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Simulating the Hysteretic Characteristics of Hard Magnetic Materials Based on Nd_{2}Fe_{14}B and Ce_{2}Fe_{14}B Intermetallics

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## Abstract

**:**

_{2}Fe

_{14}B intermetallic, like Nd

_{2}Fe

_{14}B, has the tetragonal Nd

_{2}Fe

_{14}B-type structure (space group P4

_{2}/mnm), in which Ce ions have a mixed-valence state characterized by the coexistence of trivalent 4f

^{1}and tetravalent 4f

^{0}electron states. Despite the fact that the saturation magnetization, magnetic anisotropy field, and Curie temperature of the Ce

_{2}Fe

_{14}B intermetallic are substantially lower than those of Nd

_{2}Fe

_{14}B and Pr

_{2}Fe

_{14}B, Ce

_{2}Fe

_{14}B retains the capacity of being able to be used in the manufacturing of rare-earth permanent magnets. Moreover, at low temperatures, the anisotropy field of Се

_{2}Fe

_{14}B is higher than that of Nd

_{2}Fe

_{14}B, and Се

_{2}Fe

_{14}B does not undergo the spin-reorientation transition. In this respect, studies of (Nd, Ce)-Fe-B alloys, which are intended for the improvement of the service characteristics-to-cost ratio, are very relevant. A model and algorithm for calculating the hysteresis loops of uniaxial hard magnetic materials with allowance for the K

_{1}and K

_{2}(K

_{2}> 0 and K

_{1}> 0 and K

_{1}< 0) magnetic anisotropy constants were developed and allowed us to obtain data on their effect on the parameters of hysteresis loops for a wide temperature range (0–300 K). The simulation and analysis of hysteresis loops of the quasi-ternary intermetallics (Nd

_{1−х}Се

_{х})

_{2}Fe

_{14}B (х = 0–1) was performed. Results of the simulation indicate that the alloying of the Nd

_{2}Fe

_{14}B intermetallic with Ce to x = 0.94 (1) does not completely eliminate the negative effect of spin-reorientation phase transition on the residual magnetization of the (Nd

_{1−х}Ce

_{х})

_{2}Fe

_{14}B intermetallic and (2) slightly decreases the slope of magnetization reversal curve.

## 1. Introduction

_{2}Fe

_{14}B-type structure (space group P4

_{2}/mnm) [3]. The unit cell contains four formula units comprising 68 atoms: there are six crystallographic iron sites (16k

_{1}, 16k

_{2}, 8j

_{1}, 8j

_{2}, 4c, 4e), two rare-earth metal sites (4f, 4g), and one boron site (4g). Table 1 shows the lattice parameters of the R

_{2}Fe

_{14}B compounds with R = Nd, Pr, and Ce and their principal magnetic characteristics. This shows that both the a and c lattice parameters of the Ce

_{2}Fe

_{14}B compound are slightly lower than those of the Nd

_{2}Fe

_{14}B and Pr

_{2}Fe

_{14}B compounds [4].

_{2}Fe

_{14}B compound is dominated by the rare-earth atoms that occupy two inequivalent sites, 4f and 4g, of the tetragonal structure [8,9]. One Nd site (g) strongly prefers the [001] direction at ambient temperature and dictates the macroscopic easy-axis direction. The other Nd site (f) (containing half of all the Nd atoms) reduces the intrinsic stability by favoring alignment along [110]-type directions (basal plane). The results indicate that coercivity may be enhanced by preferential chemical doping of Nd f sites. Nd

_{2}Fe

_{14}B is characterized by the uniaxial state at temperatures above the spin-reorientation temperature T

_{sr}. Ce

_{2}Fe

_{14}B exhibits the uniaxial magnetic anisotropy over the whole temperature range of ferromagnetic ordering [10]. Its uniaxial anisotropy is higher than that of R

_{2}Fe

_{14}B with R =La, Lu, Y [10], but lower than that of Nd

_{2}Fe

_{14}B at temperatures substantially higher than the spin-reorientation temperature. It was predicted [11] that, theoretically, Ce atoms in the (Nd

_{1−x}Ce

_{x})

_{2}Fe

_{14}B compounds occupy the 4g positions (large in volume); this is explained by atomic size effects. However according to [8], 4f position (smaller in volume) is preferred for the Ce atoms.

_{1−x}Ce

_{x})

_{2}Fe

_{14}B. At present, no experimental data confirming the assumption are available. In turn, when assuming that Ce atoms prefer 4g positions, it is possible to conclude that the cone opening will be smaller for low substitutions of Ce for Nd. Because of this, T

_{sr}varies slightly for low Ce contents. A small change in T

_{sr}for the Ce substitution to x = 0.3 was observed in [12]. According to the data from [8], the magnetocrystalline anisotropy of Nd

_{2}Fe

_{14}B remains when Ce substitutes for Nd up to 20%. The further increase in the Ce content decreases the uniaxial magnetic anisotropy energy.

_{2}Fe

_{14}B is attributed mainly to the magnetism of Fe [8]. Alloying Ce on R-sites in (Nd

_{1−x}Ce

_{x})

_{2}Fe

_{14}B barely affects the Fe moments. Nd magnetization changes orientation at x > 0.5, with a larger antiferromagnetic moment on 4f sites compared to 4g sites. Such a transition of the Nd magnetic state causes an overall reduction of the net magnetization of the cell, and is the reason for instability at higher percentage Ce [11]. The magnetic moment of rare-earth sublattice is mainly determined by Nd atoms. The temperature behavior of magnetization of R

_{2}Fe

_{14}B with R = Nd, Ce was studied in [12].

_{2}Fe

_{14}B and Ce

_{2}Fe

_{14}B intermetallics differ substantially although the compounds have the same crystal structure. The existence of the CeFe

_{2}phase determines the principal difference in the ternary Nd-Fe-B and Ce-Fe-B phase diagrams [3,13]. Cerium ions in the Ce

_{2}Fe

_{14}B intermetallic have a mixed-valence state, namely, the trivalent 4f

^{1}and tetravalent 4f

^{0}electron states coexist [14].

_{2}Fe

_{14}B alloys is due to the lower magnetic properties of Ce

_{2}Fe

_{14}B as compared to those of Nd

_{2}Fe

_{14}B (see Table 1). However, according to data from [18,19], an anomalous increase in the coercive force was found by studying the effect of Ce substitution for Nd on the magnetic properties and microstructure of sintered magnets. Pathak et al. [20] reported that the substitution of 20% Ce for Nd in the ternary Nd

_{2}Fe

_{14}B alloy allowed the authors to reach a sufficiently high coercive force (H

_{ci}= 10 kOe), which exceeds that of Nd

_{2}Fe

_{14}B (H

_{ci}= 8.3 kOe).

_{2}Fe

_{14}B compound at 4.2 K is in the [110] plane and makes the angle θ ≈ 30° with the c axis. As the temperature increases, the transition to the collinear structure takes place at the spin-reorientation temperature T

_{sr}= 135–138 K [21]. Below this temperature, the magnetic moment deviates from the c axis, the first magnetic anisotropy constant K

_{1}passes zero and changes the sign from positive to negative, whereas the second magnetic anisotropy constant remains positive (K

_{2}> 0). As a result, below T

_{sr}, the experimental magnetization reversal curves in negative magnetic fields exhibit a bending, which increases with decreasing temperature. In this case, the residual magnetization and maximum energy product decrease abruptly.

_{2}Fe

_{14}B are substantially lower than those of Nd

_{2}Fe

_{14}B, whereas, at cryogenic temperatures, the magnetic anisotropy field of Се

_{2}Fe

_{14}B is markedly higher than that of Nd

_{2}Fe

_{14}B. Moreover, it is of importance that Ce

_{2}Fe

_{14}B does not have a spin-reorientation transition. Thus, it is reasonable to expect that the partial substitution of Ce for Nd in the Nd

_{2}Fe

_{14}B compound can lead to the improvement of the hysteretic characteristics of permanent magnets based on the quasi-ternary (Nd, Ce)

_{2}Fe

_{14}B intermetallics.

_{2−x}Се

_{x}Fe

_{14}B single crystals are available in [22], where the evolution of the spin-reorientation temperature as a function of the Ce concentration up to x = 0.4 is considered. The spin-reorientation temperature decreases by only about 6% when Ce substitutes for 38% of Nd. It is shown that T

_{sr}is suppressed much more rapidly for higher x. It is likely that the population of REM sites (4g or 4f) is responsible for the spin-reorientation [22].

_{1−x}Ce

_{x})

_{2}Fe

_{14}B intermetallics in order to determine the optimum alloyed compositions. The simulation and analysis of hysteresis loops of the (Nd

_{1−x}Ce

_{x})

_{2}Fe

_{14}B (x = 0–1) compounds are performed for a temperature range of 0 to 300 K.

## 2. Algorithm and Model for Calculating the Hysteresis Loops of Hard Magnetic Materials with the Uniaxial Tetragonal Lattice

_{s}at each point of a crystalline ferromagnet is simultaneously oriented along certain crystallographic directions (EMAs). In order to rotate I

_{s}to another direction, magnetic field

**H**should be applied along this direction, and work should be done. This work makes sense of the anisotropy energy Е

_{а}, which, for magnets with the uniaxial tetragonal lattice, is given by the expression:

_{1}and К

_{2}are the first and second magnetic anisotropy constants, respectively, and φ

_{a}is the angle made by the EMA and I

**. The anisotropy constants are proportional to the work, which should be done to rotate the magnetization from the EMA direction to the hard magnetization axis direction.**

_{s}**also determined by the magnetic field energy:**

_{s}_{H}is the angle made by the vectors. In the end, I

**takes a position corresponding to the minimum summary energy:**

_{s}_{1−x}Ce

_{x})

_{2}Fe

_{14}B intermetallics with the uniaxial tetragonal lattice, the following initial parameters are inputted: α is the angle made by an arbitrary plane and the X-axis [100] and the external magnetic field

**H**is applied in this plane, θ

_{H}is the angle made by the H field direction and Z-axis [001]. The following parameters are counted and outputted: θ is the angle made by the Z-axis [0001] and EMA, φ

_{a}is the angle made by the EMA and

**I**

_{s}; and angle φ

_{H.}

**I**

_{s}for an arbitrary vector

**H**, at which the total energy is minimal (Equation (3)), allows us to calculate the projection of the magnetization on the field direction I = I

**сos(φ**

_{s}_{H}) and to construct the magnetization curve I = I(Н).

## 3. Results and Discussion

#### 3.1. Determination of Magnetic Anisotropy Constants, Normalized Ratio of Anisotropy Constants (K_{2}/|K_{1}|), and Angle of the EMA cone in Calculating Hysteresis Loops of the (Nd_{1−x} Ce_{x})_{2}Fe_{14}B Intermetallics with 0 ≤ x ≤1

_{1}and K

_{2}of Nd

_{2}Fe

_{14}B in a temperature range of 0–500 K are available in [24]. The sign of K

_{1}alternates at the spin-reorientation transition temperature T

_{sr}= 135 K. Below this temperature, the preferred direction of EMA begins to deviate from the c axis direction (Z-axis [001]) of the tetragonal crystal lattice, and the angle (θ) of the EMA cone for each temperature is given by the expression:

_{1}and K

_{2}of Ce

_{2}Fe

_{14}B in a temperature range of 0–300 K are available in [25]. Unlike the magnetic anisotropy constants of Nd

_{2}Fe

_{14}B, K

_{1}and K

_{2}of Ce

_{2}Fe

_{14}B remain positive within the 0–300 К temperature range. The absolute values of K

_{1}and K

_{2}of Ce

_{2}Fe

_{14}B are substantially lower than those of Nd

_{2}Fe

_{14}B.

_{1−x}Ce

_{x})

_{2}Fe

_{14}B with 0 ≤ x ≤1 at different temperatures, the literature data on the anisotropy constants of Nd

_{2}Fe

_{14}B and Ce

_{2}Fe

_{14}B and the following linear expressions were used:

_{2}to К

_{1}and the θ angle (made by the EMA and c axis) were calculated by the expressions:

_{1−x}Ce

_{x})

_{2}Fe

_{14}B were considered: (1) x = 0, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50, 0.55, 0.60, 0.65, 0.70, 0.75; (2) x = 0.80, 0.85; 0.90, 0.91, 0.92 and (3) x = 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99, 1.00.

_{1−x}Ce

_{x})

_{2}Fe

_{14}B with x = 0.05, 0.90, and 0.97 are given in Figure 1, as an example. The compositional dependences of the spin-reorientation temperature (T

_{SR}) of (Nd

_{1−x}Ce

_{x})

_{2}Fe

_{14}B are given in Figure 2.

_{1}and К

_{2}constants decrease on average by ~75% and ~99%, respectively. The temperature corresponding to the maximum of the normalized К

_{2}/|К

_{1}| ratio decreases from 135 to 63 K. The temperature T

_{sr}also decreases from 135 to 63 K. The highest value of the θ angle at 0 K decreases from 30.15° (x = 0) to 24.77° (x = 0.75).

_{1−х}Ce

_{х})

_{2}Fe

_{14}B increases in the range x = 0.80–0.92 (second composition range), the К

_{1}and К

_{2}values additionally decrease on average by ~61% and ~63%, respectively. In turn, the temperature corresponding to the maximum of normalized К

_{2}/|К

_{1}| ratio shifts from 59 to 25 K. The temperature Т

_{sr}also decreases from 59 to 25 K. The largest value of the θ angle at 0 K monotonically decreases from 23.12° (x = 0.80) to 11.03° (x = 0.92).

_{1−х}Ce

_{х})

_{2}Fe

_{14}B intermetallic increases from x = 0.93 to x = 1.00, the anisotropy constant К

_{2}additionally decreases by an average ~35%, whereas the К

_{1}constant changes the sign from negative to positive. The Т

_{sr}temperature decreases from 13 to 0 K as the cerium content increases to x = 0.94. The temperature corresponding to the maximum of normalized К

_{2}/|К

_{1}| ratio also decreases from 13 to 0 K. The highest value of the θ angle at 0 K decreases from 7.34° to 0°(at x ≥ 0.94).

#### 3.2. Simulation of Magnetization Curves and Hysteresis Loops of (Nd_{1−x} Ce_{x})_{2}Fe_{14}B with x = 0.00–1.00

_{1−x}Ce

_{x})

_{2}Fe

_{14}B with x = 0–1.00 show that, as the applied magnetization reversing field reaches the coercive force Н

_{c}(at which the abrupt overturn of I

_{S}takes place), the decrease in the hysteresis loop squareness is observed; the “rounding” becomes more substantial as the temperature decreases. Below Т

_{sr}, the decrease starts in the positive fields. Figure 3 shows the temperature dependences of the normalized and unnormalized residual magnetization (I

_{r}/I

_{s}and I

_{r}) for the three composition ranges of the (Nd

_{1−х}Ce

_{х})

_{2}Fe

_{14}B compounds.

_{r}/I

_{s}ratio monotonically decreases with decreasing temperature. This is caused by the deviation of I

_{s}from the EMA in the applied magnetic field H and the transition to the EMA cone below Т

_{sr}. Only for the high-cerium contents (x ≥ 0.94), the I

_{r}/I

_{s}ratio remains unchanged and equal to 0.9999 (<1.0000) for a certain temperature range.

_{sr}(x), I

_{r}/I

_{s}remains unchanged and equal to 0.9999 for all these compositions. This means that, after saturation, the magnetization remains parallel to the EMA and Z-axis as the external field decreases to zero. The monotonic progressive decrease in I

_{r}/I

_{s}is observed simultaneously with decreasing temperature below T

_{sr}(x) and Ce content x. For each Ce content x, the temperature dependences I

_{r}(T, x) have the maximum value in the range of Т

_{sr}(x) (Figure 3b). The monotonic shift of the I

_{r}maximum to low temperatures from I

_{r}(T = 135 K, x = 0) =191.6 A m

^{2}/kg to I

_{r}(T = 102 K, x = 0.75) = 161.5 A m

^{2}/kg correlates with a similar shift of the maximum of normalized K

_{2}/|K

_{1}| ratio and a decrease in T

_{sr}(x) with increasing cerium content x (Figure 1 and Figure 2). The maximum in these dependences is related to the competition of two physical phenomena, such as the monotonic increase in the total magnetic moment of R

_{2}Fe

_{14}B intermetallics (where R = Nd or Ce) with decreasing temperature and the spin-reorientation below the T

_{sr}temperature.

_{sr}(x), I

_{r}/I

_{s}remains equal to 0.9999 for all these compositions, i.e., the magnetization after saturation remains parallel to the EMA and Z-axis as the magnetizing field decreases to zero. The monotonic and progressive decrease in I

_{r}/I

_{s}is observed with simultaneously decreasing temperature below T

_{sr}(x) and Ce content x. For each Ce content x, the temperature dependences I

_{r}(T, x) also have the maximum in the range of Т

_{sr}(x) (Figure 3d). The monotonic shift of the I

_{r}maximum to the low temperature range from I

_{r}(T = 100 K, x = 0.80) =159.6 A m

^{2}/kg to I

_{r}(T = 25 K, x = 0.92) = 156.6 A m

^{2}/kg correlates with the similar shift of the normalized K

_{2}/|K

_{1}| ratio and a decrease in T

_{sr}(x) with increasing cerium content x (Figure 1 and Figure 2). However, in this range of cerium concentrations, the temperature dependences of I

_{r}overlap; the overlapping was not observed in the range x = 0–0.75 (Figure 3d). This interesting and anomalous change in I

_{r}is also related to the “stronger” competition of a monotonic increase in the total magnetic moment of R

_{2}Fe

_{14}B intermetallics (where R = Nd or Ce) with decreasing temperature and the spin-reorientation below T

_{sr}. Owing to the nature of the change in magnetic properties, the composition range x = 0.80–0.92 is intermediate between neodymium-based intermetallics and cerium-based intermetallics R

_{2}Fe

_{14}B.

_{1−х}Ce

_{х})

_{2}Fe

_{14}B intermetallics with x = 0.93–1.00. As the temperature decreases from 300 K to 0 K, I

_{r}/I

_{s}remains equal to 0.9999 for all alloys with x ≥ 0.94, i.e., after saturation, the magnetization remains parallel to the EMA and Z-axis as the magnetizing field decreases to zero. Only for the alloys with 0.93 ≤ x< 0.94, the monotonic and progressive decrease in I

_{r}/I

_{s}is observed as the simultaneous decrease in temperature below T

_{sr}(x) and decrease in the Ce concentration take place. As x increases from 0.94 to 1.00, the considered maximum of I

_{r}shifts to zero and degenerates owing to the decreasing effect of the spin-reorientation. This effect also correlates with a similar shift of the normalized K

_{2}/|K

_{1}| ratio and a decrease in T

_{sr}(x) with increasing cerium concentration x (Figure 1 and Figure 2).

_{2}Fe

_{14}B and is characterized by anomalous change in the residual magnetization.

_{c}of the (Nd

_{1−х}Ce

_{х})

_{2}Fe

_{14}B compounds with x = 0–1.00.

_{c}reaches the maximum value in the range of Т

_{sr}(x) (Figure 5a). The monotonic shift of the Н

_{c}maximum to the low-temperature range from Н

_{c}(T = 135 K, x = 0) =7.2 MA/m to Н

_{c}(T = 102 K, x = 0.75) = 3.2 MA/m correlates with a similar shift of the maximum of normalized K

_{2}/|K

_{1}| ratio, shift of the maximum of I

_{r}, and a decrease in T

_{sr}(x) with increasing cerium concentration x (Figure 1 and Figure 3). The maximum in these dependences is related to the competition of two physical phenomena, such as the monotonic increase in the magnetic anisotropy field H

_{A}of R

_{2}Fe

_{14}B intermetallics (where R = Nd or Ce) with decreasing temperature and the spin-reorientation below T

_{sr}, that facilitates the magnetization reversal process in an external magnetic field.

_{c}reaches the maximum value in a temperature range of Т

_{sr}(x). The monotonic shift of the Н

_{c}maximum to the low-temperature range from Н

_{c}(T = 100 K, x = 0.80) =2.9 MA/m to Н

_{c}(T = 25 K, x = 0.92) = 2.2 MA/m correlates with a similar shift of the maximum of the normalized K

_{2}/|K

_{1}| ratio, a shift of the maximum of I

_{R}, and a decrease in T

_{sr}(x) with increasing cerium concentration x (Figure 1, Figure 2 and Figure 3).

_{c}progressively shifts from 25 to ~10 K. For x > 0.94, the maximum shifts to 0 K and degenerates. This effect also correlates with a similar shift of the normalized K

_{2}/|K

_{1}| ratio, shift of the maximum of I

_{r}, and a decrease in T

_{sr}(x) with increasing cerium concentration x (Figure 1 and Figure 3).

## 4. Conclusions

_{1−х}Ce

_{х})

_{2}Fe

_{14}B compounds with х = 0–1.00 and their hysteresis loops in a wide temperature range of 300 to 0 K.

_{2}Fe

_{14}B with cerium to its contents x = 0.94 (1) does not lead to the complete elimination of the negative effect of spin-reorientation phase transition on the residual magnetization of the (Nd

_{1−х}Ce

_{х})

_{2}Fe

_{14}B compounds and (2) slightly decreases the slope of the magnetization curve and almost does not lead to the improvement of the squareness of the back of hysteresis loop.

_{1−х}Ce

_{х})

_{2}Fe

_{14}B with cerium to its contents x = 0.94 does not allow the temperature stability of hysteretic characteristics of (Nd

_{1−х}Ce

_{х})

_{2}Fe

_{14}B-based permanent magnets to be increased in order to ensure their operation at low temperatures without losing magnetic properties.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Temperature dependences of the (

**a**) К

_{1}and К

_{2}magnetic anisotropy constants, (

**b**) normalized К

_{2}/|К

_{1}| ratio, and (

**c**) θ angle for (Nd

_{1−x}Ce

_{x})

_{2}Fe

_{14}B with x = 0.05 (first composition range), 0.90 (second composition range), and 0.97 (third composition range).

**Figure 2.**Compositional dependences of spin-reorientation temperature (T

_{sr}) of (Nd

_{1−x}Ce

_{x})

_{2}Fe

_{14}B: (blue) hypothetical (linear) trend and (red) results of simulation (this work).

**Figure 3.**Temperature dependences of the normalized and unnormalized residual magnetization (I

_{r}/I

_{s}and I

_{r}) of the (Nd

_{1−х}Ce

_{х})

_{2}Fe

_{14}B compounds with different x: (

**a**) and (

**b**) 0, 0.15, 0.25, 0.35, 0.45, 0.55, 0.65, 0.75; (

**c**) and (

**d**) 0.80, 0.85, 0.90, 0.91, 0.92; (

**e**) and (

**f**) 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99, 1.00.

**Figure 4.**Compositional dependence of the residual magnetization I

_{r}for the (Nd

_{1−х}Ce

_{х})

_{2}Fe

_{14}B compounds at T = 0 K.

**Figure 5.**Temperature dependences of the coercive force Н

_{c}of the (Nd

_{1−х}Ce

_{х})

_{2}Fe

_{14}B compounds with different x: (

**a**) 0, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40, 0.45, 0.50, 0.55, 0.60, 0.65, 0.70, 0.75; (

**b**) 0.80, 0.85, 0.90, 0.91, 0.92; and (

**c**) 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99, 1.00.

**Table 1.**Saturation magnetization I

_{s}, magnetic anisotropy field H

_{A}, lattice parameters a and c, and Curie temperature T

_{C}of the R

_{2}Fe

_{14}B compounds with R = Ce, Pr, Nd at room temperature.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kolchugina, N.B.; Zheleznyi, M.V.; Savchenko, A.G.; Menushenkov, V.P.; Burkhanov, G.S.; Koshkid’ko, Y.S.; Ćwik, J.; Dormidontov, N.A.; Skotnicova, K.; Kursa, M.; Prokofev, P.A. Simulating the Hysteretic Characteristics of Hard Magnetic Materials Based on Nd_{2}Fe_{14}B and Ce_{2}Fe_{14}B Intermetallics. *Crystals* **2020**, *10*, 518.
https://doi.org/10.3390/cryst10060518

**AMA Style**

Kolchugina NB, Zheleznyi MV, Savchenko AG, Menushenkov VP, Burkhanov GS, Koshkid’ko YS, Ćwik J, Dormidontov NA, Skotnicova K, Kursa M, Prokofev PA. Simulating the Hysteretic Characteristics of Hard Magnetic Materials Based on Nd_{2}Fe_{14}B and Ce_{2}Fe_{14}B Intermetallics. *Crystals*. 2020; 10(6):518.
https://doi.org/10.3390/cryst10060518

**Chicago/Turabian Style**

Kolchugina, Natalia B., Mark V. Zheleznyi, Aleksandr G. Savchenko, Vladimir P. Menushenkov, Gennadii S. Burkhanov, Yurii S. Koshkid’ko, Jacek Ćwik, Nikolai A. Dormidontov, Katerina Skotnicova, Miroslav Kursa, and Pavel A. Prokofev. 2020. "Simulating the Hysteretic Characteristics of Hard Magnetic Materials Based on Nd_{2}Fe_{14}B and Ce_{2}Fe_{14}B Intermetallics" *Crystals* 10, no. 6: 518.
https://doi.org/10.3390/cryst10060518