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Ab initio calculation of lattice dynamics and thermodynamic properties of beryllium
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Image of FIG. 1.
FIG. 1.

(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.

Image of FIG. 2.
FIG. 2.

(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.

Image of FIG. 3.
FIG. 3.

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.

Image of FIG. 4.
FIG. 4.

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

Image of FIG. 5.
FIG. 5.

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).

Image of FIG. 6.
FIG. 6.

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.

Image of FIG. 7.
FIG. 7.

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

Image of FIG. 8.
FIG. 8.

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

Image of FIG. 9.
FIG. 9.

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).

Image of FIG. 10.
FIG. 10.

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.

Image of FIG. 11.
FIG. 11.

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

Image of FIG. 12.
FIG. 12.

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

Image of FIG. 13.
FIG. 13.

Debye temperature Θ D as a function of temperature at different volumes.


Generic image for table
Table I.

The equilibrium volume V 03/atom), lattice parameters a and c, axial ratio c/a, zero pressure bulk modulus B 0 (GPa), and pressure derivative B′.

Generic image for table
Table II.

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.

Generic image for table
Table III.

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.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Ab initio calculation of lattice dynamics and thermodynamic properties of beryllium