Predicted crystal structures of molybdenum under high pressure

Accepted Manuscript Predicted crystal structures of molybdenum under high pressure Bing Wang, Guang Biao Zhang, Yuan Xu Wang PII: DOI: Reference:

S0925-8388(12)02220-7 http://dx.doi.org/10.1016/j.jallcom.2012.12.006 JALCOM 27436

To appear in: Received Date: Revised Date: Accepted Date:

6 July 2012 3 December 2012 4 December 2012

Please cite this article as: B. Wang, G.B. Zhang, Y.X. Wang, Predicted crystal structures of molybdenum under high pressure, (2012), doi: http://dx.doi.org/10.1016/j.jallcom.2012.12.006

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Predicted crystal structures of molybdenum under high pressure Bing Wang1 , Guang Biao Zhang1 , and Yuan Xu Wang1,2∗ 1

Institute for Computational Materials Science,

School of Physics and Electronics, Henan University, Kaifeng 475004, People’s Republic of China and 2

Guizhou Provincial Key Laboratory of Computational Nano-Material Science, Institute of Applied Physics, Guizhou Normal College, Guiyang, 550018, People’s Republic of China

Abstract The high-pressure structures of molybdenum (Mo) at zero temperature have been extensively explored through the newly developed particle swarm optimization (PSO) algorithm on crystal structural prediction. All the experimental and earlier theoretical structures were successfully reproduced in certain pressure ranges, validating our methodology in application to Mo. A doublehexagonal close-packed (dhcp) structure found by Mikhaylushkin et al. [Rhys. Rev. Lett. 101, (2008) 049602] is confirmed by the present PSO calculations. The lattice parameters and physical properties of the dhcp phase were investigated based on first principles calculations. The phase transition occurs only from bcc phase to dhcp phase at 660 GPa and at zero temperature. The calculated acoustic velocities also indicate a transition from the bcc to dhcp phases for Mo. More intriguingly, the calculated density of states (DOS) shows that the dhcp structure remains metallic. The calculated electron density difference (EDD) reveals that its valence electrons are localized in the interstitial regions. Keywords: Metals, Phase transition, Electronic structure

∗

Corresponding author at: School of Physics and Electronics, Henan University, Kaifeng 475004, China. Phone: +86-378-3881488, Fax: +86-378-3881488. E-mail: [email protected]

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1. INTRODUCTION

The high-pressure phase transition of molybdenum (Mo) has attracted great attention due to its wide scientific implications under high pressure, such as being used as a calibration of the ruby fluorescence technique for ultrahigh-pressure static experiments.[1–3] At ambient condition, Mo adopts a simple body-centered-cubic (bcc) structure and melts at 2890 K.[4] Hxison el at. found a sharp break in acoustic velocity of Mo by shock-wave (SW) in the Hugoniot curve at about 210 GPa and 4000 K, which was interpreted to be indicative of a solid-solid phase transition, but the structure of the final phase was not determined.[5] Later, Errandonea et al. [6] suggested that the shock transition may be due to melting. The bcc phase of Mo was found to be stable up to 272 GPa at 300 K in the diamond anvil cell (DAC) experiment [7], which was different from the SW measurements. Pettifor reported that the ambient bcc phase was found to be stable over a pressure range up to 420 GPa in DAC.[8] By further x-ray diffraction investigation, Ruoff et al. showed that bcc Mo was stable up to 560 GPa at room temperature.[9] As a result, all experimental studies consistently found the bcc structure to be stable up to the highest pressure reached, i.e., no phase transition was detected. Ab initio calculation is a powerful tool to explore the phase transition of metals under high pressure. Several theoretical studies have been performed to investigate the phase transition of Mo. Moriarty have reported a body-centered cubic (bcc) → hexagonal-close-packed (hcp) transition at 420 GPa and then a hcp → fcc transition at 620 GPa.[1] Belonoshko et al. found that below 700 GPa the bcc phase was more stable than the hcp phase; above 700 GPa, the fcc structure was most stable.[10] Cazorla et al. investigated the melting curve of Mo for the pressure range 0-400 GPa by ab inito calculations and found the DFT melting curve is consistent with shock data up to 400 GPa. [11] Mikhaylushkin et al. found that dhcp Mo is stable above 660 GPa at low temperature, and Mo may melt from the ω phase, not the hcp phase at high pressure.[12] Zeng et al. showed that bcc Mo was stable up to 703 ± 19 GPa [13]and 706 GPa [14] at zero temperature, and then it transformed to the fcc phase. Recently, Cazorla et al. noted that the high-pressure/high-temperature solid phase of Mo indicated by shock experiment is not fcc or hcp, but they did not rule out the possibility of other stable crystal phases.[15] However, direct confirmation of these calculated transition pressures with experiment is impossible, as the current maximum static pressure

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reached experimentally is 560 GPa. It is known that crystal structure is the basis for deep understanding of any physical properties. Up to now, the structure of Mo at high pressure is not clear. Therefore, it is desirable to explore the crystal structures and physical property of Mo under high pressure. In this work, we investigate the effect of ultrahigh pressure on the structures of Mo at zero temperature by the ab initio particle swarm optimization (PSO) algorithm on crystal structural prediction, [16] which requires only the chemical compositions for a given compound at given pressure. This method has been successfully applied to many other systems.[17– 23] As a result, the PSO calculations predict a dhcp structure above 660 GPa and at zero temperature. The dhcp structure has been found in the high-pressure phases of K, Rb, Cs, Mg, and Na.[24–26]

2. COMPUTATIONAL DETAIL

To search for potential crystal structures, the PSO technique implemented in the Crystal Structure Analysis by Particle Swarm Optimization (CALYPSO) package was employed at 0-900 GPa with 1-4 formula units (f.u.) up to 16 atoms in each simulation cell.[16] The underlying calculations were performed with the projector-augmented wave (PAW) method [27–29] implemented in the Vienna ab initio simulation package (VASP) [30–33]. The generalized gradient approximation (GGA) [34] was used to describe the exchangecorrelation function. The all-electron projector augmented wave method was adopted with 4d5 5s1 treated as valence electrons for Mo. Geometry optimization was performed using the conjugate gradient algorithm method with a plane-wave cutoff energy of 700 eV. The calculations were conducted with 20×20×20, 16×16×10, 19×19×6, and 16×16×16 k-mesh for bcc, hcp, dhcp, and fcc phase, respectively. For the hexagonal structures, Γ centered k mesh was used, and for other structures, Monkhorst-Pack k-points were used to ensure that all structures are well-converged to better than 1 meV/atom. The structures were relaxed with respect to both lattice parameters and atomic positions.

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3. RESULTS AND DISCUSSIONS 3.1 Structural determination and feature

The simulations were carried out within a pressure range of 0-900 GPa at zero temperature. The experimental and earlier theoretical structures were successfully reproduced in certain pressure ranges, validating our methodology in application to Mo. At 0-650 GPa, we found that the bcc phase possesses the lowest energy. Our calculated lattice parameter (a = 3.1507 ˚ A) agrees well with the experimental value of 3.1468 ˚ A[4], confirming the accuracy of the present calculation. No phase transition occurs in this pressure range. At 700 GPa, the PSO calculations predicted a double-hexagonal close-packed (dhcp) structure (shown in Fig. 1). By first-principles calculations, Mikhaylushkin et al. suggested that Mo should have a dhcp structure above 660 GPa, which is confirmed by our PSO calculations. However, since no lattice parameters have been reported in Ref. [12], it is difficult to compare our structure with them in detail. Our calculations show that this dhcp structure has a reasonable c/a ratio (= 3.154), which is close to the ideal value of c/a = 3.266 in the normal dhcp structure that corresponds to the densest packing of spherical particles. In this dense structure, Mo atoms locate at two different sites: 2c (1/3, 2/3, 1/4) and 2a (0, 0, 0). The stacking of closed-packed layers of Mo atoms is ABACABAC... (Fig. 1). However, the fcc structure, which has been previous proposed by other authors [13, 14], has a higher enthalpy than the dhcp structure. At 0 K, the Gibbs free energy is equal to the enthalpy H. Fig. 2a shows the calculated enthalpies curves of the newly predicted stable phases relative to bcc Mo. In Fig. 2a, the enthalpies of the most energetically competitive structures are compared over the pressure range up to 1000 GPa. For comparison, the ideal hcp phase was also considered. From Fig. 2a, it is clearly seen that the bcc → dhcp transition occurs at 660 GPa. That is to say, the experimental bcc structure is stable up to 660 GPa, above which the dhcp structure has a lower enthalpy. This is different from the previous results.[1, 10] It is worth mentioning that the present calculation correctly predicts the bcc → fcc phase transition at 706 GPa, which is in good agreement with the previous theoretical results [13, 14], demonstrating the reliability of the present theoretical method. The hcp phase becomes more stable than the bcc phase above 740 GPa, which is consistent with the previous results [12]. However,

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the fcc phase and the bcc phase are less stable than the dhcp phase above 660 GPa. We also calculated the static energy-volume (E-V) curves (Fig. 2b) and pressure-volume (P-V) curves (Fig. 3). From the static energy differences, we can see that the bcc phase is stable above 7.75 ˚ A3 . It means that bcc Mo transforms to dhcp Mo under compression. The fcc phase and the hcp phase become more stable than the bcc phase below 7.56 ˚ A3 and 7.39 ˚ A3 , respectively. But the dhcp phase is more stable than the fcc phase and the hcp phase below ˚3 . This is in good agreement with the results of calculated enthalpy curves. From 7.75 A Fig. 3, the compressibility of bcc Mo is in good agreement with the previous results[14].

3.2 Acoustic velocities

Elastic moduli are crucial to engineering applications. Theoretical polycrystalline elastic modulus can be determined from independent elastic constants. Single crystal elastic constants were determined from evaluation of stress tensor generated by small strain. Then bulk modulus B and shear modulus G were estimated by using the Voigt-ReussHill approximation.[35] The elastic moduli determine long-wavelength vibrational modes, or sound waves, in a solid. The isotropically averaged aggregate velocities can be obtained using vl = [(B + 4G/3)/ρ]1/2

(1)

vb = [(B/ρ)]1/2

(2)

vt = [(G/ρ)]1/2

(3)

where υ l , υ b , and υ t are the compressional, bulk, and shear acoustic velocities, respectively. The aggregate velocities are shown in Fig. 4 (from 450 GPa to 900 GPa). At zero pressure, the υ l , υ b , and υ t of bcc Mo are 6.608 km/s, 5.127 km/s, and 3.611 km/s, respectively, which are in good agreement with the previous results (υ l is 6.44 km/s and υ b is 5.04 km/s) [36]. From Fig. 4, there is a break around 650 GPa ∼ 750 GPa for the compressional acoustic velocity υ l and shear acoustic velocity υ t of bcc Mo. The break in the acoustic velocities can be attributed to a change in elastic properties associated with a solid-solid phase transition, which agrees well with the calculated enthalpy curves and energy-volume (E-V) curves data. However, the acoustic velocities of dhcp Mo have no break in the range of pressure, which confirms that the high-pressure phase of Mo should be the dhcp phase. 5

3.3 Chemical bonding and electronic structure

Under pressure, metals exhibit increasingly shorter interatomic distances. Intuitively, this response is expected to be accompanied by an increase in the widths of valence and conduction bands and hence more pronounced free-electron-like behaviour. The nearest Mo-Mo distance of the dhcp structure is 2.18 ˚ A at 700 GPa, which implies a strong valencevalence overlap (the 5s and 4d orbital radii in Mo are 1.52 ˚ A and 0.70 ˚ A, respectively.) between neighboring Mo atoms and core exclusion already plays a significant role to the valence electron. As a result, the valence electrons are repelled into the lattice interstices. Electron density difference (EDD) maps have been generated by subtracting the superpositioned, non-interacting, atomic electron densities from the calculated electron density of dhcp Mo at 700 GPa, shown in Fig. 1. Charge accumulation between atoms indicates the form of a covalent bond. From Fig. 1, the most intriguing feature of EDD is that there is a strong electron localization: the charge accumulation occurs in the open interstitial regions. This suggest that repulsion between the interstitial electron pairs is a major structureforming interaction. It also indicates that the overlaps of valence and core orbitals plays a critical role in the structure-forming and the electronic states of dhcp Mo. To probe the electronic properties and chemical bonding features of the bcc and dhcp phases, we calculated total and partial density of states (DOS) at 0 and 700 GPa, respectively, shown in Fig. 5. Fig. 5a highlights that the bcc and dhcp structures are metallic by an evidence of the finite electron DOS at the Fermi level. The total DOS at the Fermi level for the dhcp structure is 1.12 eV−1 at 0 GPa, which is larger than that (0.64 eV−1 ) of the bcc phase. However, at 700 GPa, the total DOS at the Fermi level for the dhcp structure (0.43 eV−1 ) is smaller than that of the bcc phase (0.49 eV−1 ). This suggests that the metallicity of the dhcp structure is becoming weaker than the bcc structure. This is apparently contrary to the increased metallicity observed in groups III-VII elements and is in good agreement with the alkalis and some other metals (Mg, K, Na, Ca). [24–26, 37] From Fig. 5b, it is found that the DOS of these two structures are mainly dominated by the d electrons. The shapes of d state and p state for the dhcp phase are similar, which indicates that there is a strong hybridization between them. The most intriguing feature of the partial DOS (Fig. 5b) is the growing of p state for the dhcp phase. The weakening of metallicity is attributed to the drop of 4d states in relative to the 4p states and the increased p-d hybridization upon

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compression.

4. CONCLUSION

In summary, using the new developed PSO technique on crystal structure prediction, we successfully predicted the crystal structures of Mo under various pressure and at zero temperature. In good agreement with experiment, we reproduced the bcc structure at lower pressure, which is stable to 660 GPa. Above 660 GPa, a dhcp structure is predicted by the PSO method, which confirms the previous theoretical report.[12] Such dhcp structure is more stable than bcc structure above 660 GPa. We also found that the previous proposed fcc structure has a higher energy than that of the dhcp structure above 706 GPa. As a result, phase transition occurs only from bcc phase to dhcp phase at 660 GPa and at zero temperature. Our calculated acoustic velocities confirmed the high-pressure phase transition at 660 GPa. Moreover, the electron density difference indicates that the valence electrons of dhcp Mo are mostly localized in the interstitial sites. The present results can stimulate future experimental and theoretical study of Mo under high pressure.

Acknowledgments

This research was sponsored by the National Natural Science Foundation of China (No. 21071045), the Program for New Century Excellent Talents in University (No. NCET-100132), and the fund of Henan University (No. SBGJ090508).

[1] J.A. Moriarty, Phys. Rev. B 45, (1992) 2004. [2] H.K. Mao, P.M. Bell, J.W. Shaner, D.J. Steinberg, J. Appl. Phys. 49, (1978) 3276. [3] D. Santamar´ıa-P´erez, M. Ross, D. Errandonea, G.D. Mukherjee, M. Mezouar, R. Boehler, J. Chem. Phys. 130, (2009) 124509. [4] H.T. Tsai, A. Muan, J. Am. Ceram. Soc. 75, (1992) 1412. [5] R.S. Hixson, D.A. Boness, J.W. Shaner, J.A. Moriarty, Phys. Rev. Lett. 62, (1989) 637. [6] D. Errandonea, B. Schwager, R. Ditz, C. Gessmann, R. Boehler, M. Ross, Phys. Rev. B 63, (2001) 132104.

7

[7] Y.K. Vohra, A.L. Ruoff, Phys. Rev. B 42, (1990) 8651. [8] D.G. Pettifor, Bonding and Structure of Molecules and solids (Clarendon Press, Oxford, 1995). [9] A.L. Ruoff, H. Xia, Q. Xia, Rev. Sci. Instrum. 63, (1992) 4342. [10] A.B. Belonoshko, S.I. Simak, A.E. Kochetov, B. Johansson, L. Burakovsky, D.L. Preston, Phys. Rev. Lett. 92, (2004) 195701. [11] C. Cazorla, M.J. Gillan, S. Taioli, D. Alf`e, J. Chem. Pgys. 126, (2007) 194502. [12] A.S. Mikhaylushkin, S.I. Simak, L. Burakovsky, S.P. Chen, B. Johansson, D.L. Preston, D.C. Swift, A.B. Belonoshko, Phys. Rev. Lett. 101, (2008) 049602. [13] Z.Y. Zeng, C.E. Hu, L.C. Cai, X.R. Chen, F.Q. Jing, J. Phys. Chem. B 114, (2010) 298. [14] Z.Y. Zeng, C.E. Hu, X.R. Chen, X.L. Zhang, L.C. Cai, F.Q. Jing, Phys. Chem. Chem. Phys. 13, (2011) 1669. [15] C. Cazorla, D. Alf`e, M.J. Gillan, Phys. Rev. B 85, (2012) 064113. [16] Y.C. Wang, J. Lv, Y.M. Ma, Phys. Rev. B 82, (2010) 094116. [17] J. Lv, Y.C. Wang, L. Zhu, Y.M. Ma, Phys. Rev. Lett. 106, (2011) 015503. [18] Y.C. Wang, H.Y. Liu, J. Lv, L. Zhu, H. Wang, Y.M. Ma, Nature Commun. 2, (2011) 563. [19] L. Zhu, Z.W. Wang, Y.C. Wang, G.T. Zou, H.K. Mao, Y.M. Ma, PNAS 109, (2012) 751. [20] L. Zhu, H. Wang, Y.C. Wang, J. Lv, Y.M. Ma, Q.L. Cui, Y.M. Ma, G.T. Zou, Phys. Rev. Lett. 106, (2011) 145501. [21] Q.Q. Wang, Z.S. Zhao, L.F. Xu, L.M. Wang, D.L. Yu, Y.J. Tian, J.L. He, J. Phys. Chem. C 115, (2011) 19910. [22] M.G. Zhang, H.Y. Yan, G.T. Zhang, H. Wang, J. Phys. Chem. C 116, (2012) 4293. [23] Z.S. Zhao, B. Xu, X.F. Zhou, L.M. Wang, B. Wen, J.L. He, Z.Y. Liu, H.T. Wang, Y.J. Tian, Phys. Rev. Lett. 107, (2011) 215502. [24] Y.M. Ma, A.R. Oganov, Y. Xie, Phys. Rev. B 78, (2008) 014102. [25] P.F. Li, G.Y. Gao, Y.C. Wang, Y.M. Ma, J. Phys. Chem. C 114, (2010) 21745. [26] Y.M. Ma, M. Eremets, A.R. Oganov, Y. Xie, I. Trojan, S. Medvedev, A.O. Lyakhov, M. Valle, V. Prakapenka, Nature 458, (2009) 182. [27] P.E. Bl¨ ochl, Phys. Rev. B 50, (1994) 17953. [28] G. Kresse, D. Joubert, Phys. Rev. B 59, (1999) 1758. [29] P. Hohenberg, W. Kohn, Phys. Rev. B 136, (1964) 864. [30] G. Kresse, J. Hafner, Phys. Rev. B 47, (1993) 558.

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[31] G. Kresse, J. Hafner, Phys. Rev. B 49, (1994) 14251. [32] G. Kresse, J. Furthm¨ uller, Comput. Mater. Sci. 6, (1996) 15. [33] G. Kresse, J. Furthm¨ uller, Phys. Rev. B 54, (1996) 11169. [34] J.P. Perdew, K.Burke, M. Ernzerhof, Phys. Rev. Lett. 77, (1996) 3865. [35] R. Hill, Proc. Phys. Soc. A. 65, (1952) 349. [36] Z.Y. Zeng, C.E. Hu, X. Liu. L.C. Cai, F.Q. Jing, Appl. Phy. Lett. 99, (2011) 191906. [37] A.R. Oganov, Y.M. Ma, Y. Xu, I. Errea, A. Bergara, A.O. Lyakhov, PNAS 107, (2010) 7646.

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FIGURE CAPTIONS Fig. 1. (Colour online) Left: Crystal structure of dhcp Mo (space group P63 /mmc). Lattice parameters at 700 GPa: a = b = 2.2276 ˚ A and c = 7.0253 ˚ A. There are two inequivalent atomic positions: Mo1 at 2a (0, 0, 0) and Mo2 at 2c (1/3, 2/3, 1/4). Right: Difference charge density of dhcp Mo plotted in the 110 plane at 700 GPa. Fig. 2. (a) Enthalpy curves (relative to the Mo bcc structure) as function of pressure at 0 K. (b) Static energies (relative to the Mo bcc structure) as function of unit cell volume. Fig. 3.(Colour online) Isotherms of the bcc, dhcp, hcp, and fcc Mo at 0 K (V0 is the equilibrium volume at 0 K and 0 GPa), compared with the theoretical results of bcc [14]. Fig. 4. Acoustic velocity of Mo as function of pressure. Fig. 5. (Colour online) Calculated total and partial electronic DOS plots for Mo bcc and dhcp structure. The Fermi level was set to zero.

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FIG. 1: (Colour online) Left: Crystal structure of dhcp Mo (space group P 63 /mmc). Lattice parameters at 700 GPa: a = b = 2.2276 ˚ A and c = 7.0253 ˚ A. There are two inequivalent atomic positions: Mo1 at 2a (0, 0, 0) and Mo2 at 2c (1/3, 2/3, 1/4). Right: Difference charge density of dhcp Mo plotted in the 110 plane at 700 GPa.

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ΔHdhcp-bcc

ΔH (eV/atom)

0.4

ΔHfcc-bcc

0.2

706 GPa 740 GPa

ΔHhcp-bcc

0 -0.2

660 GPa

-0.4 0

ΔEstatic(eV/atom)

0.6

200

800

1000

ΔEfcc-bcc

0.4

ΔEdhcp-bcc

0.2

ΔEhcp-bcc

0

400 600 (a) Pressure (GPa)

3

7.39 Å

3

-0.2

7.56 Å

7.75 Å

3

-0.4 7

8

9 3 (b) Volume (Å )

10

11

12

FIG. 2: (a) Enthalpy curves (relative to the Mo bcc structure) as function of pressure at 0 K. (b) Static energies (relative to the Mo bcc structure) as function of unit cell volume.

12

FIG. 1: (Colour online) Left: Crystal structure of dhcp Mo (space group P 63 /mmc). Lattice parameters at 700 GPa: a = b = 2.2276 ˚ A and c = 7.0253 ˚ A. There are two inequivalent atomic positions: Mo1 at 2a (0, 0, 0) and Mo2 at 2c (1/3, 2/3, 1/4). Right: Difference charge density of dhcp Mo plotted in the 110 plane at 700 GPa.

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Acoustic velocity (m/s)

12000 10000 8000

bcc vl

Transition Zone

dhcp vl bcc vb

6000

dhcp vb bcc vt dhcp vt

4000 2000 450

500

550

600

650

700

750

800

850

Pressure (GPa)

FIG. 4: Acoustic velocity of Mo as function of pressure.

14

900

10 bcc 0 GPa dbcp 0 GPa dhcp 700 GPa bcc 700 GPa

(a) Total DOS

8

DOS (states/eV/4 atoms)

6 4 2 0 -16

-12

-8

-4

0

4

8

12

16

20

24

28

5 dhcp-d dhcp-p dhcp-s bcc-d bcc-p bcc-s

(b) Partial DOS (700 GPa)

4 3 2 1 0 -16

-12

-8

-4

0

4

8

12

16

20

24

28

Energy (eV)

FIG. 5: (Colour online) Calculated total and partial electronic DOS plots for Mo bcc and dhcp structure. The Fermi level was set to zero.

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Graphical Abstract (for review)

ΔHdhcp-bcc

ΔH (eV/atom)

0.4

ΔHfcc-bcc

0.2

706 GPa 740 GPa

ΔHhcp-bcc

0 -0.2

660 GPa

-0.4 0

ΔEstatic(eV/atom)

0.6

200

800

1000

ΔEfcc-bcc

0.4

ΔEdhcp-bcc

0.2

ΔEhcp-bcc

0

400 600 (a) Pressure (GPa)

3

7.39 Å

3

-0.2

7.56 Å

7.75 Å

3

-0.4 7

8

9 3 (b) Volume (Å )

10

11

12


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