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First-Principles Calculations on Structural Property and Anisotropic Elasticity of γ_{1}-Ti_{4}Nb_{3}Al_{9} under Pressure

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Crystal Structure of ${\gamma}_{1}$-Ti${}_{4}$Nb${}_{3}$Al${}_{9}$ Phase

#### 2.2. Computational Details

#### 2.3. Calculations of Elastic Constants and Related Properties

## 3. Results and Discussion

#### 3.1. Bulk Properties at Zero Pressure

#### 3.2. Pressure Dependence of Structural Property

#### 3.3. Pressure Dependence of Elastic Properties

#### 3.4. Pressure Dependence of Acoustic and Related Properties

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**The ratios of lattice parameters as a function of pressure for the Ti${}_{4}$Nb${}_{3}$Al${}_{9}$ phase.

**Figure 4.**Difference between elastic constant ${C}_{66}$ and pressure P as a function of pressure for the Ti${}_{4}$Nb${}_{3}$Al${}_{9}$ phase.

**Figure 5.**Bulk (B), shear (G) and Young’s (E) moduli as a function of pressure for the Ti${}_{4}$Nb${}_{3}$Al${}_{9}$ phase.

**Figure 6.**Ratio of bulk modulus to shear modulus (

**a**) and Poisson’s ratio (

**b**) as a function of pressure for the Ti${}_{4}$Nb${}_{3}$Al${}_{9}$ phase.

**Figure 7.**Percentage anisotropy in compressibility (

**a**) and shear (

**b**), and universal anisotropy (

**c**) as a function of pressure for the Ti${}_{4}$Nb${}_{3}$Al${}_{9}$ phase.

**Figure 8.**Directional bulk modulus (

**a**) and Young’s modulus (

**b**) along principle crystallographic axes as a function of pressure for the Ti${}_{4}$Nb${}_{3}$Al${}_{9}$ phase.

**Figure 9.**Longitudinal and transverse sound velocities in [100] (

**a**), [001] (

**b**) and [110] (

**c**) directions as a function of pressure for the Ti${}_{4}$Nb${}_{3}$Al${}_{9}$ phase.

**Figure 10.**Minimum thermal conductivities in [100], [001] and [110] directions as a function of pressure for the Ti${}_{4}$Nb${}_{3}$Al${}_{9}$ phase.

**Figure 11.**Polycrystal longitudinal, transverse and average sound velocities as a function of pressure for the Ti${}_{4}$Nb${}_{3}$Al${}_{9}$ phase.

**Figure 12.**Polycrystal Debye temperature (

**a**) and minimum thermal conductivity (

**b**) as a function of pressure for the Ti${}_{4}$Nb${}_{3}$Al${}_{9}$ phase.

**Table 1.**Calculated lattice constants a, c (in Å) and elastic constants (in GPa) of Ti${}_{4}$Nb${}_{3}$Al${}_{9}$ at zero pressure.

Method | a | c | ${\mathit{C}}_{11}$ | ${\mathit{C}}_{12}$ | ${\mathit{C}}_{13}$ | ${\mathit{C}}_{33}$ | ${\mathit{C}}_{44}$ | ${\mathit{C}}_{66}$ |
---|---|---|---|---|---|---|---|---|

Present | 5.6510 | 8.2051 | 219.36 | 56.83 | 83.43 | 185.91 | 104.86 | 24.71 |

Exp. [4,5] | 5.58–5.84 | 8.15–8.45 | ||||||

Exp. [3] | 5.607 | 8.270 | ||||||

Theo. [17] | 5.651 | 8.205 | 222.71 | 60.27 | 87.99 | 187.36 | 104.77 | 23.06 |

**Table 2.**Optimized lattice parameters a, c (in Å), c/a, cell volume (in Å${}^{3}$) and mass density $\rho $ (in g/cm${}^{3}$) of Ti${}_{4}$Nb${}_{3}$Al${}_{9}$ under different pressures.

P | a | c | $\mathit{c}/\mathit{a}$ | V | $\mathit{\rho}$ |
---|---|---|---|---|---|

0 | 5.6510 | 8.2051 | 1.4520 | 262.02 | 4.5194 |

10 | 5.5224 | 8.0042 | 1.4494 | 244.10 | 4.8511 |

20 | 5.4209 | 7.8525 | 1.4486 | 230.75 | 5.1317 |

30 | 5.3379 | 7.7269 | 1.4475 | 220.16 | 5.3785 |

40 | 5.2685 | 7.6168 | 1.4457 | 211.42 | 5.6009 |

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**MDPI and ACS Style**

Zeng, X.; Peng, R.; Yu, Y.; Hu, Z.; Wen, Y.; Song, L.
First-Principles Calculations on Structural Property and Anisotropic Elasticity of *γ*_{1}-Ti_{4}Nb_{3}Al_{9} under Pressure. *Materials* **2018**, *11*, 2025.
https://doi.org/10.3390/ma11102025

**AMA Style**

Zeng X, Peng R, Yu Y, Hu Z, Wen Y, Song L.
First-Principles Calculations on Structural Property and Anisotropic Elasticity of *γ*_{1}-Ti_{4}Nb_{3}Al_{9} under Pressure. *Materials*. 2018; 11(10):2025.
https://doi.org/10.3390/ma11102025

**Chicago/Turabian Style**

Zeng, Xianshi, Rufang Peng, Yanlin Yu, Zuofu Hu, Yufeng Wen, and Lin Song.
2018. "First-Principles Calculations on Structural Property and Anisotropic Elasticity of *γ*_{1}-Ti_{4}Nb_{3}Al_{9} under Pressure" *Materials* 11, no. 10: 2025.
https://doi.org/10.3390/ma11102025