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[Masamichi Nishino](https://orcid.org/0000-0002-2060-2303), [Rachida Lamouri](https://orcid.org/0000-0003-1820-5277), Hisazumi Akai

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[Atomistic model analysis of the spin reorientation transition in                    <math>                      <mrow>                        <msub>                          <mrow>                            <mo>(</mo>                            <msub>                              <mi>Nd</mi>                              <mrow>                                <mn>1</mn>                                <mo>−</mo>                                <mi>x</mi>                              </mrow>                            </msub>                            <msub>                              <mi>Dy</mi>                              <mi>x</mi>                            </msub>                            <mo>)</mo>                          </mrow>                          <mn>2</mn>                        </msub>                        <msub>                          <mi>Fe</mi>                          <mn>14</mn>                        </msub>                        <mi>B</mi>                      </mrow>                    </math>                    systems](https://mdr.nims.go.jp/datasets/6c5773ff-b6c7-4ff2-8ad5-1073ab4e63c2)

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d(dm-dt)-x(Poly3).epsAtomistic model analysis of the spin reorientation transition in (Nd1−xDyx)2Fe14BsystemsMasamichi Nishino,1, ∗ Rachida Lamouri,1 and Hisazumi Akai21Research Center for Materials Nanoarchitectonics,National Institute for Materials Science, 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan2Graduate School of Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka 565-0871, Japan(Dated: January 27, 2026)Neodymium (Nd) magnets (Nd2Fe14B) are important permanent magnets due to their strong co-ercive force, which contributes to high-efficiency energy conversion technologies. This coercivity isoften enhanced by substituting dysprosium (Dy). Therefore, understanding the magnetic propertiesof Dy-substituted systems, (Nd1−xDyx)2Fe14B, is essential. We investigate the spin reorientationtransition in (Nd1−xDyx)2Fe14B using a recently developed atomistic modeling approach. This mod-eling method captures the microscopic mechanisms of magnetic interactions and temperature effects,including thermal fluctuations. We study the x dependence of the spin reorientation transition tem-perature (TSR) and the canting angle (θ) of the total magnetization using an importance-samplingMonte Carlo method, based on a model with microscopic parameters derived primarily from first-principles calculations. Our estimates of TSR and θ are consistent with experimental results. Wealso compare our results with those obtained from a previous mean-field-like study and show sig-nificant differences, particularly at higher Dy concentrations. Our model more accurately capturesexperimental trends in this regime. We attribute this improvement to more accurate representationsof canting angles and anisotropy energies of the constituent atoms. Additionally, we investigate ananomalous behavior in the total magnetization of (Nd1−xDyx)2Fe14B. We find a non-differentiablepoint in the magnetization at TSR, which becomes more pronounced with increasing x, peaking atx = 0.5. We discuss the origin of this anomaly in detail through analysis of the magnetic propertiesof the constituent atoms.I. INTRODUCTIONControlling the magnetic properties of permanentmagnets is crucial for achieving high energy conversionefficiency. Neodymium (Nd) magnets [1–10], whose mainphase is Nd2Fe14B, are particularly important due totheir high coercive fields [11–20]. These magnets arewidely used in motors, generators, electronic devices, andother applications. The increasing demand for electricvehicle motors and related technologies is expected tofurther expand the applications of Nd magnets.However, Nd magnets suffer from reduced coerciv-ity at elevated temperatures. To overcome this is-sue, heavy rare-earth elements such as dysprosium(Dy) are often added to enhance coercivity for practi-cal use. The formation of Dy-rich shells, specifically(Nd1−xDyx)2Fe14B, plays a key role in improving co-ercivity [21–27]. Therefore, investigating the magneticproperties of (Nd1−xDyx)2Fe14B is of great significance.Theoretical studies of the magnetic properties of per-manent magnets have often been conducted using contin-uum models [28], which rely on a small number of macro-scopic magnetic parameters such as the exchange stiffnessconstant (A) and magnetic anisotropy energy (K). Whilethese models are advantageous for simulating large sys-tems, they inherently ignore microscopic details of thecrystal structure and magnetic interactions due to coarsegraining. Furthermore, as shown in Ref. [29], continuum∗ Corresponding author: nishino.masamichi@nims.go.jpmodels struggle to accurately incorporate temperatureeffects and thermal fluctuations.Temperature-dependent properties of R2Fe14B com-pounds have also been studied using mean-field (MF)approximations [30, 31]. However, MF approaches oftenyield inaccurate magnetic properties because of oversim-plified modeling and the neglect of thermal fluctuations.Recently, atomistic modeling approaches have been de-veloped to investigate the detailed magnetic propertiesof permanent magnets. These models account for mi-croscopic magnetic interactions that reflect the atomic-scale crystal structure. They also allow for an accu-rate treatment of temperature effects, including thermalfluctuations and the dynamics toward thermal equilib-rium [32, 33]. Given the structural and interactionalcomplexity of R2Fe14B compounds, atomistic modelingis particularly well suited for studying their magneticproperties. Atomistic model studies have successfullyelucidated both qualitative and quantitative aspects ofNd magnets at zero and finite temperatures [17, 34–51].We recently investigated the thermodynamic proper-ties of Dy2Fe14B using an atomistic model [52] and com-pared them with those of Nd2Fe14B, finding good agree-ment with experimental results. We also studied themechanism of coercivity enhancement due to Dy sub-stitution in Nd magnets [49]. Our findings indicate thatthe crystal electric field energy barrier of Dy atoms ismore resistant to thermal fluctuations at high tempera-tures, which contributes to the coercivity enhancement,in addition to the difference in the magnetic interactionbetween rare earth and iron atoms, that is, antiferro-2magnetic coupling between Dy and Fe moments whileferromagnetic coupling between Nd and Fe moments.Nd2Fe14B exhibits a spin reorientation (SR) tran-sition at a low temperature, which has been exten-sively studied [7–9, 53–58]. The SR transition in(Nd1−xDyx)2Fe14B has also attracted attention [54, 57,59]. In this study, we focus on the SR transition inthese compounds. The only theoretical analysis avail-able for the x-dependent behavior is a mean-field-likemodel by Lim et al. [54], which considered Nd, Dy, andFe in a fitted mean-field framework without employinga self-consistent field method. Here, we study the x-dependent magnetic properties related to the SR tran-sition using an atomistic modeling approach. We showthat our estimates of the SR transition temperature, TSR,and the canting angle of the total magnetic moment, θ,as functions of x are consistent with experimental obser-vations [54, 57]. We also highlight discrepancies at largex between our results and those of Lim et al.’s MF-likemethod [54], and provide a detailed discussion of thesedifferences.Furthermore, we investigate an anomalous phe-nomenon in the magnetization near TSR. This anomalywas first reported by Hirosawa in Nd2Fe14B [7]. How-ever, its x-dependence has not been systematically stud-ied. Here, we clarify this behavior using the atomisticmodeling. We find that the anomaly in total magnetiza-tion becomes more pronounced with increasing x, peaksat x = 0.5, and then decreases. We attribute the anomalyto unusual behavior of the Nd magnetic moments.The rest of this paper is organized as follows. InSec. II, we describe the atomistic model. Section IIIpresents the method to study thermodynamic quantities.In Sec. IVA, we investigate the magnetic properties ofthe SR transition. In Sec. IVB, we analyze the cantingangles of magnetic moments. In Sec. IVC, we explore theorigin of the anomalous magnetization behavior. Finally,Sec. V provides a summary of our findings.II. MODELWe employ the following atomistic Hamiltonian for the(Nd1−xDyx)2Fe14B system:H =−∑i<j2Jijsi · sj −Fe∑iDi(szi )2 (1)+Nd,Dy∑i∑l,mΘl,iAml,i〈rl〉iÔml,iHere, Jij denotes the exchange interaction between theith and jth atoms. Di represents the anisotropy constantfor the ith Fe atom. The third term corresponds to thecrystal electric field (CEF) energy of the rare-earth atoms(Nd and Dy), where Θl,i, Aml,i, 〈rl〉i, and Ôml,i denote theStevens factor, the coefficient of the spherical harmon-ics of the crystalline electric field, the radial expectationvalue, and the Stevens operator, respectively. We con-sider l = 2, 4, 6 and m = 0 (diagonal operators), whichare the dominant contributions.For Fe and B atoms, si represents the magnetic mo-ment at site i. For Nd and Dy atoms, si representsthe moment of the valence (5d and 6s) electrons, whichare strongly coupled to the 4f electron moment, J i =gTJ iµB, where gT is the Lande g-factor and J i is thetotal angular momentum. The total moment for eachrare-earth atom is given by Si = si+J i. For Nd atoms,J = L− S = 9/2 and gT = 8/11, where L and S are theorbital and spin angular momenta, respectively. For Dyatoms, J = L + S = 15/2 and gT = 4/3. For Fe and Batoms, we define Si = si. It should be noted that si of aNd or Dy atom and Si(= si) of an Fe atom are antiferro-magnetically coupled. However, the total moment Si ofa Nd atom is ferromagnetically coupled to that of an Featom, whereas that of a Dy atom is antiferromagneticallycoupled to Fe [49].Monte Carlo simulations are performed to investigatethermodynamical properties of the (Nd1−xDyx)2Fe14Bsystem as a function of x. Since x represents the concen-tration of Dy atoms among the rare-earth sites, largersystems are needed to accurately evaluate thermody-namic quantities for a random distribution of Dy atoms.In this study, we adopt a system consisting of 14,688atoms, corresponding to 6×6×6 unit cells with periodicboundary conditions.First, magnetic moments and exchange interactionsfor Nd2Fe14B and Dy2Fe14B are estimated with theAkaiKKR code based on the Korringa-Kohn-Rostoker(KKR) first-principles method [60], using a single unitcell structure of each compound with periodic boundaryconditions. The Liechtenstein method [61] is employedto estimate Jij . In the calculations, standard muffin-tin-type potentials and the local density approximationare used. For the electronic states of the rare-earth ele-ments, the open-core approximation is applied. The ob-tained data are summarized in Figs. S1–S3 of the Sup-plemental Material (SM) [62]. Accurate first-principlesestimation of Jij in R2Fe14B remains challenging [63].The applicability of the AkaiKKR method to R2Fe14Bhas been examined and is generally regarded as rea-sonable [52, 63]. Magnetization values calculated usingthis method show agreement with experimental measure-ments. Although the Curie temperatures are somewhatoverestimated, the dependence on R reproduces the ex-perimental trend. These results support the validity ofthis method for R2Fe14B.Because the system containing 14,688 atoms is toolarge to directly compute magnetic moments and ex-change interactions using the first-principles method, weadopt the following strategy to determine the exchangeinteractions:For Nd–Nd, Nd–Fe, and Nd–B bonds, we use valuesestimated in Nd2Fe14B. For Dy–Dy, Dy–Fe, and Dy–Bbonds, we use values from Dy2Fe14B. For Nd–Dy bonds,we use the average of the Nd–Nd and Dy–Dy values. For3(a) (b)ca (b)ba FIG. 1. (a) Side view and (b) top view of the model structureof (Nd1−xDyx)2Fe14B, illustrated for a 3× 3× 3 supercell asan expample. Red, orange, blue, and yellow spheres representNd, Dy, Fe, and B atoms, respectively. Lattice constants forNd2Fe14B are la = lb = 8.80 Å and lc = 12.20 Å, and forDy2Fe14B are la = lb = 8.76 Å and lc = 12.01 Å.Fe–Fe, B–B, and Fe–B bonds, we use a weighted averageof the values from Nd2Fe14B and Dy2Fe14B, with a ra-tio of 1− x to x for each bond. Since the correspondingexchange interaction values in Nd2Fe14B and Dy2Fe14Bare very close (see Fig. S1– Fig. S3 in SM), this strategyis expected to be a good approximation.For computing magnetizations, we use si values fromNd2Fe14B and Dy2Fe14B for Nd and Dy atoms, re-spectively. For Fe and B atoms, si is determined bya weighted average of the values from Nd2Fe14B andDy2Fe14B using the ratio 1 − x : x. The values of sifor Fe and B atoms are also very close in both systems,supporting the validity of this approximation. For rare-earth atoms, Aml values reported by Yamada et al. [53]for R2Fe14B are used, along with 〈rl〉 from Ref. [64]. TheR ions occupy the Wyckoff 4f and 4g sites. In accor-dance with Yamada et al. [53], the same set of CEF coef-ficients is employed for both positions. Determining thedifference between the CEF coefficients at the 4f and 4gsites is a nontrivial problem, with published estimatesranging from negligible to appreciable [6]. Consequently,this work does not address the possible differences in theCEF energy between the two sites. The anisotropy con-stants Di for six types of Fe sites are taken from first-principles results reported in Ref. [65]. The parametervalues for the CEF coefficients and the anisotropy en-ergies of Fe atoms are given in Tables S1–S3 in the SM.The anisotropy energies of Nd and Dy atoms are expectedto be nearly unaffected by substitution, as these atomsare surrounded by Fe atoms, with the 4f electrons con-tributing to anisotropy and the 5d/6s electrons mainlycontributing to exchange interactions. 0 0.5 1 1.5 2 2.5 0  200  400  600  800  1000  1200(a)Mz (µ B)T (K)x=0x=0.1x=0.2x=0.4x=0.5x=0.6x=0.8x=0.9x=0.95x=1 0 0.5 1 1.5 2 2.5 0  200  400  600  800  1000  1200(b)M (µ B)T (K)x=0x=0.1x=0.2x=0.4x=0.5x=0.6x=0.8x=0.9x=0.95x=1 FIG. 2. Temperature dependences of (a) the z-component ofthe total magnetization Mz and (b) the total magnetizationmagnitude M for (Nd1−xDyx)2Fe14B systems over the fulltemperature range.III. METHODWe use a Metropolis importance-sampling MonteCarlo (MC) method to study equilibrium magnetizationsat finite temperatures. The per-site magnetizations Mz,Mxy, and M for the (Nd1−xDyx)2Fe14B model are de-fined asMz =1N〈∣∣∣∣∣N∑i=1Szi∣∣∣∣∣〉, (2)Mxy =1N〈√√√√(N∑i=1Sxi)2+(N∑i=1Syi)2〉, (3)andM =1N〈√√√√(N∑i=1Sxi)2+(N∑i=1Syi)2+(N∑i=1Szi)2〉,(4)4 0 0.5 1 1.5 2 2.5 0  0.2  0.4  0.6  0.8  1M (µ B)xour dataLim et al.FIG. 3. Magnetization M of (Nd1−xDyx)2Fe14B systems(blue circles) as a function of concentration x at T = 4 K. Theblue solid line represents a linear fit using the least-squaresmethod. Red squares indicate experimental values of M atT = 4.2 K reported by Lim et al. [54]., respectively. Here, N is the number of all atoms (allspins) in the (Nd1−xDyx)2Fe14B model, and 〈〉 denotesthermal average.We also define species-specific magnetizations for Nd,Dy, and Fe atoms.m(A) =1NA〈√√√√(NA∑i=1Sxi)2+(NA∑i=1Syi)2+(NA∑i=1Szi)2〉,(5)where A denotes Nd, Dy, or Fe and NA is the number ofatom A in the (Nd1−xDyx)2Fe14B model.We perform 200,000 Monte Carlo steps (MCS) forequilibration, followed by 400,000 to 1,600,000 MCSfor measurements, with more steps applied near theSR temperature. Figures 1(a) and (b) show the sideand top views, respectively, of the model structure of(Nd1−xDyx)2Fe14B for a 3× 3× 3 supercell as an exam-ple.IV. RESULTSA. Spin reorientation transitionThe temperature (T ) dependences of Mz and M for(Nd1−xDyx)2Fe14B systems are shown in Figs. 2 (a) and(b), respectively. A cusp appears at TSR ≃ 134 K in theMz–T curve for x = 0 (Nd2Fe14B), indicating a spin re-orientation (SR) transition. As x increases, the cusp inMz becomes broader and the SR transition temperature,TSR, becomes lower. Both M and Mz decrease with in-creasing of x, especially at lower temperatures. Although 0.5 1 1.5 2 0  50  100  150  200(a)Mz (µ B)T (K) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0  50  100  150  200(b)Mxy (µ B)T (K)x=0x=0.1x=0.2x=0.4x=0.5x=0.6x=0.8x=0.9x=0.95x=1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0  50  100  150  200(b)Mxy (µ B)T (K)x=0x=0.1x=0.2x=0.4x=0.5x=0.6x=0.8x=0.9x=0.95x=1FIG. 4. Temperature dependences of (a) Mz and (b) Mxy for(Nd1−xDyx)2Fe14B systems at low temperatures.M appears to change smoothly with temperature at thisscale in Fig. 2(b), careful inspection around TSR revealsthat TSR is a non-differentiable point. This feature isdiscussed in Sec. IVC.Experimentally, TSR for Nd2Fe14B has been reportedto range from 126 to 150 K [7–9, 53–57], and it decreaseswith increasing x [54, 57]. Our simulation results are con-sistent with these observations. The Curie temperatureTc remains almost constant with x due to the similar-ity in 2Jijsi ·sj values between Nd2Fe14B and Dy2Fe14B(see Fig. S1–Fig. S3 in SM). Our estimated Tc ≃ 870 Kis a little overestimated compared to experimental val-ues (∼ 600 K) [4, 6, 7], due to a small overestimationin Jij . The validity of the estimation by AkaiKKR hasbeen discussed [52, 63]. The estimated Curie tempera-tures are a little overestimated, but the R dependencecaptures the experimental trend. The calculated magne-tization of R2Fe14B generally agrees with experimentalvalues. In this paper, we focus on magnetic propertiesassociated with the SR transition.Figure 3 shows the concentration (x) dependence ofmagnetization M for (Nd1−xDyx)2Fe14B systems at T =4 K. We find that M exhibits a linear dependence onx. Experimentally measured values of M at T = 4.2 K,5 0 20 40 60 80 100 120 140 0  0.2  0.4  0.6  0.8  1TSR (K)xour dataLim et al.Kim et al.MF-likeFIG. 5. Concentration (x) dependence of the spin reorien-tation transition temperature (TSR) for (Nd1−xDyx)2Fe14Bsystems (blue circles). Blue solid line is a linear fitting us-ing the least squares method. Experimental data from Limet al. [54] and Kim et al. [57] are shown by red squares andmagenta triangles, respectively. 0 10 20 30 40 0  0.2  0.4  0.6  0.8  1θ (deg)xour dataLim et al.Kim et al.MF-likeFIG. 6. Canting angle (θ) of the total magnetic mo-ment from the c axis as a function of concentration (x) for(Nd1−xDyx)2Fe14B systems at T = 4K (blue circles). Ex-perimental data at T = 4.2K from Lim et al. [54] and Kimet al. [57] are shown by red squares and magenta triangles,respectively. The MF-like theoretical prediction by Lim etal. [54] is also plotted as an orange solid line.reported by Lim. et al. [54], are also plotted for compar-ison. We find that our estimation is in good agreementwith the experimental values. Figures 4 (a) and (b) showdetailed profiles of Mz and Mxy near the SR transitionpoint, respectively. It is difficult to identify the SR tran-sition for large x solely from the Mz–T curve. However,by analyzing the Mxy–T curve, the SR transition tem-perature TSR can be clearly detected as a function of x.Notably, the SR transition is observed even for small Ndconcentrations, down to x = 0.95.Figure 5 illustrates the concentration (x) dependenceof TSR for (Nd1−xDyx)2Fe14B systems. We find that TSRdecreases approximately linearly with increasing x. A lin-ear fit obtained by the least squares method, constrainedto pass through the point (x = 1, TSR = 0K), is shown bythe blue solid line and serves as a good approximation toour simulation results (blue circles). Experimental valuesof TSR for x ≤ 0.6 reported by Lim et al. [54] and thosefor x ≤ 0.75 reported by Kim et al. [57] are shown by redsquares and magenta triangles, respectively. In addition,an estimation of TSR as a function of x using a “MF-like”approach by Lim et al. is also given in Fig. 5 as a ref-erence [54]. In their approach, three-sites, i.e., Nd, Dy,and Fe sites were considered under several assumptions.The temperature dependences of the anisotropy constantof the Fe site and its moment mFe were assumed to besimilar to that in Y2F14B, with a modification due tothe difference in the Curie temperature, where Y ions isnon magnetic. The molecular-field vector H (T) at theNd site and H′ (T) at the Dy site were antiparallel andproportional to the Fe moment vector mFe, in which theR-R interaction was ignored. The values of H and H ′ at0 K were adjustable parameters. The total magnetizationwas determined by the condition that the total free en-ergy should be minimum with respect to the polar angleof mFe for zero external field. Therefore, unlike a usualMF theory based on self consistent field equations, theirmethod is a phenomenological treatment using effectivefields for Nd, Dy, and Fe sites.We find that our estimation of TSR is close to the ex-perimental values by Lim et al. and reasonably consis-tent with those by Kim et al. In Lim’s MF-like result,TSR decreases with increasing x and after x ≃ 0.7, itrapidly drops, reaching TSR = 0 at x ≃ 0.79. Our sim-ulation result differs from the MF-like result, especiallyin the high x regime. In Kim’s experiment, TSR ≃ 56Kat x = 0.75 was given. This temperature is much largerthan TSR ≃ 26K of the MF-like estimate. Our estimate,TSR ≃ 35K, lies between these two values. In the nextsubsection, we investigate the canting angle of the totalmoment and discuss its characteristics.B. Feature of canting angleFigure 6 depicts the concentration (x) dependence ofthe canting angle (θ) of the total magnetic moment fromthe c axis for (Nd1−xDyx)2Fe14B systems at T = 4K,calculated using the following relation:θ = tan−1 MxyMz. (6)Experimental data at T = 4.2K by Lim et al. [54] andKim et al. [57] are also plotted, along with a MF-likeestimation by Lim et al. [54] (orange solid line) forcomparison.The canting angle of the total moment in Nd2Fe14B,i.e., x = 0, has been experimentally estimated to be θ ≃6 0 10 20 30 40 0  0.2  0.4  0.6  0.8  1q (deg)xNdDyFe(a) 0 50 100 150 200 250 0  10  20  30  40  50  60  70  80  90Energy (K)q (deg) NdDyFe (x7)ca, bqFeqDyqNd(b) (c)FIG. 7. (a) Concentration x dependence of the canting anglebetween the magnetic moment and c-axis for Fe, Dy, andNd atoms. (b) Canting angles of Fe, Dy, and Nd atoms asa function of x. (c) Anisotropy energies as functions of thecanting angle θ from the c axis for Nd, Dy, and Fe atoms.The anisotropy energy of Fe has been magnified seven timesfor clarity.30–33 deg. at T = 4.2K [55]. Our simulation yields acanting angle of θ = 34 deg. for x = 0, which is in goodagreement with the experimental values.The estimated value of θ decreases gradually with in-creasing x. Although our values are slightly larger thanthe experimental results reported by Lim et al. and Kimet al., they successfully capture the overall trend withrespect to x. In the MF-like analysis by Lim et al., thecanting angle of the magnetic moment decreases rapidlyand disappears at x ≃ 0.79 as corresponding to theiranalysis of the x dependence of TSR. However, in ourresult the cant of the moment remains at such large xand it gradually decrease toward x = 1. At x = 0.75,the experimental canting angle reported by Kim et al. isapproximately θ ≃ 15 deg. Compared to θ ≃ 6 deg. fromthe MF-like model, our estimate of θ ≃ 18 deg. is muchcloser to the experimental value. Considering both thex-dependences of TSR and θ, our results suggest that TSRdoes not vanish at x ≃ 0.79, making our estimation moreconsistent with experimental observations.To investigate the canting behavior of the individualatomic moments, we present in Figs. 7(a) and (b) theconcentration (x) dependence of the canting angle (θ)from the c axis for Nd, Dy, and Fe atoms at T = 4K,estimated using the relation θ = tan−1(mxymz).With a decrease of the Nd concentration (i.e., an in-crease of x), θNd gradually decreases from θ = 36 deg.and still maintain a large cant (≃ 27 deg.) even at a verylow Nd concentration such as x = 0.9. In contrast, θDyshows a strong dependence on x. As the Dy concentra-tion decreases (i.e., as x decreases), θDy increases fromθ = 0 deg. and reaches approximately 22 deg. at x = 0.1.θFe exhibits the highest sensitivity to x among thethree elements and takes values between θNd and θDy.As the concentration of Nd or Dy increases, θFe tends toapproach the corresponding value of θNd or θDy. Theseobservations provides evidence that even at high x, theNd moments remain canted at very low temperatures,thereby driving the SR transition.We consider these canting angle dependencies from theperspective of the anisotropy energies of the constituentatoms. Figure 7(c) presents the per-site CEF energies forthe rare-earth atoms (R=Nd or Dy), given by,HCEF(R) =∑l,mΘlAml 〈rl〉Ôml . (7)and per-site anisotropy energy (ground state energy) ofFe atoms, expressed asHa = −1NFeFe∑iDi(szi )2, (8)as a function of the angle θ. In both plots, the energyminimum is set to zero, and Ha is multiplied by a factorof 7 for better visual clarity.When only exchange interactions are considered, thetotal energy is minimized when θNd = θFe = θDy =0. However, the results discussed above suggest thatthis configuration is not energetically favorable due tothe large CEF energy of Nd atoms, approximatelyHCEF(Nd) ≃ 250 K at θNd = 0. Therefore, the actualminimum total energy is determined by a trade-off be-tween the exchange interactions and the anisotropy en-ergies of each atomic species. Since the CEF energy forDy atoms and the anisotropy energy of Fe atoms haverelatively shallow curvatures near their respective min-ima and increase gradually with θ, the system allowssome deviation of Fe and Dy moments from their idealalignment. Consequently, the Nd moments maintain asignificant canting even at high Dy concentrations (x),enabling the SR transition to persist across a wide com-position range.C. Anomalous Behavior in MagnetizationIn this subsection, we study the origin of the anoma-lous behavior in magnetization. In Figs. 8 (a), (b), and(c), temperature dependence of M is illustrated by athick dashed line for x = 0, 0.4, and 0.8, respectively.7 1.9 1.95 2 2.05 2.1 2.15 2.2 0  50  100  150  200(a)M (µ B)T (K)MFit for x ≤  TSRFit for x ≥  TSR 1.4 1.45 1.5 1.55 1.6 0  50  100  150  200(b)M (µ B)T (K)MFit for x ≤  TSRFit for x ≥  TSR 0.88 0.9 0.92 0.94 0.96 0.98 0  50  100  150  200(c)M (µ B)T (K)MFit for x ≤  TSRFit for x ≥  TSRFIG. 8. Anomaly of the magnetization M of(Nd1−xDyx)2Fe14B systems for (a) x = 0, (b) x = 0.4,and (c) x = 0.8. The thick dashed curves represent thetemperature dependence of M . The blue and red curves arefitted functions for T ≤ TSR and T ≥ TSR, respectively.The curves of M are well fitted by least-squares fits us-ing cubic functions in the regions below and above TSR,shown by blue and red lines, respectively. We observe anon-differentiable point in M at TSR for x = 0, 0.4, and0.8. In a similar manner, non-differentiable points at TSRare found for all x values (x 6= 0). To quantify the degreeof this anomaly, we evaluate dM/dT as a function of Twith varying x.Figure 9 shows dM/dT as a function of temperaturefor various values of x. Fig. 10 presents x dependence ofthe gap in dM/dT , defined as ∆dM/dT at TSR. The solidline in Fig. 10 represents a cubic function fitted using theleast-squares method. We find that as x increases from0, the gap increases, reaches a maximum around x = 0.5,and then decreases toward x = 1.0. In the following, weexamine the origin of this behavior.In Figs. 11 (a), (b), and(c), the temperature depen-dences of m(Fe), m(Dy), and m(Nd) are plotted, respec-tively, for various values of x. We find that m exhibitsno anomaly for Fe and Dy atoms, but a clear anomalyis observed for Nd atoms across all values of x. As xincreases, m(Nd) smoothly extends toward lower tem--0.002-0.0015-0.001-0.0005 0 0.0005 0.001 0.0015 0  50  100  150  200dM/dT (µ B/K)T (K)x=0x=0.1x=0.2x=0.4x=0.5x=0.6x=0.8x=0.9x=0.95x=1FIG. 9. Temperature dependences of dMdTfor(Nd1−xDyx)2Fe14B systems. 0 0.2 0.4 0.6 0.8 0  0.2  0.4  0.6  0.8  1∆(dM/dT) (x10-3 µB/K)xFIG. 10. Concentration (x) dependence of ∆dMdTat TSR for(Nd1−xDyx)2Fe14B systems.peratures and reaches TSR, but shows a sharp increasebelow TSR. Interestingly, this behavior becomes morepronounced with increasing x. It is also noteworthy thatthe magnitude of the Nd moment at absolute zero tem-perature remains nearly constant regardless of x. Thisindicates that the temperature at which the Nd momentsexhibit anomalous behavior depends on the surroundingenvironment, specifically, the concentration of Dy atoms,but the ground-state magnitude of the Nd moment re-mains independent of the Dy concentration. As a result,the anomaly in the Nd moment at the SR transition tem-perature becomes more pronounced as the Dy concentra-tion increases.We estimate dm(Nd)/dT as a function of T for all val-ues of x. Following the same procedure as for estimatingdM/dT , we find the temperature dependence of m(Nd)for all x is well described by least-squares fitting of cu-bic functions in the regions below and above TSR. We8 1.8 1.9 2 2.1 2.2 2.3 0  50  100  150  200(a)m(Fe) (µB)T (K)x=0x=0.1x=0.2x=0.4x=0.5x=0.6x=0.8x=0.9x=0.95x=1 8 9 10 0  50  100  150  200(b)m(Dy) (µB)T (K)x=0.1x=0.2x=0.4x=0.5x=0.6x=0.8x=0.9x=0.95x=1 2.2 2.4 2.6 2.8 0  50  100  150  200(c)m(Nd) (µB)T (K)x=0x=0.1x=0.2x=0.4x=0.5x=0.6x=0.8x=0.9x=0.95FIG. 11. Temperature dependences of (a) m(Fe) (b) m(Dy),and (c) m(Nd) for (Nd1−xDyx)2Fe14B systems.thus obtain dm(Nd)/dT from the slopes. We plot thegap of dm(Nd)/dT at TSR, denoted ∆dm(Nd)/dT , as afunction of x by blue circles in Fig. 12. We find that∆dm(Nd)/dT increases rapidly increasing with x. Since∆dm(Nd)/dT is a per-site quantity, the total gap for Ndatoms, denoted ∆(Nd)total, is related by:∆(Nd)total =NNd ×∆dm(Nd)dT(9)∝(1− x)×∆dm(Nd)dT. (10) 0 1 2 3 4 0  0.2  0.4  0.6  0.8  1 0 0.1 0.2 0.3 0.4 0.5∆(dm(Nd)/dT) (x10-2 µB/K) (1-x) ∆(dm(Nd)/dT) (x10-2 µB/K) xFIG. 12. Concentration (x) dependences of ∆dm(Nd)dT(blue circles) and ∆dm(Nd)dT(1 − x) (red circles) at TSR for(Nd1−xDyx)2Fe14B systems.In Fig. 12, (1 − x) × ∆dm(Nd)dTis given with red cir-cles. The red solid line represents a cubic function fittedby the least-squares method. We find that the x depen-dence of (1−x)×∆dm(Nd)/dT at TSR closely resemblesthat of ∆dM/dT at TSR. Since the ratio of NNd to N(the total number of atoms) is 2(1 − x) : 17, the or-der of ∆dm(Nd)dT× 2(1−x)17 is expected to be comparableto that of ∆dM/dT . For example, ∆dm(Nd)dT× 2(1−x)17 ≃0.47 × 10−3µB/K for x = 0.5, which is the same orderas ∆dM/dT ≃ 0.67 × 10−3µB/K for x = 0.5. Theseconsiderations indicate that the origin of the anomaly inthe total magnetization M lies in the behavior of the Ndmagnetic moments, regardless of their concentration.V. SUMMARYNeodymium magnets (Nd2Fe14B) are important per-manent magnets and are often employed with dyspro-sium substitution to enhance their coercive force. There-fore, it is important to study the magnetic propertiesof (Nd1−xDyx)2Fe14B. In this work, we investigated thespin-reorientation (SR) transition in (Nd1−xDyx)2Fe14Busing a recently developed atomistic modeling approach.This method treats microscopic details of magnetic inter-actions and temperature effects appropriately. We em-ployed the Metropolis importance-sampling Monte Carlomethod for the (Nd1−xDyx)2Fe14B model, using micro-scopic parameters primarily derived from first-principlescalculations. We analyzed the temperature and con-centration (x) dependences of the magnetization andits components under zero external field. Our simula-tions yielded the spin-reorientation transition tempera-ture (TSR) and canting angle (θ) of the total magnetiza-tion as functions of x that are in good agreement withexperimental observations [54, 57]. We also found a sig-9nificant deviation in TSR at high Dy concentrations (x)between our results and those of the mean-field (MF)-liketheory by Lim et al. [54]. Based on our analysis of thecanting angles and anisotropy energies of the constituentatoms, we conclude that our estimations provide a morephysically reasonable description.Furthermore, we investigated an anomalous feature inthe magnetization near TSR. We found that the anomalyin the total magnetization becomes more pronouncedwith increasing x, reaches a maximum at x = 0.5, andthen diminishes for larger x. To elucidate the origin ofthis behavior, we analyzed the magnetic moments of theconstituent atoms and identified the Nd moments as theprimary source of the anomaly. As the Dy concentra-tion increases, the temperature at which the Nd momentanomaly appears shifts to lower values. However, themagnitude of the Nd moment rapidly increases belowTSR, and at sufficiently low temperatures, it convergesto nearly the same value for all x. In this regime, thecanting angle of the Nd moments changes only slightly.Consequently, the anomaly in the Nd moment at TSR be-comes more significant with increasing Dy concentration,leading to the observed anomaly in the total magnetiza-tion M , which peaks at x = 0.5.ACKNOWLEDGMENTSThe authors would like to thank Dr. Hirosawa forinsightful discussions on the experimental results of(Nd1−xDyx)2Fe14B, Prof. Miyashita for useful theoreti-cal discussions, and Dr. Toga for helpful discussions onthe magnetic parameters. This work was supported byGrants-in-Aid for Scientific Research B (No. 24K01332)from MEXT. 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