#
Impact of Antiphase Boundaries on Structural, Magnetic and Vibrational Properties of Fe_{3}Al

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## Abstract

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## 1. Introduction

## 2. Methods

## 3. Results

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Visualizations of (super-)cells that were used in our theoretical study. Part (

**a**) shows a rhombohedral primitive unit cell of Fe${}_{3}$Al with 4 atoms and includes a naming convention of Fe sublattices when Fe${}^{\mathrm{II}}$ sites are twice more abundant than the Fe${}^{\mathrm{I}}$ sites. Part (

**b**) exhibits a 16-atom cube-shaped elementary cell containing 4 formula units. A 64-atom supercell, as an 1 × 1 × 4 multiple of the 16-atom elementary cell, is visualized in part (

**c**) and includes red vectors defining a shift characterizing the studied type of antiphase boundaries (APBs). When applying the APB shift to the central half of the 64-atom supercell in part (c) an APB-containing supercell shown in part (

**d**) is formed. To (i) keep the stoichiometry and (ii) apply the 〈111〉 shift to all atoms in the middle half of (c), one atomic plane is cyclically shifted as schematically marked by a series of green arrows (an APB shift applies also to this plane). There are two APB interfaces per each 64-atom supercell shown in part (c). One is formed by two adjacent layers containing both Fe${}^{\mathrm{I}}$ and Al atoms (the Fe${}^{\mathrm{I}}$-Al APB interface) and the second one consisting of two planes of Fe${}^{\mathrm{II}}$ atoms (the Fe${}^{\mathrm{II}}$-Fe${}^{\mathrm{II}}$ APB interface).

**Figure 2.**Local magnetic moments of Fe atoms (

**a**) and inter-layer distances (

**b**) for the 64-atom supercell with the examined antiphase boundaries. The values are compared with those in the bulk (horizontal dashed lines) and the positions of APBs are schematically indicated by red vertical dashed-dotted lines. The line in part (b) connecting the data points is added only to guide the eye.

**Figure 3.**Schematic visualizations of 192-atom supercells (1 × 1 × 12 multiples of the 16-atom cell shown in Figure 1b) that we used for examining possible long-range interactions between the two APB interfaces. The studied Fe${}^{\mathrm{II}}$-Fe${}^{\mathrm{II}}$ and Fe${}^{\mathrm{I}}$-Al APB interfaces were separated by one 16-atom Fe${}_{3}$Al unit, i.e., 4 atomic planes (

**a**), two 16-atom Fe${}_{3}$Al units (

**b**), three 16-atom Fe${}_{3}$Al units (

**c**), four 16-atom Fe${}_{3}$Al units (

**d**), five 16-atom Fe${}_{3}$Al units (

**e**) and six 16-atom Fe${}_{3}$Al units, i.e., 24 atomic planes (

**f**).

**Figure 4.**Computed properties of 192-atom 1 × 1 × 12 supercells of Fe${}_{3}$Al with different separation between the two APB interfaces (see Figure 3). In particular we present the APB interface energy (averaged over both types of interfaces) in part (

**a**), the total magnetic moment relative to the magnetic moment of the same amount of bulk Fe${}_{3}$Al without any APBs (

**b**), a relative volume with respect to that of APB-free Fe${}_{3}$Al bulk (

**c**) and lattice parameters (and their ratio) again relative to the corresponding values in the APB-free Fe${}_{3}$Al bulk (

**d**). The averaged APB interface energies obtained using the 192-atom supercells are compared with the value computed using the 64-atom supercell (shown in Figure 2d), which has the same separation of APBs interfaces but three times lower separation of periodically repeated images (black horizontal dashed line).

**Figure 5.**Calculated local magnetic moments of Fe atoms in 192-atom 1 × 1 × 12 supercells of Fe${}_{3}$Al with the APB interfaces (shown in Figure 3) separated by one 16-atom Fe${}_{3}$Al unit (

**a**), two units (

**b**), three (

**c**), four (

**d**), five (

**e**) and six 16-atom Fe${}_{3}$Al units (

**f**). For the separation of 2 units in part (b), the computed values (black circles) are compared with those (red triangles) shown in Figure 2a obtained for the 64-atom supercell presented in Figure 1d that has the same distance between the APBs interfaces.

**Figure 6.**Calculated phonon frequencies and the corresponding densities of vibrational states for the defect-free Fe${}_{3}$Al (

**a**) and Fe${}_{3}$Al with the studied two types of APB interfaces (

**b**).

**Figure 7.**The Helmholtz free energy F (

**a**) and vibrational entropy S (

**b**) of the defect-free bulk Fe${}_{3}$Al together with changes induced in them by the studied APBs in (

**c**,

**d**), respectively. Also shown are the harmonic phonon energy E (

**e**) and constant-volume heat capacity ${C}_{\mathrm{v}}$ (

**f**) of the perfect bulk Fe${}_{3}$Al accompanied by the APB-induced changes in parts (

**g**,

**h**), respectively. The elementary entity for defining one mol is the 64-atom supercell.

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**MDPI and ACS Style**

Friák, M.; Černý, M.; Všianská, M.; Šob, M.
Impact of Antiphase Boundaries on Structural, Magnetic and Vibrational Properties of Fe_{3}Al. *Materials* **2020**, *13*, 4884.
https://doi.org/10.3390/ma13214884

**AMA Style**

Friák M, Černý M, Všianská M, Šob M.
Impact of Antiphase Boundaries on Structural, Magnetic and Vibrational Properties of Fe_{3}Al. *Materials*. 2020; 13(21):4884.
https://doi.org/10.3390/ma13214884

**Chicago/Turabian Style**

Friák, Martin, Miroslav Černý, Monika Všianská, and Mojmír Šob.
2020. "Impact of Antiphase Boundaries on Structural, Magnetic and Vibrational Properties of Fe_{3}Al" *Materials* 13, no. 21: 4884.
https://doi.org/10.3390/ma13214884