f电子作为芯层电子处理的模拟方法能够较好地描述稀土金属在常压下的弹性性质, 但在描述高压下的结构相变及弹性性质演化趋势时, 将f电子作为价电子并考虑电子关联效应修正的处理方法则更为有效. 本文数据集可在
https://doi.org/10.57760/sciencedb.j00213.00150 中访问获取.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/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 athttps://doi.org/10.57760/sciencedb.j00213.00150 .
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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
Fig. 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.
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[27]
Fig. 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[27].
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[11–13,34]进行比较, 图中虚线是实验给出的Ce的γ-α相变压力点
Fig. 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 CL as a function of pressure. (d) Transverse wave velocity CT as a function of pressure. Comparisons are made with existing experimental results [11-13,34]. The dashed line in the figure marks the experimentally reported γ-α phase transition pressure.
图 4 (a) B/G随压强的变化关系, 并与已有的实验结果[11–13,34]进行比较; (b) s价电子数随压强的变化关系; (c) d, f价电子数随压强变化关系; (d) 柯西压力C12–C44随压强的变化关系
Fig. 4. (a) B/G ratio as a function of pressure. Comparisons with existing experimental results[11–13,34] 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.
图 5 Yb的fcc-bcc相焓差随压强的变化
Fig. 5. Enthalpy difference between fcc and bcc phase for Yb as a function of pressure.
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
Fig. 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 CL as a function of pressure. (d) Transverse wave velocity CT as a function of pressure. Comparisons with existing experimental results[9] are provided.
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
Fig. 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.
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 Expt. 14.83 12.86 24.1 10.2 19.4 — — 1.15 [29] 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 PBE+U 12.52 8.40 16.46 10.55 16.34 — — 1.49 [22] Expt. 12.20 7.56 16.20 10.21 13.80 — — 1.61 This Work 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 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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