#
Microstructural Modeling of the Magnetization Process in Ni_{2}MnGa Alloy Polytwin Crystals

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Relations of Micromagnetism

## 3. Statement of the Problem

## 4. Results of Numerical Simulation

- (1)
- isotropic polycrystal—martensitic twins are randomly oriented (17 twinned crystals are located at angles ${\varphi}_{j}={0}^{\circ},\dots ,{90}^{\circ}$ according to Figure 2 and occupy equal areas);
- (2)
- texture-oriented polycrystal—there is a predominant direction of martensitic structures orientation (3 twinned crystals are located at angles ${\varphi}_{j}=40.{11}^{\circ},\phantom{\rule{3.33333pt}{0ex}}45.{84}^{\circ},\phantom{\rule{3.33333pt}{0ex}}51.{57}^{\circ}$ according to Figure 2 and occupy equal areas)—structure 1;
- (3)
- texture-oriented polycrystal—there is a predominant direction of martensitic structures orientation (5 twinned crystals are located at angles ${\varphi}_{j}=34.{38}^{\circ},\phantom{\rule{3.33333pt}{0ex}}40.{11}^{\circ},\phantom{\rule{3.33333pt}{0ex}}45.{84}^{\circ},\phantom{\rule{3.33333pt}{0ex}}51.{57}^{\circ},\phantom{\rule{3.33333pt}{0ex}}57.{30}^{\circ}$ and occupy equal areas)—structure 2.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Dependences of the average value of the magnetization projection on the axis along which the external magnetic field is directed on the applied field for different directions of application of the external magnetic field $\varphi $ (in degrees).

**Figure 3.**Simulated magnetization curves from article [21]. In this work, the angles between the applied magnetic field and the x axis were set in contrast of our case. If we consider the angles between this field and the y axis, then the curves will correspond to the following angles: A—${90}^{\circ}$, B—${70}^{\circ}$, C—${45}^{\circ}$, D—${20}^{\circ}$, E—${0}^{\circ}$. A1, A2 are the sections of the curve A; B1, B2 are the sections of the curve B and so on. These curves are given to qualitatively confirm the correctness of the results presented in Figure 2, and are discussed after discussing the curves in Figure 2.

**Figure 4.**Evolution of the magnetization vector upon application of an external magnetic field along the y axis: (

**a**)—${\tilde{H}}_{0}=0$, (

**b**)—${\tilde{H}}_{0}=0.25$, (

**c**)—${\tilde{H}}_{0}=0.3$, (

**d**)—${\tilde{H}}_{0}=0.32$, (

**e**)—${\tilde{H}}_{0}=0.33$, (

**f**)—${\tilde{H}}_{0}=1.5$.

**Figure 5.**Evolution of the magnetization vector when an external magnetic field is applied at an angle of $28.{65}^{\circ}$ to the y axis: (

**a**)—${\tilde{H}}_{0}=0$, (

**b**)—${\tilde{H}}_{0}=0.19$, (

**c**)—${\tilde{H}}_{0}=0.2$, (

**d**)—${\tilde{H}}_{0}=0.65$, (

**e**)—${\tilde{H}}_{0}=0.75$, (

**f**)—${\tilde{H}}_{0}=1.5$.

**Figure 6.**Evolution of the magnetization vector when an external magnetic field is applied at an angle of $45.{84}^{\circ}$ to the y axis: (

**a**)—${\tilde{H}}_{0}=0$, (

**b**)—${\tilde{H}}_{0}=0.15$, (

**c**)—${\tilde{H}}_{0}=0.16$, (

**d**)—${\tilde{H}}_{0}=0.17$, (

**e**)—${\tilde{H}}_{0}=1.1$, (

**f**)—${\tilde{H}}_{0}=1.5$.

**Figure 7.**Evolution of the magnetization vector upon application of an external magnetic field along the x axis: (

**a**)—${\tilde{H}}_{0}=0$, (

**b**)—${\tilde{H}}_{0}=0.11$, (

**c**)—${\tilde{H}}_{0}=0.13$, (

**d**)—${\tilde{H}}_{0}=1.5$.

**Figure 10.**Magnetization curves obtained numerically and with the generalized Fröhlich-Kenelly formula.

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**MDPI and ACS Style**

Rogovoy, A.A.; Stolbova, O.S.
Microstructural Modeling of the Magnetization Process in Ni_{2}MnGa Alloy Polytwin Crystals. *Magnetochemistry* **2022**, *8*, 78.
https://doi.org/10.3390/magnetochemistry8080078

**AMA Style**

Rogovoy AA, Stolbova OS.
Microstructural Modeling of the Magnetization Process in Ni_{2}MnGa Alloy Polytwin Crystals. *Magnetochemistry*. 2022; 8(8):78.
https://doi.org/10.3390/magnetochemistry8080078

**Chicago/Turabian Style**

Rogovoy, Anatoli A., and Olga S. Stolbova.
2022. "Microstructural Modeling of the Magnetization Process in Ni_{2}MnGa Alloy Polytwin Crystals" *Magnetochemistry* 8, no. 8: 78.
https://doi.org/10.3390/magnetochemistry8080078