**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 e fforts [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 *P*42/*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 (4*f*, <sup>4</sup>*g*), and one boron site (4*g*). Table 1 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].

**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.


The anisotropy of the Nd2Fe14B compound is dominated by the rare-earth atoms that occupy two inequivalent sites, 4*f* and 4*g*, 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 Tsr. 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 (Nd1−<sup>x</sup>Cex)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 (Nd1−<sup>x</sup>Cex)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, Tsr varies slightly for low Ce contents. A small change in Tsr 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 (Nd1−<sup>x</sup>Cex)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 4*f* 1 and tetravalent 4*f* 0 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 [16–18], 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 e ffect 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 Nd2Fe14 B alloy allowed the authors to reach a su fficiently 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 Nd2Fe14 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.

At room temperature, the magnetic properties of C<sup>е</sup>2Fe14B are substantially lower than those of Nd2Fe14B, whereas, at cryogenic temperatures, the magnetic anisotropy field of C<sup>е</sup>2Fe14B 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 e ffect of Ce substitution for Nd on the spin-reorientation transition temperature of the Nd2−<sup>x</sup>C<sup>е</sup>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 (4*g* or 4*f*) 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 (Nd1−<sup>х</sup>Ceх)2Fe14B intermetallics in order to determine the optimum alloyed compositions. The simulation and analysis of hysteresis loops of the (Nd1-хC<sup>е</sup>х)2Fe14B (х = 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 *Is* at each point of a crystalline ferromagnet is simultaneously oriented along certain crystallographic directions (EMAs). In order to rotate *Is* 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*<sup>a</sup>*, which, for magnets with the uniaxial tetragonal lattice, is given by the expression:

$$E\_4 = -K\_1 \cos^2(\varphi) - K\_2 \cos^4(\varphi),\tag{1}$$

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<sup>s</sup>*. 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 e ffect 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 *Is* also determined by the magnetic field energy:

$$E\_H = -H \, I\_s \cos(\varphi),\tag{2}$$

where ϕ *H* is the angle made by the vectors. In the end, *Is* takes a position corresponding to the minimum summary energy:

$$E = E\_a + E\_H \tag{3}$$

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 (Nd1−<sup>x</sup>Cex)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 *Is*; 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* = *Is*<sup>с</sup>os( 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 (Nd1*−*<sup>x</sup> Cex)2Fe14B Intermetallics with 0* ≤ *x* ≤*1*

Temperature dependences of the magnetic anisotropy constants *K*1 and *K*2 of Nd2Fe14B in a temperature range of 0–500 K are available in [24]. The sign of *K*1 alternates at the spin-reorientation transition temperature *Tsr* = 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:

$$
\sin^2(\theta(T)) = -\frac{K\_1(T)}{2\ K\_2(T)}\tag{4}
$$

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

The temperature dependences of the magnetic anisotropy constant *K*1 and *K*2 of Ce2Fe14B in a temperature range of 0–300 K are available in [25]. Unlike the magnetic anisotropy constants of Nd2Fe14B, *K*1 and *K*2 of Ce2Fe14B remain positive within the 0–300 Кtemperature range. The absolute values of *K*1 and *K*2 of Ce2Fe14B are substantially lower than those of Nd2Fe14B.

To calculate the anisotropy constants of the quasi-ternary intermetallics (Nd1−<sup>x</sup>Cex)2Fe14B with 0 ≤ *x* ≤1 at di fferent temperatures, the literature data on the anisotropy constants of Nd2Fe14B and Ce2Fe14B and the following linear expressions were used:

$$K\_1(T, \mathbf{x}) = (1 - \mathbf{x}) \ K\_1(T)\_{\text{Nd}\_2 \text{F} \mathbf{v}\_{\text{4}} \mathbf{a}} + \mathbf{x} \ K\_1(T)\_{\text{Cx2} \text{F} \mathbf{v}\_{\text{4}} \mathbf{a}} \tag{5}$$

$$K\_2(T, \mathbf{x}) = (1 - \mathbf{x}) \ K\_2(T)\_{Nd\_2Fe\_{14}B} + \mathbf{x} \ K\_2(T)\_{Co\_2Fe\_{14}B} \tag{6}$$

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

$$\frac{|K\_2(T\_\prime, \infty)|}{|K\_1(T\_\prime, \infty)|}\tag{7}$$

at  $K\_1(T, x) < 0$  
$$\theta(T) = \arcsin\left(\sqrt{-\frac{K\_1(T)}{2\ K\_2(T)}}\right) \tag{8}$$

at *<sup>K</sup>*1(*<sup>T</sup>*, *x*) ≥ 0

$$
\theta(T, \mathbf{x}) = 0 \tag{9}
$$

The following composition ranges of (Nd1−<sup>x</sup>Cex)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 (Nd1−<sup>x</sup> Cex)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 (*TSR*) of (Nd1−<sup>x</sup>Cex)2Fe14B are given in Figure 2.

(a)

(b) 

(c)

**Figure 1.** Temperature dependences of the (**a**) К1 and К2 magnetic anisotropy constants, (**b**) normalized К*2*/|К*1*| ratio, and (**c**) θ angle for (Nd1−<sup>x</sup> Cex)2Fe14B 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 (*Tsr*) of (Nd1−<sup>x</sup> Cex)2Fe14B: (blue) hypothetical (linear) trend and (red) results of simulation (this work).

It is seen from Figures 1 and 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 К*2*/|К*1*| ratio decreases from 135 to 63 K. The temperature *Tsr* 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 (Nd1−<sup>х</sup>Ceх)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 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).

It is seen from the dependences given in Figures 1 and 2 that, as the cerium content in the (Nd1−<sup>х</sup>Ceх)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 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 (Nd1*−*<sup>x</sup> Cex)2Fe14B with x* = *0.00–1.00*

The simulated hysteresis loops for (Nd1−<sup>x</sup> Cex)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 *IS* 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 (*Ir*/*Is* and *Ir*) for the three composition ranges of the (Nd1−<sup>х</sup>Ceх)2Fe14B compounds.

For all quasi-ternary compositions, the *Ir*/*Is* ratio monotonically decreases with decreasing temperature. This is caused by the deviation of *Is* 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 *Ir*/*Is* 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)*, *Ir*/*Is* 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 *Ir*/*Is* is observed simultaneously with decreasing temperature below *Tsr(x)* and Ce content *x*. For each Ce content *x,* the temperature dependences *Ir(T, x)* have the maximum value in the range of Т*sr(x)* (Figure 3b). The monotonic shift of the *Ir* maximum to low temperatures from *Ir*(*T* = 135 K, *x* = 0) =191.6 A m<sup>2</sup>/kg to *Ir*(*T* = 102 K, *x* = 0.75) = 161.5 A m<sup>2</sup>/kg correlates with a similar shift of the maximum of normalized *K2*/|*K1*| ratio and a decrease in *Tsr(x)* with increasing cerium content *x* (Figures 1 and 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 *Tsr* temperature.

**Figure 3.** Temperature dependences of the normalized and unnormalized residual magnetization (*Ir*/*Is* and *Ir*) of the (Nd1−<sup>х</sup>Ceх)2Fe14B compounds with di fferent *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.

For the composition range *x* = 0.80–0.92 (Figure 3c), as the temperature decreases from 300 K to Т*sr(x)*, *Ir*/*Is* 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 *Ir*/*Is* is observed with simultaneously decreasing temperature below *Tsr(x)* and Ce content *x*. For each Ce content *x,* the temperature dependences *Ir(T, x)* also have the maximum in the range of Т*sr(x)* (Figure 3d). The monotonic shift of the *Ir* maximum to the low temperature range from *Ir*(*T* = 100 K, *x* = 0.80) =159.6 A m<sup>2</sup>/kg to *Ir*(*T* = 25 K, *x* = 0.92) = 156.6 A m<sup>2</sup>/kg correlates with the similar shift of the normalized *K2*/|*K1*| ratio and a decrease in *Tsr(x)* with increasing cerium content *x* (Figures 1 and 2). However, in this range of cerium concentrations, the temperature dependences of *Ir* overlap; the overlapping was not observed in the range *x* = 0–0.75 (Figure 3d). This interesting and anomalous change in *Ir* 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 *Tsr*. 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 (Nd1−<sup>х</sup>Ceх)2Fe14B intermetallics with *x* = 0.93–1.00. As the temperature decreases from 300 K to 0 K, *Ir*/*Is* 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 *Ir*/*Is* is observed as the simultaneous decrease in temperature below *Tsr(x)* and decrease in the Ce concentration take place. As *x* increases from 0.94 to 1.00, the considered maximum of *Ir* 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 *Tsr(x)* with increasing cerium concentration *x* (Figures 1 and 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 4.** Compositional dependence of the residual magnetization *Ir* for the (Nd1−<sup>х</sup>Ceх)2Fe14B compounds at *T* = 0 K.

Figure 5 shows temperature dependences of the coercive force Н*c* of the (Nd1−<sup>х</sup>Ceх)2Fe14B compounds with *x* = 0–1.00.

**Figure 5.** Temperature dependences of the coercive force Н*c* of the (Nd1−<sup>х</sup>Ceх)2Fe14B compounds with di fferent *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.

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 *Ir*, and a decrease in *Tsr(x)* with increasing cerium concentration *x* (Figures 1 and 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 *HA* of R2Fe14B intermetallics (where R = Nd or Ce) with decreasing temperature and the spin-reorientation below *Tsr*, 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 *IR*, and a decrease in *Tsr(x)* with increasing cerium concentration *x* (Figures 1–3).

For the compositions with *x* = 0.93–0.94 (Figure 5c), the maximum of *Hc* progressively shifts from 25 to ~10 K. For *x* > 0.94, the maximum shifts to 0 K and degenerates. This e ffect also correlates with a similar shift of the normalized *K2*/|*K1*| ratio, shift of the maximum of *Ir*, and a decrease in *Tsr(x)* with increasing cerium concentration *x* (Figures 1 and 3).
