^{1,2,a)}, Zhao-Yi Zeng

^{1,2}, Lin Zhang

^{1,b)}, Xiang-Rong Chen

^{2}, Ling-Cang Cai

^{1}and Dario Alfè

^{3}

### Abstract

We report a detailed first-principles calculation to investigate the structures,elastic constants, and phase transition of Ti. The axial ratios of both and are nearly constant under hydrostatic compression, which confirms the latest experimental results. From the high pressureelastic constants, we find that the is unstable when the applied pressures are larger than 24.2 GPa, but the is mechanically stable at all range of calculated pressure. The calculated phonon dispersion curves agree well with experiments. Under compression, we captured a large softening around point of . When the pressure is raised to 35.9 GPa, the frequencies around the point along and in transverse acoustical branches become imaginary, indicating a structural instability. Within quasiharmonic approximation, we obtained the full phase diagram and accurate thermal equations of state of Ti. The phase transition at zero pressure occurs at 146 K and 1143 K, respectively. The predicted triple point is at 9.78 GPa, 931 K, which is close to the experimental data. Our thermal equations of state confirm the available experimental results and are extended to a wider pressure and temperature range.

The authors would like to thank Dr. Yan Bi for the helpful discussions. This research is supported by the National Key Laboratory Fund for Shock Wave and Detonation Physics Research under Grant Nos. 9140C6701010901 and 9140C6702030802, the National Natural Science Foundation of China under Grant Nos. 10776029 and 10776022.

I. INTRODUCTION

II. DETAILS OF CALCULATIONS AND MODELS

III. RESULTS AND DISCUSSION

A. Structural properties

B. Elastic properties

C. Phonon dispersions

D. Phase diagram

E. Thermal EOS

IV. CONCLUSIONS

### Key Topics

- High pressure
- 27.0
- Elasticity
- 19.0
- Equations of state
- 18.0
- Phonons
- 17.0
- Phase diagrams
- 14.0

## Figures

Zero temperature isotherms of Ti. The solid lines are the present work. The solid and open symbols are experimental data from Vohra and Spencer (Ref. 11) and Errandonea *et al.* (Ref. 17), respectively.

Zero temperature isotherms of Ti. The solid lines are the present work. The solid and open symbols are experimental data from Vohra and Spencer (Ref. 11) and Errandonea *et al.* (Ref. 17), respectively.

Lattice parameters , , and axial ratio as a function of pressure for (a) and (b) . The solid and open circles are the experimental data from Zhang *et al.* (Ref. 16) and Errandonea *et al.* (Ref. 17), respectively.

Lattice parameters , , and axial ratio as a function of pressure for (a) and (b) . The solid and open circles are the experimental data from Zhang *et al.* (Ref. 16) and Errandonea *et al.* (Ref. 17), respectively.

The aggregate sound velocities (, , and ) of - and vs pressure at 0 K.

The aggregate sound velocities (, , and ) of - and vs pressure at 0 K.

Phonon dispersion curves of (a) -, (b) -, and (c) at 0 GPa and 0 K. The solid circles in (a) and (c) are the experimental data measured by Stassis *et al.* (Ref. 42) and Petry *et al.* (Ref. 43), respectively. The experimental data for were measured at 1293 K.

Phonon dispersion curves of (a) -, (b) -, and (c) at 0 GPa and 0 K. The solid circles in (a) and (c) are the experimental data measured by Stassis *et al.* (Ref. 42) and Petry *et al.* (Ref. 43), respectively. The experimental data for were measured at 1293 K.

Phonon dispersion curves of (a) and (b) at different pressures.

Phonon dispersion curves of (a) and (b) at different pressures.

Mode Grüneisen parameter of and at zero pressure.

Mode Grüneisen parameter of and at zero pressure.

Phase diagram of Ti. The thin solid and thick grey lines are the present results. The dashed line and dashed-dotted line are the theoretical results from Mei *et al.* (Ref. 8) and Henning *et al.* (Ref. 6), respectively. The dashed-dotted-dotted and dotted lines are the experimental data from Zhang *et al.* (Ref. 14) and Young (Ref. 12), respectively.

Phase diagram of Ti. The thin solid and thick grey lines are the present results. The dashed line and dashed-dotted line are the theoretical results from Mei *et al.* (Ref. 8) and Henning *et al.* (Ref. 6), respectively. The dashed-dotted-dotted and dotted lines are the experimental data from Zhang *et al.* (Ref. 14) and Young (Ref. 12), respectively.

Thermal EOS of (a) -, (b) -, and , together with the experimental data (solid symbols) at the same condition (Ref. 16). The solid circles in the right plane of (b) are the experimental data at 8.1 GPa (Ref. 16), the grey line in the right plane of (b) locate at .

Thermal EOS of (a) -, (b) -, and , together with the experimental data (solid symbols) at the same condition (Ref. 16). The solid circles in the right plane of (b) are the experimental data at 8.1 GPa (Ref. 16), the grey line in the right plane of (b) locate at .

The thermal pressure as a function of volume and temperature. The solid triangles, circles, and squares in the left panel of the figure are the calculated data at 900, 673, and 300 K for (Ref. 16). The grey line in the right panel of the figure is the linear fitted results for the calculated data (open circles) (Ref. 16).

The thermal pressure as a function of volume and temperature. The solid triangles, circles, and squares in the left panel of the figure are the calculated data at 900, 673, and 300 K for (Ref. 16). The grey line in the right panel of the figure is the linear fitted results for the calculated data (open circles) (Ref. 16).

The isothermal bulk modulus and adiabatic bulk modulus vs (a) temperature at 0 GPa and (b) pressure. The solid and open stars are the experimental data from Ogi *et al.* (Ref. 46) and Fisher *et al.* (Ref. 49), respectively. The two grey vertical lines in (a), including Figs. 11(a) and 12(a), located at 146 K and 1143 K, respectively.

The isothermal bulk modulus and adiabatic bulk modulus vs (a) temperature at 0 GPa and (b) pressure. The solid and open stars are the experimental data from Ogi *et al.* (Ref. 46) and Fisher *et al.* (Ref. 49), respectively. The two grey vertical lines in (a), including Figs. 11(a) and 12(a), located at 146 K and 1143 K, respectively.

The first and second order pressure derivative of ( and ) vs (a) temperature at 0 GPa and (b) pressure.

The first and second order pressure derivative of ( and ) vs (a) temperature at 0 GPa and (b) pressure.

Entropy of -, -, and as functions of (a) temperature at 0 GPa, compared with the experimental data from NIST-JANAF (open diamonds) (Ref. 50) and (b) pressure.

Entropy of -, -, and as functions of (a) temperature at 0 GPa, compared with the experimental data from NIST-JANAF (open diamonds) (Ref. 50) and (b) pressure.

## Tables

The equilibrium axial ratio , volume , zero pressure bulk modulus (GPa), pressure derivative , (1/GPa), and static energy (eV/atom).

The equilibrium axial ratio , volume , zero pressure bulk modulus (GPa), pressure derivative , (1/GPa), and static energy (eV/atom).

The calculated elastic modulus (GPa), Poisson’s Ratio of Ti under pressure (GPa) at 0 K. The experimental data (Refs. 28 and 39) for were measured at room pressure (RP) and 293 K, and for were at 1273 K.

The calculated elastic modulus (GPa), Poisson’s Ratio of Ti under pressure (GPa) at 0 K. The experimental data (Refs. 28 and 39) for were measured at room pressure (RP) and 293 K, and for were at 1273 K.

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