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The equation of state of tantalum (Ta) has been investigated to 100 GPa and 3,000 K using the first-principles molecular dynamics method. A large volume dependence of the thermal pressure of Ta was revealed from the analysis of our data. A significant temperature dependence of the calculated effective Grüneisen parameters was confirmed at high pressures. This indicates that the conventional approach to analyze thermal properties using the Mie-Grüneisen approximation is likely to have a significant uncertainty in determining the equation of state for Ta, and that an intrinsic anharmonicity should be considered to analyze the equation of state.

Equations of state (EOS) for some elemental metals have been used as an internal pressure gauge in X-ray diffraction high-pressure studies using diamond anvil cell [

In this study, we used density functional theory to investigate the thermal pressure of Ta. We also used the experimental data to determine the room temperature EOS of Ta. The combination of the first-principles molecular dynamics calculations and the high-pressure experiments led us to determine a reliable EOS over a wide range of pressures and temperatures.

The first-principles calculations carried out in this study were based on density functional theory using the VASP package [^{2} and 5p^{6}6s^{2}5d^{3} of the Ta atom are treated as frozen core and valence electron, respectively. We used a 128-atom supercell with Γ-point Brillouin zone sampling and a time step of 1 fs for the first-principles molecular dynamics simulation at constant volume. Simulations were run in the constant

The EOS of a solid can be described in a general form as a functional relationship between the pressure, volume, and temperature as
_{total}(V,T)_{st}(V,300)_{th}(V,T)_{0}_{T0}_{0}_{T0}′_{T0}_{th}

The value of _{0}_{0}_{T}_{T}_{V}^{2} ^{2})_{V}_{0}_{T})/B_{T0}

It is likely that the experimental uncertainty is related to the differential stress in the sample and the credibility of the pressure standard. In the case of diamond anvil cell experiments, the differential stress is accumulated as the pressure increases because the diamond anvils apply a uniaxial compression in the sample chamber. Therefore, the soft materials should be used as the pressure transmitting medium to reduce the differential stress of the sample. It is known that one of best pressure transmitting mediums is helium. Therefore, an experimental data set from Dewaele

The bulk modulus, _{T0}_{T0}′_{0}^{3}, _{T0}_{T0}′_{0}

The first-principle molecular dynamics method has been used to calculate the thermoelastic properties of Ta under extreme high-pressure and high-temperature conditions. _{0}_{,} (_{T}_{V}^{2} ^{2})_{V}^{−5} (K^{−1}), −0.0050 (GPaK^{−1}), and 2.11 × 10^{−7} (GPa^{2}K^{−2}), respectively. _{th}

The Mie-Grüneisen-Debye EOS has been frequently used in previous studies on the EOS of solids. The Mie-Grüneisen approximation is valid if the quasiharmonic term is dominant in the thermal pressure. However, it is known that the anharmonic term, which is not included in the Mie-Grüneisen approximation, is not negligible at high temperatures. Therefore, we assessed the effect of the anharmonicity on the EOS for Ta. We used the Grüneisen parameter to investigate the anharmonicity. The effective Grüneisen parameter can be written as follows:
_{qh}(V)

If the anharmonicity is negligible, then the effective Grüneisen parameter does not change at high temperatures. Therefore, we calculated the effective Grüneisen parameter at different volumes and temperatures. The Grüneisen parameter was obtained directly in our calculations from:
_{th}

We have investigated the EOS of Ta, which has a body-centred cubic structure, using the first-principles molecular dynamics method. We used the high-pressure experimental data to determine the compressibility at room temperature, and used the generalized gradient approximation (GGA) and the projector augmented-wave method (PAW) in simulations to calculate the thermal pressure. A Vinet EOS fitted to the room temperature data yielded an isothermal bulk modulus of _{T0}_{T0}_{0}_{T}_{V}^{2}^{2})_{V}^{−}^{5} (K^{−1}), −0.0050 (GPaK^{−1}), and 2.11 × 10^{−7} (GPa^{2}K^{−2}), respectively. The temperature dependence of the calculated effective Grüneisen parameters at high pressures indicates that the conventional Mie-Grüneisen approximation is not suitable for the analysis of thermoelastic properties of Ta, and that the intrinsic anharmonicity is non-negligible at high pressures.

This work made use of the super computer system of JAMSTEC and of the computer systems of the Earthquake Information Center of the Earthquake Research Institute. This work was partially supported by Grant-in-Aid for Scientific Research from JSPS and the Earthquake Research Institute cooperative research program, Japan.

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Pressure-volume data for Ta. The squares and diamonds denote the volume from experiments [

A plot of the thermal pressure (_{th}

(a) Thermal linear expansion at ambient pressure. The solid line denotes the regression fit from experimental data [_{0} is the length at 300 K. (b) Calculated temperature dependence of the thermal expansion coefficient. The results are shown for 0, 50, and 100 GPa.

The pressure difference between previous EOS based on experimental data and our EOS combined thermal pressure calculated by first-principles and the room-temperature compression from experimental data. The lines denote pressures of previous EOS [

Anharmonic effects on the Grüneisen parameter. The solid squares, circles, and triangles denote the calculated Grüneisen parameter at 1,000, 2,000, and 3,000 K using first-principles molecular dynamics calculations. Errors of calculated values are ∼20%. The dashed lines denote the curve fit using the least square method.

The model thermoelastic parameters of Ta. The Vinet equation of state was used to calculate the parameters of Ta. Key: _{T0}_{T0}′_{0}_{0}

_{0}^{3}) |
18.016 |

_{T0} |
194 |

_{T0}′ |
3.740(11) |

_{0}^{−5} K^{−1}) |
1.47(2) |

_{T}/δT)_{V}^{−1}) |
−0.0050(1) |

^{2}P/δT^{2)}V^{−7}GPa^{2}K^{−2}) |
2.11(31) |

Pressure-Temperature-Volume table for Ta from this study. The unit of pressure is GPa.

_{0} |
|||||||
---|---|---|---|---|---|---|---|

0.00 | 0.00 | 0.58 | 2.10 | 3.73 | 5.46 | 7.30 | 9.24 |

0.02 | 4.07 | 4.63 | 6.10 | 7.67 | 9.36 | 11.14 | 13.04 |

0.04 | 8.54 | 9.08 | 10.50 | 12.03 | 13.66 | 15.39 | 17.23 |

0.06 | 13.46 | 13.98 | 15.35 | 16.82 | 18.40 | 20.08 | 21.87 |

0.08 | 18.88 | 19.37 | 20.68 | 22.10 | 23.63 | 25.25 | 26.99 |

0.10 | 24.84 | 25.30 | 26.56 | 27.92 | 29.39 | 30.97 | 32.65 |

0.12 | 31.38 | 31.83 | 33.04 | 34.34 | 35.75 | 37.27 | 38.90 |

0.14 | 38.60 | 39.03 | 40.17 | 41.42 | 42.78 | 44.24 | 45.80 |

0.16 | 46.55 | 46.96 | 48.04 | 49.23 | 50.53 | 51.93 | 53.44 |

0.18 | 55.32 | 55.70 | 56.72 | 57.86 | 59.09 | 60.43 | 61.88 |

0.20 | 65.00 | 65.35 | 66.32 | 67.38 | 68.56 | 69.84 | 71.22 |

0.22 | 75.69 | 76.02 | 76.92 | 77.92 | 79.03 | 80.25 | 81.57 |

0.24 | 87.50 | 87.81 | 88.64 | 89.58 | 90.63 | 91.78 | 93.04 |

0.26 | 100.59 | 100.86 | 101.63 | 102.51 | 103.49 | 104.57 | 105.76 |