*Ab initio*calculation of lattice dynamics and thermodynamic properties of beryllium

^{1,2}, Ling-Cang Cai

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

^{2,3,b)}, Fu-Qian Jing

^{1,2}and Dario Alfè

^{4}

### Abstract

We investigate the phase transition,elastic constants, phonon dispersion curves, and thermal properties of beryllium (Be) at high pressures and high temperatures using density functional theory. By comparing the Gibbs free energy, in the quasiharmonic approximation (QHA), of hexagonal-closed-packed (hcp) with those of the face-centered cubic (fcc) and body-centered-cubic (bcc) we find that the hcp Be is stable up to 390 GPa, and then transforms to the bcc Be. The calculated phonon dispersion curves are in excellent agreement with experiments. Under compression, the phonon dispersion curves of hcp Be do not show any anomaly or instability. At low pressure the phonon dispersion of bcc Be display imaginary along Γ-*N* in the *T* _{1} branches. Within the quasiharmonic approximation, we predict the thermal equation of state and other properties including the thermal expansion coefficient, Hugoniot curves, heat capacity, Grüneisen parameter, and Debye temperature.

The authors would like to thank Dr. Zhao-Yi Zeng for the helpful discussions. We also thank the support by the National Natural Science Foundation of China under Grant Nos. 10776029/A06 and 11174214, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 20090181110080 and the Science Foundation of China Academy of Engineering Physics under Grant No. 2010A0101001.

I. INTRODUCTION

II. COMPUTATIONAL METHOD

III. RESULTS AND DISCUSSION

A. Structural and elastic properties

B. Phonon dispersions and phase transition

C. Thermodynamic properties

IV. CONCLUSIONS

### Key Topics

- Elasticity
- 26.0
- High pressure
- 26.0
- Phonon dispersion
- 18.0
- Phonons
- 14.0
- Equations of state
- 13.0

## Figures

(a) Static energy per atom of the bcc, fcc, and hcp phases as function of volume. (b) Enthalpy per atom of the bcc and fcc phases relative to the bcc enthalpy as function of pressure.

(a) Static energy per atom of the bcc, fcc, and hcp phases as function of volume. (b) Enthalpy per atom of the bcc and fcc phases relative to the bcc enthalpy as function of pressure.

(a) Bulk modulus *B*, shear modulus *G*, and Young’s modulus *E* vs pressure at zero temperature. (b) Aggregate sound velocities (*V* _{P}, *V* _{B}, and *V* _{S}) vs pressure at zero temperature.

(a) Bulk modulus *B*, shear modulus *G*, and Young’s modulus *E* vs pressure at zero temperature. (b) Aggregate sound velocities (*V* _{P}, *V* _{B}, and *V* _{S}) vs pressure at zero temperature.

Phonon dispersion curves of (a) hcp Be, (b) bcc Be at 0 GPa and 0 K. The solid circles in (a) are neutron diffraction data (Ref. 19) measured at 80 K.

Phonon dispersion curves of (a) hcp Be, (b) bcc Be at 0 GPa and 0 K. The solid circles in (a) are neutron diffraction data (Ref. 19) measured at 80 K.

The phonon dispersion curves of the hcp (a) and bcc (b) Be shown along high-symmetry directions at different volumes.

The phonon dispersion curves of the hcp (a) and bcc (b) Be shown along high-symmetry directions at different volumes.

Phase diagram of Be at high pressure and temperature. The dotted line is the hcp-bcc boundary calculated by Robert *et al.* (Ref. 21), the dashed line is the theoretical melting curve by Robert *et al.* (Ref. 21).

Phase diagram of Be at high pressure and temperature. The dotted line is the hcp-bcc boundary calculated by Robert *et al.* (Ref. 21), the dashed line is the theoretical melting curve by Robert *et al.* (Ref. 21).

Free energy from the phonons *F* _{phon} (a) and electronic excitations *F* _{elec} (b) vs volume of hcp Be at temperatures from 300 to 3500 K.

Free energy from the phonons *F* _{phon} (a) and electronic excitations *F* _{elec} (b) vs volume of hcp Be at temperatures from 300 to 3500 K.

Isothermal compression curves at different temperatures, compared with experimental data (Refs. 6 and 8).

Isothermal compression curves at different temperatures, compared with experimental data (Refs. 6 and 8).

Calculated thermal pressures of hcp Be (a) as a function of volume and (b) temperature.

Calculated thermal pressures of hcp Be (a) as a function of volume and (b) temperature.

Volume–pressure (a) and temperature–pressure (b) relations on Hugoniot curves obtained from the QHA, in comparison with experimental data (Ref. 43) and other calculations (Refs. 14, 44, and 45).

Volume–pressure (a) and temperature–pressure (b) relations on Hugoniot curves obtained from the QHA, in comparison with experimental data (Ref. 43) and other calculations (Refs. 14, 44, and 45).

Thermal expansion coefficient α_{ V } as a function of temperature (a) and pressure (b). The solid squares and solid triangle are taken form Gordon *et al.* (Ref. 46) and Grimvall *et al.* (Ref. 47), respectively.

Thermal expansion coefficient α_{ V } as a function of temperature (a) and pressure (b). The solid squares and solid triangle are taken form Gordon *et al.* (Ref. 46) and Grimvall *et al.* (Ref. 47), respectively.

Heat capacity *C* _{P} as a function of temperature at different pressure, together with the experimental data (Ref. 48).

Heat capacity *C* _{P} as a function of temperature at different pressure, together with the experimental data (Ref. 48).

Variation of the Grüneisen parameter γ with temperature (a) and pressure (b).

Variation of the Grüneisen parameter γ with temperature (a) and pressure (b).

Debye temperature Θ_{ D } as a function of temperature at different volumes.

Debye temperature Θ_{ D } as a function of temperature at different volumes.

## Tables

The equilibrium volume *V* _{0} (Å^{3}/atom), lattice parameters *a* and *c*, axial ratio *c*/*a*, zero pressure bulk modulus *B* _{0} (GPa), and pressure derivative *B*′.

The equilibrium volume *V* _{0} (Å^{3}/atom), lattice parameters *a* and *c*, axial ratio *c*/*a*, zero pressure bulk modulus *B* _{0} (GPa), and pressure derivative *B*′.

The calculated elastic constants and anisotropies Δ_{ p } *, Δ* _{ S1}, and Δ_{ S2} for the three types of elastic waves of hcp Be, compared with the experimental data and the other theoretical results. *C* _{ ij }, *B*, *G*, and *E* are in GPa.

The calculated elastic constants and anisotropies Δ_{ p } *, Δ* _{ S1}, and Δ_{ S2} for the three types of elastic waves of hcp Be, compared with the experimental data and the other theoretical results. *C* _{ ij }, *B*, *G*, and *E* are in GPa.

Value of the parameters obtained in the linear fits to our *C* _{ ij }(*P*) results. *a* _{ ij } is given in units of GPa, and *b* _{ ij } is given in GPa/K.

Value of the parameters obtained in the linear fits to our *C* _{ ij }(*P*) results. *a* _{ ij } is given in units of GPa, and *b* _{ ij } is given in GPa/K.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content