# Crystal-Structure Analysis with Moments of the Density-of-States: Application to Intermetallic Topologically Close-Packed Phases

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Moments of the Density-of-States

**Figure 1.**The third and fourth moment gives rise to a skewing (

**left**) and a bimodal shape (

**right**) of the DOS.

#### 2.2. Computational Details

**Table 1.**Frank–Kasper polyhedra with coordination numbers Z of the considered TCP phases, ordered by increasing average Z. The list indicates the multiplicity of the different Wyckoff positions with the same Z. The values in parentheses indicate coordination polyhedra that are not Frank–Kasper polyhedra.

Structure | fcc | χ | C14 | C15 | C36 | μ | A15 | σ | bcc |
---|---|---|---|---|---|---|---|---|---|

Z12 | (1) | 12 | 2, 6 | 4 | 4, 6, 6 | 1, 6 | 2 | 2, 8 | - |

Z13 | - | (12) | - | - | - | - | - | - | - |

Z14 | - | - | - | - | - | 2 | 6 | 8, 8 | (1) |

Z15 | - | - | - | - | - | 2 | - | 4 | - |

Z16 | - | 1, 4 | 4 | 2 | 4, 4 | 2 | - | - | - |

〈Z〉 | 12.00 | 13.10 | 13.33 | 13.33 | 13.33 | 13.39 | 13.50 | 13.57 | 14.00 |

#### 2.3. Moments of TCP Phases

**Figure 2.**Second to sixth moments (colors) computed with a canonical d-valent TB model for DFT-relaxed crystal structures of Ta. Several values of the same moment (symbols) reflect the different environments of different Wyckoff positions. (This color coding is used in all following figures.)

**Table 2.**Similarity matrix of TCP phases in terms of the distance in moments space ${\Delta}_{ij}$. The entries are ordered by increasing difference to bcc. The symmetric lower-left part is omitted for brevity. The grey scale reflects the numerical entries and is included to guide the eye.

${\Delta}_{ij}^{\left(6\right)}\xb7100$ | bcc | χ | σ | A15 | fcc | μ | C14 | C36 | C15 |
---|---|---|---|---|---|---|---|---|---|

bcc | 0.000 | 0.226 | 0.304 | 0.362 | 0.885 | 1.365 | 1.619 | 1.982 | 2.301 |

χ | - | 0.000 | 0.220 | 0.315 | 0.808 | 1.301 | 1.550 | 1.922 | 2.247 |

σ | - | - | 0.000 | 0.290 | 0.662 | 1.124 | 1.378 | 1.747 | 2.069 |

A15 | - | - | - | 0.000 | 0.818 | 1.161 | 1.403 | 1.769 | 2.091 |

fcc | - | - | - | - | 0.000 | 0.655 | 0.886 | 1.236 | 1.546 |

μ | - | - | - | - | - | 0.000 | 0.257 | 0.623 | 0.947 |

C14 | - | - | - | - | - | - | 0.000 | 0.375 | 0.702 |

C36 | - | - | - | - | - | - | - | 0.000 | 0.327 |

C15 | - | - | - | - | - | - | - | - | 0.000 |

## 3. Moments Analysis of Volume Changes and Internal Relaxations

#### 3.1. Influence of Band Filling across TM Series

**Figure 3.**Trend of second to sixth moments of unary C14 (

**top left**), C15 (

**top right**), A15 (

**bottom left**) and σ (

**bottom right**) phases across the 5d TM series. The values were computed from the DFT-relaxed unit cells and scaled such that $\langle {\mu}^{\left(2\right)}\rangle =1$. (for color coding, see Figure 2.)

**Figure 4.**Comparison of second to sixth moments of the χ phase across the 4d (

**left**) and 5d (

**right**) TM series using the DFT-relaxed unit cells with scaling such that $\langle {\mu}^{\left(2\right)}\rangle =1$. (for color coding, see Figure 2.)

#### 3.2. Atomic-Volume Differences in Compounds: V-Ta

**Figure 5.**Influence of considerable atomic-size difference on moments of DFT-relaxed unit cells of A15 (

**left**) and C15 (

**middle**). Variation of second moment for different Wyckoff positions in A15 and C15 (

**right**). (for color coding, see Figure 2.)

**Figure 6.**Variation of moments due to internal relaxation in the V-Ta compound system for the TCP phases C14 (

**left**) and σ (

**right**). The moments of the DFT-relaxed unit cells were scaled such that $\langle {\mu}^{\left(2\right)}\rangle =1$. (for color coding, see Figure 2.)

#### 3.3. Influence of Magnetism: Fe${}_{2}$Nb-C14

**Figure 7.**Variation of moments of C14-Fe${}_{2}$Nb (

**left**) with spin configuration of the Wyckoff sites 2a, 6h and 4f (

**right**), as well as site-resolved close-up on the second moment (

**middle**). The ordering is according to structural stability [51] starting with the energetically most favorable configuration (UD)(UD)U that is also indicated in the crystal structure (right). (for color coding, see Figure 2.)

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Hammerschmidt, T.; Ladines, A.N.; Koßmann, J.; Drautz, R. Crystal-Structure Analysis with Moments of the Density-of-States: Application to Intermetallic Topologically Close-Packed Phases. *Crystals* **2016**, *6*, 18.
https://doi.org/10.3390/cryst6020018

**AMA Style**

Hammerschmidt T, Ladines AN, Koßmann J, Drautz R. Crystal-Structure Analysis with Moments of the Density-of-States: Application to Intermetallic Topologically Close-Packed Phases. *Crystals*. 2016; 6(2):18.
https://doi.org/10.3390/cryst6020018

**Chicago/Turabian Style**

Hammerschmidt, Thomas, Alvin Noe Ladines, Jörg Koßmann, and Ralf Drautz. 2016. "Crystal-Structure Analysis with Moments of the Density-of-States: Application to Intermetallic Topologically Close-Packed Phases" *Crystals* 6, no. 2: 18.
https://doi.org/10.3390/cryst6020018