Simulating the Hysteretic Characteristics of Hard Magnetic Materials Based on Nd₂Fe₁₄B and Ce₂Fe₁₄B Intermetallics

Abstract

The Ce₂Fe₁₄B intermetallic, like Nd₂Fe₁₄B, has the tetragonal Nd₂Fe₁₄B-type structure (space group P4₂/mnm), in which Ce ions have a mixed-valence state characterized by the coexistence of trivalent 4f¹ and tetravalent 4f⁰ electron states. Despite the fact that the saturation magnetization, magnetic anisotropy field, and Curie temperature of the Ce₂Fe₁₄B intermetallic are substantially lower than those of Nd₂Fe₁₄B and Pr₂Fe₁₄B, Ce₂Fe₁₄B 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 Се₂Fe₁₄B is higher than that of Nd₂Fe₁₄B, and Се₂Fe₁₄B 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 K₁ and K₂ (K₂ > 0 and K₁ > 0 and K₁ < 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−хСе_х)₂Fe₁₄B (х = 0–1) was performed. Results of the simulation indicate that the alloying of the Nd₂Fe₁₄B 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−хCe_х)₂Fe₁₄B intermetallic and (2) slightly decreases the slope of magnetization reversal curve.

1. Introduction

In recent years, the low cost of Ce-containing Nd-Fe-B permanent magnets, in some areas being a potential alternative to those based on expensive rare-earth elements (Nd, Pr, Dy, Tb), has stimulated considerable research efforts [1,2].

Similar to the other R-Fe-B systems, the Ce-Fe-B system is characterized by the formation of the ternary intermetallic compound having a 2:14:1 stoichiometry and tetragonal Nd₂Fe₁₄B-type structure (space group P4₂/mnm) [3]. The unit cell contains four formula units comprising 68 atoms: there are six crystallographic iron sites (16k₁, 16k₂, 8j₁, 8j₂, 4c, 4e), two rare-earth metal sites (4f, 4g), and one boron site (4g).

Table 1. Saturation magnetization I_s, magnetic anisotropy field H_A, lattice parameters a and c, and Curie temperature T_C of the R₂Fe₁₄B compounds with R = Ce, Pr, Nd at room temperature.

Compound	Is, T	HA, kOe	TC, К	а, nm	с, nm	Reference
Nd2Fe14B	1.61	73	585	0.880	1.220	[4,5]
Pr2Fe14B	1.56	87	565	0.880	1.223	[4,6]
Ce2Fe14B	1.17	26	424	0.876	1.211	[4,7]

shows the lattice parameters of the R₂Fe₁₄B compounds with R = Nd, Pr, and Ce and their principal magnetic characteristics. This shows that both the a and c lattice parameters of the Ce₂Fe₁₄B compound are slightly lower than those of the Nd₂Fe₁₄B and Pr₂Fe₁₄B compounds [4].

The anisotropy of the Nd₂Fe₁₄B 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. Nd₂Fe₁₄B is characterized by the uniaxial state at temperatures above the spin-reorientation temperature T_sr. Ce₂Fe₁₄B exhibits the uniaxial magnetic anisotropy over the whole temperature range of ferromagnetic ordering [10]. Its uniaxial anisotropy is higher than that of R₂Fe₁₄B with R =La, Lu, Y [10], but lower than that of Nd₂Fe₁₄B at temperatures substantially higher than the spin-reorientation temperature. It was predicted [11] that, theoretically, Ce atoms in the (Nd_1−xCe_x)₂Fe₁₄B 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.

Taking into account the fact that Ce prefers 4f positions, the progressive substitution of Ce for Nd should decrease the easy magnetization axis (EMA) cone opening. Thus, the formation of a uniaxial state can occur even in the presence of a small amount of Nd in (Nd_1−xCe_x)₂Fe₁₄B. 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 Nd₂Fe₁₄B remains when Ce substitutes for Nd up to 20%. The further increase in the Ce content decreases the uniaxial magnetic anisotropy energy.

The uniaxial anisotropy found in Ce₂Fe₁₄B is attributed mainly to the magnetism of Fe [8]. Alloying Ce on R-sites in (Nd_1−xCe_x)₂Fe₁₄B 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 R₂Fe₁₄B with R = Nd, Ce was studied in [12].

The metallurgical behavior, heat treatment conditions, fundamental characteristics, and magnetic properties of the Nd₂Fe₁₄B and Ce₂Fe₁₄B intermetallics differ substantially although the compounds have the same crystal structure. The existence of the CeFe₂ phase determines the principal difference in the ternary Nd-Fe-B and Ce-Fe-B phase diagrams [3,13]. Cerium ions in the Ce₂Fe₁₄B intermetallic have a mixed-valence state, namely, the trivalent 4f¹ and tetravalent 4f⁰ electron states coexist [14].

According to data from [15], the hysteretic properties of Nd-Ce-Fe-B magnets decrease as the Ce content increases. However, the squareness of the magnetization reversal curve remains high [16,17,18], and the magnetic characteristics remain adequate when the Ce content does not exceed 10%.

Usually, it is assumed that the decrease in the magnetic characteristics of Ce-containing Nd₂Fe₁₄B alloys is due to the lower magnetic properties of Ce₂Fe₁₄B as compared to those of Nd₂Fe₁₄B (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 Nd₂Fe₁₄B alloy allowed the authors to reach a sufficiently high coercive force (H_ci = 10 kOe), which exceeds that of Nd₂Fe₁₄B (H_ci = 8.3 kOe).

Currently, the development of high-coercivity, high-performance permanent magnets operating in a wide temperature range, in particular at low and cryogenic temperatures, is of importance. Nd-Fe-B magnets are unsuitable for operation at such temperatures. The EMA of the Nd₂Fe₁₄B 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 K₁ passes zero and changes the sign from positive to negative, whereas the second magnetic anisotropy constant remains positive (K₂ > 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.

At room temperature, the magnetic properties of Се₂Fe₁₄B are substantially lower than those of Nd₂Fe₁₄B, whereas, at cryogenic temperatures, the magnetic anisotropy field of Се₂Fe₁₄B is markedly higher than that of Nd₂Fe₁₄B. Moreover, it is of importance that Ce₂Fe₁₄B does not have a spin-reorientation transition. Thus, it is reasonable to expect that the partial substitution of Ce for Nd in the Nd₂Fe₁₄B compound can lead to the improvement of the hysteretic characteristics of permanent magnets based on the quasi-ternary (Nd, Ce)₂Fe₁₄B intermetallics.

Data on the effect of Ce substitution for Nd on the spin-reorientation transition temperature of the Nd_2−xСе_xFe₁₄B 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].

Taking into account the absence of systematic data on this problem in the literature, the present study is aimed at the simulation of hysteretic properties of the (Nd_1−xCe_x)₂Fe₁₄B intermetallics in order to determine the optimum alloyed compositions. The simulation and analysis of hysteresis loops of the (Nd_1−xCe_x)₂Fe₁₄B (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

The resulting magnetization vector I_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:

$$E_a=-K_1 co{s}^2\left(\phi \right)-K_2 co{s}^4\left(\phi \right),$$

(1)

where К₁ and К₂ are the first and second magnetic anisotropy constants, respectively, and φ_a is the angle made by the EMA and I_s. 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.

To take into account the cooperative effect of magnetic anisotropy and magnetic field, we considered a single crystal in the form of a plain disk oriented along a certain crystallographic plane. When the magnetic field is applied along the plane, the magnetization is also within the plane. It is assumed that the magnetization of the body is uniform and domains are absent.

When applying magnetic field Н, the position of I_s also determined by the magnetic field energy:

$$E_H=-H I_s cos\left(\phi \right),$$

(2)

where φ_H is the angle made by the vectors. In the end, I_s takes a position corresponding to the minimum summary energy:

$$E=E_a+E_H$$

(3)

In terms of the model and algorithm [23] (program Hysteresis developed by Associate Prof. V.L. Stolyarov) used for the simulation of magnetic hysteresis loops of (Nd_1−xCe_x)₂Fe₁₄B 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.

Calculation of the position of the magnetization vector 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_sсos(φ_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 (K₂/|K₁|), and Angle of the EMA cone in Calculating Hysteresis Loops of the (Nd_1−x Ce_x)₂Fe₁₄B Intermetallics with 0 ≤ x ≤1

Temperature dependences of the magnetic anisotropy constants K₁ and K₂ of Nd₂Fe₁₄B in a temperature range of 0–500 K are available in [24]. The sign of K₁ 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:

$$si{n}^2\left(\theta \left(T\right)\right)=-\frac{K_1\left(T\right)}{2 K_2\left(T\right)}$$

(4)

At 4.2 K, this angle reaches ~30° [24].

The temperature dependences of the magnetic anisotropy constant K₁ and K₂ of Ce₂Fe₁₄B in a temperature range of 0–300 K are available in [25]. Unlike the magnetic anisotropy constants of Nd₂Fe₁₄B, K₁ and K₂ of Ce₂Fe₁₄B remain positive within the 0–300 К temperature range. The absolute values of K₁ and K₂ of Ce₂Fe₁₄B are substantially lower than those of Nd₂Fe₁₄B.

To calculate the anisotropy constants of the quasi-ternary intermetallics (Nd_1−xCe_x)₂Fe₁₄B with 0 ≤ x ≤1 at different temperatures, the literature data on the anisotropy constants of Nd₂Fe₁₄B and Ce₂Fe₁₄B and the following linear expressions were used:

$$K_1\left(T, x\right)=\left(1-x\right) K_1{\left(T\right)}_{N{d}_{2}F{e}_{14}B}+x K_1{\left(T\right)}_{C{e}_{2}F{e}_{14}B}$$

(5)

$$K_2\left(T, x\right)=\left(1-x\right) K_2{\left(T\right)}_{N{d}_{2}F{e}_{14}B}+x K_2{\left(T\right)}_{C{e}_{2}F{e}_{14}B}$$

(6)

The ratio of anisotropy constants К₂ to К₁ and the θ angle (made by the EMA and c axis) were calculated by the expressions:

$$\frac{K_2\left(T, x\right)}{\left|K_1\left(T, x\right)\right|}$$

(7)

at $K_1\left(T, x\right)<0$

$$\theta \left(T\right)=arcsin\left(\sqrt{-\frac{K_1\left(T\right)}{2 K_2\left(T\right)}}\right)$$

(8)

at $K_1\left(T, x\right)\ge 0$

$$\theta \left(T, x\right)=0$$

(9)

The following composition ranges of (Nd_1−xCe_x)₂Fe₁₄B 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.

The calculated temperature dependences of the magnetic anisotropy constants, their ratio, and temperature dependences of the θ angle for (Nd_1−x Ce_x)₂Fe₁₄B 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−xCe_x)₂Fe₁₄B are given in Figure 2.

It is seen from Figure 1 and Figure 2 that, as Ce substitutes for Nd in the range x = 0–0.75 (first composition range), the values of К₁ and К₂ constants decrease on average by ~75% and ~99%, respectively. The temperature corresponding to the maximum of the normalized К₂/|К₁| 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).

As the Ce content in (Nd_1−хCe_х)₂Fe₁₄B increases in the range x = 0.80–0.92 (second composition range), the К₁ and К₂ values additionally decrease on average by ~61% and ~63%, respectively. In turn, the temperature corresponding to the maximum of normalized К₂/|К₁| 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).

It is seen from the dependences given in Figure 1 and Figure 2 that, as the cerium content in the (Nd_1−хCe_х)₂Fe₁₄B intermetallic increases from x = 0.93 to x = 1.00, the anisotropy constant К₂ additionally decreases by an average ~35%, whereas the К₁ 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 К₂/|К₁| 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)₂Fe₁₄B with x = 0.00–1.00

The simulated hysteresis loops for (Nd_1−x Ce_x)₂Fe₁₄B 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−хCe_х)₂Fe₁₄B compounds.

For all quasi-ternary compositions, the I_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.

For the composition range x = 0–0.75 (Figure 3a), as the temperature decreases from 300 K to Т_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 m²/kg to I_r(T = 102 K, x = 0.75) = 161.5 A m²/kg correlates with a similar shift of the maximum of normalized K₂/|K₁| 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 R₂Fe₁₄B intermetallics (where R = Nd or Ce) with decreasing temperature and the spin-reorientation below the T_sr temperature.

For the composition range x = 0.80–0.92 (Figure 3c), as the temperature decreases from 300 K to Т_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 m²/kg to I_r(T = 25 K, x = 0.92) = 156.6 A m²/kg correlates with the similar shift of the normalized K₂/|K₁| 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 R₂Fe₁₄B 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 R₂Fe₁₄B.

Figure 3e shows the temperature dependences of the normalized residual magnetization for the third composition range of the (Nd_1−хCe_х)₂Fe₁₄B 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 K₂/|K₁| ratio and a decrease in T_sr(x) with increasing cerium concentration x (Figure 1 and Figure 2).

The considered features of the change of the residual magnetization clearly manifest themselves in the concentration dependence at T = 0 K. Figure 4 shows that the composition range x = 0.80–0.92 is intermediate between neodymium-based intermetallics and cerium-based intermetallic R₂Fe₁₄B and is characterized by anomalous change in the residual magnetization.

Figure 5 shows temperature dependences of the coercive force Н_c of the (Nd_1−хCe_х)₂Fe₁₄B compounds with x = 0–1.00.

It is seen that, for each Ce content x = 0–0.75, the coercive force Н_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 K₂/|K₁| 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 R₂Fe₁₄B 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.

As is seen, for each Ce content x = 0.80–0.92 (Figure 5b), the coercive force Н_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 K₂/|K₁| 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).

For the compositions with x = 0.93–0.94 (Figure 5c), the maximum of H_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 K₂/|K₁| 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

The following are our conclusions, which are inferred based on the data obtained in simulating the magnetization reversal process of the (Nd_1−хCe_х)₂Fe₁₄B compounds with х = 0–1.00 and their hysteresis loops in a wide temperature range of 300 to 0 K.

1. The model and algorithm developed for simulating the hysteresis loops of ferromagnets allow us to obtain important data on the effect of uniaxial tetragonal lattice, magnetic anisotropy parameters, and temperature dependences of the magnetic anisotropy constants on the characteristics of hysteresis loops (anisotropy field, residual magnetization, the shape of the hysteresis loop, and coercive force) in a wide temperature range.

2. Results of the simulation indicate that alloying of Nd₂Fe₁₄B 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)₂Fe₁₄B 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.

3. The alloying of (Nd_1−хCe_х)₂Fe₁₄B with cerium to its contents x = 0.94 does not allow the temperature stability of hysteretic characteristics of (Nd_1−хCe_х)₂Fe₁₄B-based permanent magnets to be increased in order to ensure their operation at low temperatures without losing magnetic properties.