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Exploring Crystal Structure, Hyperfine Parameters, and Magnetocaloric Effect in Iron-Rich Intermetallic Alloy with ThMn_{12}-Type Structure: A Comprehensive Investigation Using Experimental and DFT Calculation

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Techniques

## 3. Results and Discussion

#### 3.1. Crystal Structure

#### 3.2. Intrinsic Magnetic Properties

#### 3.3. Mössbauer Spectrometry and DFT Calculations

- The cut-off parameter was established as ${R}_{MT}\times {K}_{max}=$7, where ${K}_{max}$ denotes the magnitude of the largest K vector.
- We implemented ${G}_{max}$ = 12 Ry${}^{1/2}$ for Fourier potential expansion.
- 1200 k-points in the Brillouin zone.
- The muffin-tin radii ${R}_{MT}$ were set at 2.50, 2.14, and 2.17 Bohr for Pr, Fe, and Ti atoms, respectively.
- To distinguish between valence and core states, we adopted a cut-off energy of $E\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}$7 Ry
- Convergence was deemed achieved when the total energy difference was less than ${10}^{-4}$ Ry and the charge was below ${10}^{-4}$ electron charges.

#### 3.4. Magnetocaloric Performance

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The crystal structure of the hexagonal CaCu${}_{5}$ phase (on the left with black color) and the tetragonal ThMn${}_{12}$ phase (on the right with red color). In the middle is the relationship between the two structures.

**Figure 2.**Refined X-ray diffraction patterns of PFT compound at ambient temperature. The emergence of superstructure reflection peaks denoted by hkl (202), (002), (202) are indicative of the ThMn${}_{12}$-type structure $I4/mmm$.

**Figure 3.**Variation of the magnetization as a function of temperature at 0.13 T of the PFT alloy. The inset is the plot of $dM/dT$ versus T.

**Figure 4.**Magnetization $M\left(H\right)$ vs. magnetic field of PrFe${}_{11}$Ti measured at T = 10 K. The red line represents the fitted curve with the law of approach to saturation.

**Figure 5.**Mössbauer spectrum of PFT at $T=10\phantom{\rule{0.166667em}{0ex}}$K (

**left**) and $300\phantom{\rule{0.166667em}{0ex}}$K (

**right**). The magenta dots and the blue symbol represent the experimental data selected at 10 K and 300 K, respectively. The solid blue line represents the fit, and the solid colored lines represent the nine sextets utilized in the fitting process.

**Figure 8.**The isothermal magnetization curves as a function of the magnetic field. +10 K and +5 K mean the temperature step between two successive isotherms in the heating process.

**Figure 9.**Arrott plots (${M}^{2}$ vs. ${\mu}_{0}H/M$) for PFT compound measured at different temperatures.

**Figure 10.**Magnetic entropy change vs. temperature under several magnetic fields for PrFe${}_{11}$Ti compound.

**Table 1.**Crystallographic parameters from Rietveld refinement of PrFe${}_{11}$Ti: a and c are unit cell parameters, ($x,y,z$) are the atomic position, and ${\chi}^{2}$, ${R}_{B}$ are agreement factors.

Sample | PrFe${}_{11}$Ti | |||
---|---|---|---|---|

Space group | $I4/mmm$ | |||

Cell parameters | $a=b$ (Å) | 8.601(1) | ||

c (Å) | 4.789(1) | |||

$c/a$ | 0.556 | |||

V (Å${}^{3}$) | 354.308(1) | |||

Atom site | x | y | z | Atomic number |

Pr($2a$) | 0 | 0 | 0.344(1) | 2 |

Fe/Ti($8i$) | 0.360(1) | 0 | 0 | 6/2 |

Fe($8j$) | 0.274(1) | 0.5 | 0 | 8 |

Fe($8f$) | 0.25 | 0.25 | 0.25 | 8 |

Agreement | ${R}_{B}$ | 6.17 | ||

factors | ${R}_{F}$ | 7.5 | ||

${\chi}^{2}$ | 1.40 |

Compound | ${\mathit{T}}_{\mathit{C}}$ (K) | Ref. |
---|---|---|

YFe${}_{11}$Ti | 525 | [36] |

PrFe${}_{11}$Ti | 547 | [30] |

PrFe${}_{11}$Ti | 543 | This work |

NdFe${}_{11}$Ti | 552 | [24] |

SmFe${}_{11}$Ti | 589 | [37] |

GdFe${}_{11}$Ti | 604 | [25] |

**Table 3.**Wigner–Seitz cell volume (WSCV) and Mössbauer hyperfine parameters for PrFe${}_{11}$Ti: hyperfine field (${H}_{\mathrm{HF}}$), isomer shift ($\delta $), and quadrupole interaction ($2\epsilon $). $\langle \mathrm{HF}\rangle $ denotes the average of the hyperfine parameters.

${\mathit{\mu}}_{0}{\mathit{H}}_{\mathbf{HF}}$ (T) | $\mathit{\delta}$ (mm/s) | $2\mathit{\epsilon}$ (mm/s) | WSCV (Å${}^{3}$) | ||||
---|---|---|---|---|---|---|---|

T(K) | 10 | 300 | 10 | 300 | 10 | 300 | - |

Fe$\left\{8i\right\}$ | 31.2 | 27.1 | 0.08 | −0.08 | 0.13 | 0.09 | 13.1 |

Fe$\left\{8j\right\}$ | 28.7 | 23.8 | 0.01 | −0.10 | 0.11 | 0.09 | 11.6 |

Fe$\left\{8f\right\}$ | 26.3 | 21.8 | −0.01 | −0.12 | 0.06 | 0.04 | 10.6 |

Pr$\left\{2a\right\}$ | - | - | - | - | - | - | 23.3 |

$\langle \mathrm{HF}\rangle $ | 28.5 | 24.2 | 0.03 | −0.09 | 0.10 | 0.08 | - |

**Table 4.**Values of applied magnetic field ${\mu}_{0}\Delta H$, Curie temperature ${T}_{C}$, maximum of entropy variation $|-\Delta {S}_{M}^{\mathrm{max}}|$, relative cooling power (RCP), and the temperature-averaged entropy change parameters (TEC) of PrFe${}_{11}$Ti compared with other magnetic materials.

Sample | ${\mathit{\mu}}_{0}\Delta \mathit{H}$ | ${\mathit{T}}_{\mathit{C}}$ | $-\Delta {\mathit{S}}_{\mathit{M}}^{\mathbf{max}}$ | RCP | TEC | $\frac{\mathbf{TEC}}{{\mathit{\mu}}_{0}\Delta \mathit{H}}$ | Ref. |
---|---|---|---|---|---|---|---|

(T) | (K) | J/(kg·K) | (J/kg) | J(K·kg) ${}^{-\mathbf{1}}$ | J(K·kg·T) ${}^{-\mathbf{1}}$ | ||

PrFe${}_{11}$Ti | 0.5 | 545 | 1.1 | 23 | 1.1 | 2.2 | This |

1 | 1.9 | 48 | 1.9 | 1.9 | work | ||

1.5 | 2.5 | 70 | 2.44 | 1.6 | |||

GdFe${}_{6}$Al${}_{6}$ | 2 | 298 | 0.56 | 25 | - | - | [4] |

NdFe${}_{9.5}$Mo${}_{2.5}$ | 5 | 302 | 2.38 | - | - | - | [23] |

NdFe${}_{11}$Ti | 1.5 | 552 | 1.5 | - | - | - | [24] |

SmFe${}_{10}$V${}_{2}$ | 1.5 | 603 | 1.6 | - | - | - | |

GdFe${}_{10}$Cr${}_{2}$ | 1.5 | 580 | 1.82 | 10.5 | - | - | [25] |

ErFe${}_{9}$Mn${}_{3}$ | 1 | 312.5 | 0.7 | - | - | - | [26] |

5 | 1.92 | - | - | - |

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**MDPI and ACS Style**

Horcheni, J.; Jaballah, H.; Dhahri, E.; Bessais, L.
Exploring Crystal Structure, Hyperfine Parameters, and Magnetocaloric Effect in Iron-Rich Intermetallic Alloy with ThMn_{12}-Type Structure: A Comprehensive Investigation Using Experimental and DFT Calculation. *Magnetochemistry* **2023**, *9*, 230.
https://doi.org/10.3390/magnetochemistry9110230

**AMA Style**

Horcheni J, Jaballah H, Dhahri E, Bessais L.
Exploring Crystal Structure, Hyperfine Parameters, and Magnetocaloric Effect in Iron-Rich Intermetallic Alloy with ThMn_{12}-Type Structure: A Comprehensive Investigation Using Experimental and DFT Calculation. *Magnetochemistry*. 2023; 9(11):230.
https://doi.org/10.3390/magnetochemistry9110230

**Chicago/Turabian Style**

Horcheni, Jihed, Hamdi Jaballah, Essebti Dhahri, and Lotfi Bessais.
2023. "Exploring Crystal Structure, Hyperfine Parameters, and Magnetocaloric Effect in Iron-Rich Intermetallic Alloy with ThMn_{12}-Type Structure: A Comprehensive Investigation Using Experimental and DFT Calculation" *Magnetochemistry* 9, no. 11: 230.
https://doi.org/10.3390/magnetochemistry9110230