# Fileset [66_MT-M2024132.pdf](https://mdr.nims.go.jp/filesets/16a56c7c-04f4-4921-bea0-e20b3ab2d2c4/download) ## Creator [Masamichi Nishino](https://orcid.org/0000-0002-2060-2303), Seiji Miyashita ## Rights [In Copyright](http://rightsstatements.org/vocab/InC/1.0/) ## Other metadata [Atomistic Model Study on Magnetic Properties of Permanent Magnets—Treatment of Thermal Fluctuation and Thermal Effects, and Future Perspective—](https://mdr.nims.go.jp/datasets/4e621c9d-c03b-4a03-a167-2a7f64aca4a9) ## Fulltext Atomistic Model Study on Magnetic Properties of Permanent Magnets—Treatment of Thermal Fluctuation and Thermal Effects, and Future Perspective—Atomistic Model Study on Magnetic Properties of Permanent Magnets—Treatmentof Thermal Fluctuation and Thermal Effects, and Future Perspective—+1Masamichi Nishino1,+2 and Seiji Miyashita21National Institute for Materials Science, Tsukuba 305-0047, Japan2Graduate School of Science, The University of Tokyo, Tokyo 113-0033, JapanWe review atomistic spin model studies, a new approach for theoretical investigations, on magnetic properties of permanent magnets. Inthe atomistic modeling, the microscopic details of magnetic parameters and lattice structures are realistically considered, and the temperatureeffect, including thermal fluctuation, is properly treated based on statistical physics methods: Monte Carlo methods and stochastic Landau-Lifshitz-Gilbert equation methods. We introduce how to treat thermal effects for static and dynamical properties using these methods. Focusingespecially on neodymium permanent magnets, we discuss features of magnetization, domain wall, coercivity of a grain, nucleation and pinningfields, and dysprosium substitution effect, which were first elucidated with those methods. [doi:10.2320/matertrans.MT-M2024132](Received August 28, 2024; Accepted October 28, 2024; Published December 25, 2024)Keywords: atomistic spin model, thermal fluctuation, thermal effects, permanent magnets, stochastic Landau-Lifshitz-Gilbert (LLG) equation,Monte Carlo method1. IntroductionThe control of the coercive force (coercive field) ofpermanent magnets is an important issue for achieving highenergy conversion efficiency. Neodymium magnets (Nd-Fe-Bmagnets) [1–9] are known as powerful permanent magnetsand used in motors, generators, electrical appliances, etc.Their application is expected to expand in the future due tothe growing demand for electric vehicle motors. Neodymiummagnets have a problem with coercivity at high temperaturesand are often reinforced by adding heavy rare earths suchas dysprosium (Dy). Japan depends on foreign countries forthe rare earth resources, but due to price instability, etc.,development of high-performance permanent magnets withreduced use of rare elements is required.Against this background, the Elements Strategy InitiativeCenter for Magnetic Materials (ESICMM) project (2012–2021), a national project based at National Institute forMaterials Science (NIMS), has been carried out for 10 yearssince 2012, and coercivity research has made significantprogress in both experimental and theoretical aspects [10]. Inthis review, we mainly focus on the microscopic modelingbased on the electron theory and theoretical studies on thecoercivity at finite temperatures which were developed by theauthors and collaborators in ESICMM [11–28].Theoretical studies of the magnetic properties and coerciveforce of permanent magnets have been developed in the fieldof micromagnetics [29]. For details of micromagneticssimulations, please refer to the review by Tomohiro Tanakain the same special issue. In micromagnetics simulations,a continuum model in which the magnetization is spatiallycontinuous with nm-order block magnetization (fixedmagnitude of magnetization) as the smallest unit isconsidered, and magnetic materials are described with asmall number of macroscopic magnetic parameters, mainlyexchange stiffness constant (A) and magnetic anisotropyenergy (K ). This method has the advantage that the system(the size is on the order of µm) of the aggregate of grainsconstituting a magnetic material can be treated, and manysimulations of magnetic properties such as magnetizationreversal have been carried out. However, due to the coarsegraining of the model, the microscopic mechanism of themagnetic properties induced by atomic-scale magneticinteractions cannot be treated. In addition, there is a problemin handling thermal fluctuations and temperature effects, asdescribed below.Coercivity at finite temperatures is a phenomenon thatinvolves the collapse of a metastable magnetic state due tothermal activation. The magnetization reversal is a stochasticprocess, since it requires crossing the energy barrier and iscaused by thermal fluctuations. In order to study such effectsquantitatively, it is necessary to treat entropy effects attemperature T correctly according to statistical mechanics.In thermal equilibrium, using the energy E(i) of state i, theprobability of realization of state i, Peq(i), in the canonicalensemble is expressed asPeq ¼1Ze�¢EðiÞ: ð1ÞHere, ¢ is defined by ¢ ¼ 1kBTusing the Boltzmann constantkB. Z is a quantity called the partition function and defined byZ ¼Xallie�¢EðiÞ: ð2ÞThe sum is taken for all states. Thus, the value of the thermalequilibrium state ©Aª of any physical quantity A is given bythe average in the canonical ensemble as followshAi ¼ 1ZXalliAðiÞe�¢EðiÞ: ð3ÞWhen using the coarse-grained Hamiltonian for thecontinuum model of micromagnetics, the degrees of freedom(entropy) of the states are unclear and it is difficult to applythe above probabilities. In order to properly handle temper-+1This Paper was Originally Published in Japanese in J. Japan Inst. Met.Mater. 87 (2023) 158–172.+2Corresponding author, E-mail: nishino.masamichi@nims.go.jpMaterials Transactions, Vol. 66, No. 1 (2025) pp. 1 to 16©2024 The Japan Institute of Metals and Materials OVERVIEWhttps://doi.org/10.2320/matertrans.MT-M2024132ature effects, it is necessary to use statistical mechanicsmethods with the canonical ensemble. By constructing a spinmodel based on atomistic theory that takes into account thespin of all atoms in the system, the state and number of states(entropy) can be defined, and statistical mechanics methodscan be used to correctly incorporate temperature effects in theanalysis. In this paper, we will focus on neodymium magnetsin particular, explain the atomistic spin model, and introducestatistical mechanics methods for calculating magneticproperties at finite temperatures (including absolute zero).Various magnetic properties revealed by using the method-ologies will be discussed.2. Atomistic Spin ModelThe unit cell of Nd2Fe14B, the hard magnet phase ofneodymium magnets, is shown in Fig. 1 [30]. There are twotypes of Nd sites, six types of Fe sites and one type of B site.We adopt the following atomistic Hamiltonian as a micro-scopic model for neodymium magnets [13–15, 24]. Thissystem is a three-dimensional ferromagnet, and we treat thespins classically.H ¼ �Xi 0, only the nearest neighbor interaction wasconsidered, and g®B = 1 was the unit.In this case, J was used instead of A, D instead of K, andthe parameters F and E were defined asF ¼ J2S2J1S1; ð38ÞE ¼ J2D2J1D1: ð39ÞThe magnitudes of magnetization were set equal in regions I,II, and III («Si« = S1 = S2 = 1) and the magnetic anisotropyenergy of the hard phase was set to D1 = 0.2J1. The Curietemperature of the hard phase is TC ’ 1:5J1. Here, for themagnetic field, we used the normalized field h = H/HSW as ineq. (31). Here, HSW = 2D1/S1.Using the sLLG method, we simulated the time evolutionof the states by applying a reversed magnetic field to the(+++) state and investigated the magnetic field thresholds.(a) (b) (c)E/F(1-E )/(1+ F1/2)2Fig. 10 Threshold fields between (+++) and (+¹+) (blue upward triangles) and between (+¹+) and (¹¹¹) (red downwardtriangles) for F = 0.5 at (a) T = 0, (b) T = 0.3J1, and (c) T = 0.5J1. The dotted lines denote hNCII(0) in (34) and hDWP(0) in (35).Reprinted figure with modification from [S. Mohakud et al., Phys. Rev. B 94, 054430 (2016)]. Copyright (2016) by the AmericanPhysical Society. (online color)Atomistic Model Study on Magnetic Properties of Permanent Magnets 9Figure 10(a) shows the magnetic field thresholds for(+++)¼ (+¹+) (blue line) and (+¹+)¼ (¹¹¹) (redline) at T = 0. It can be seen that the analytical value forthe continuum model (dotted line) is a good approximation.Figure 11(a) gives the threshold field of (+¹¹) ¼ (¹¹¹)at T = 0. In this case, the magnetic field thresholds for smallE coincide with the analytical values of the continuum model,but for large E, the magnetic field thresholds shift upwardfrom the analytical values. This is due to the so-called narrowdomain wall effect, which makes the continuum approx-imation poor [12]. As seen in Figs. 10(b) and 10(c), the fieldthresholds of (+++) ¼ (+¹+) and (+¹+) ¼ (¹¹¹)decrease significantly from the field thresholds of T = 0 asthe temperature increases, and the (+¹+) region expands.As shown in Figs. 11(b) and 11(c), the threshold field of(+¹¹)¼ (¹¹¹) also decreases due to the temperatureeffect, but the E dependence becomes moderate. Especially inthe narrow domain wall situation, the magnetization state ofregions II and III has little influence on region I, and thesurface nucleation in region I becomes important [12].6.2 Nucleation and pinning fields in Nd2Fe14BThe nucleation and pinning fields were evaluated for thecases of Bloch and Néel walls, respectively, by simulatinga hard-soft-hard magnet model composed of the atomisticmodel (4) of Nd2Fe14B [20]. In the case of Bloch wall, the adirection corresponds to the x direction in Fig. 9, and the cdirection to the x direction in the case of Néel wall. Thesoft magnet phase (grain boundary) is complex with anamorphous-like structure that depends on experimentalconditions. Since the first-principles study [60, 61] has justbegun and it is difficult to use the microscopic parametersestablished at this time, we assumed the same structure asthe hard magnet phase. The parameters E and F are definedas in the previous section. However, in Nd2Fe14B, there arevarious kinds of exchange interactions and magneticanisotropy energies, which were scaled in the sameproportion. Figure 12(a) shows for the Bloch and Néeldomain walls the E-dependences of the threshold fieldbetween (+++) and (+¹+) and between (+¹+) and(¹¹¹) at T = 0K. We assumed F = 0.5. L1 = L2 = L3 =12 unit cells and the height and depth are 5 unit cells. Theoutline of the diagram is similar to that of the anisotropicHeisenberg model described in the previous section. Thethreshold field is lower in the case of the Néel wall, reflectingthe weaker exchange interaction in the c-axis. Figure 12(b)exhibits the threshold field between (+¹¹) and (¹¹¹) atT = 0K. This pinning field is not so different between thetwo domain walls as in the case of the nucleation field. WhereE is large, the threshold value increases reflecting the natureof the narrow down wall.Figure 13(a) depicts the magnetic field thresholds between(+++) and (+¹+) and between (+¹+) and (¹¹¹) atT = 0.34TC, and Fig. 13(b) shows the threshold fieldbetween (+¹¹) and (¹¹¹). Compared with the changeof the threshold field between T = 0 and T = 0.5J1(a) (b) (c)Fig. 11 Threshold fields between (+¹¹) and (¹¹¹) (red downward triangles) for F = 0.5 at (a) T = 0, (b) T = 0.3J1, and (c) T =0.5J1. Blue dashed lines denote threshold fields between (+¹+) and (¹¹¹) in Fig. 10. Reprinted figure with permission from[S. Mohakud et al., Phys. Rev. B 94, 054430 (2016)]. Copyright (2016) by the American Physical Society. (online color)(a) (b)Fig. 12 (a) Threshold fields at 0K between (+++) and (+¹+) (circles) and between (+¹+) and (¹¹¹) (triangles) with thecomparison between Bloch and Néel domain walls. (b) Threshold fields at 0K between (+¹¹) and (¹¹¹) with the comparisonbetween Bloch and Néel domain walls. Reprinted figure with permission from [I.E. Uysal et al., Phys. Rev. B 101, 094421 (2020)].Copyright (2020) by the American Physical Society. (online color)M. Nishino and S. Miyashita10(’ 0:33TC) for F = 0.5 in the anisotropic Heisenberg modelof the previous section, the threshold field of Nd2Fe14Bshows a larger decrease rate due to temperature effects. Thisis may be due to the sensitiveness to temperature effectsbecause of the weaker exchange interaction between Nd andFe compared to that between Fe and due to the property ofthe magnetic anisotropy energy of Nd, which is the origin ofthe spin rearrangement transition.7. Dysprosium Substitution EffectAs mentioned in the introduction, Nd magnets haveproblems in high-temperature properties and are oftensubstituted with Dy to suppress the decrease in coerciveforce at high temperatures. Experiments using the methodof grain boundary diffusion process have shown that thecoercivity can be enhanced without loosing remanence [62–66]. Dy-rich shells have been observed to form between thegrain boundary and core grains [66]. The effect of thiscoercivity enhancement by Dy substitution has been studiedby micromagnetics calculations using macroscopic parame-ters of the Dy-substituted phase [67, 68]. The enhancementeffect at absolute zero temperature has also been investigatedin a simplified crystal lattice model [69]. However, themicroscopic origin of the coercivity-enhancing effect of Dysubstitution, including the temperature effect, can be under-stood only by the method of atomistic modeling.Here we explain the mechanism and effect of the coercivityenhancement by Dy substitution based on an atomistic model[27]. The Hamiltonian (4) has the energy term of the crystalelectric field of eq. (40).H ¼XNd,DyiXl;m�l;iAml;ihrliiÔml;i ð40Þis composed of Nd and Dy atoms. For Dy, the coefficients ofthe crystal electric field, etc. are those of Dy. For Dy atoms,J ¼ Lþ S ¼ 15=2 and gT = 4/3. As shown in Fig. 2(b), siof Dy atoms and Si (= si) of Fe atoms are antiferromagneti-cally coupled. The magnitude of this exchange interaction isalmost the same as that between Nd and Fe atoms [27]. Inthis case, unlike Fig. 2(a), Si (¼ si þJ i) of a Dy atom and Si(= si) of an Fe atom are antiferromagnetically coupled.As shown in Fig. 14, the Nd layers are numbered fromn = 1 at the (001) surface of the hard magnet phase ofNd2Fe14B to n = 2 and n = 3* toward the interior. Weconsidered two systems: the system in Fig. 14(a) where thesurface is in contact with the vacuum (system A) and thesystem in Fig. 14(b) where the surface is in contact with thesoft magnet phase (system B). The hard magnet phaseconsists of 12 © 12 © 9 unit cells, the soft magnet phase12 © 12 © 3 unit cells, with periodic boundary conditions inthe a and b axes, and a free boundary condition for system Aand periodic boundary condition for system B in the c axis.Here, all Nd atoms from layer 1 to layer n were replaced by(a) (b)Fig. 13 (a) Threshold fields at 0.34TC between (+++) and (+¹+) (circles) and between (+¹+) and (¹¹¹) (triangles) with thecomparison between Bloch and Néel domain walls. (b) Threshold fields at 0.34TC between (+¹¹) and (¹¹¹) with the comparisonbetween Bloch and Néel domain walls. Reprinted figure with permission from [I.E. Uysal et al., Phys. Rev. B 101, 094421 (2020)].Copyright (2020) by the American Physical Society. (online color)(a) (b)n=1n=2n=3Vacuum Soft phasen=1n=2n=3Fig. 14 (a) System A. The Nd surface layers were in contact with vacuum. In this example, Nd atoms (red) in the first Nd layerwere substituted by Dy atoms (orange). The Nd layers are numbered as n = 1, 2,* . (b) System B. The Nd surface layers were incontact with a soft magnet phase. [M. Nishino et al., Phys. Rev. B 106, 054422 (2022)]. Copyright (2022) by the American PhysicalSociety. (online color)Atomistic Model Study on Magnetic Properties of Permanent Magnets 11Dy atoms, and the threshold field of magnetization reversalwas investigated. As in the previous section, the soft magnetphase was assumed to have the same structure as the hardmagnet phase, and small magnetic parameters (exchangeinteractions by a factor of 0.5 and magnetic anisotropyenergies by a factor of 0.2) were adopted as in themicromagnetics calculations. The threshold field of magnet-ization reversal was estimated by real-time simulation of thesLLG method.Figure 15 presents the n dependence of the threshold fieldand its temperature dependence for system A (dotted line)and system B (solid line). n = 0 means no Dy substitution(original system). The ratios (%) in the figures are thoseagainst the value of n = 0. Since magnetization reversaloccurs more easily in the grain boundary phase, domainwalls are more easily generated at the phase boundary, andsurface nucleation of the hard magnet phase tends to occur.Therefore, the coercive force is greatly reduced compared tothat of the vacuum surface. However, the ratio against n = 0when n increases in system B is larger than that in system A,and the coercive force increases by a large ratio with Dysubstitution in system B. In other words, the enhancementof coercivity is more intensely in system B. This is acharacteristic feature of Dy substitution, and it suggests thatDy substitution has a strong pinning effect hindering thedomain wall motion to the hard magnet phase. The rate ofincrease of coercivity (¦hC/¦n) with respect to the numberof Dy substitution layers (n) is constant at finite temperatureand almost equal for both systems.Regarding system A, we investigated n-dependence of thecoercivity for the system with the magnetic anisotropy of Ndatoms enhanced by a factor of 2 instead of Dy substitution[22], and compared it with that of Dy substitution. Wecompared the ratios of increase of n = 5 against n = 0 for0.34TC, 0.46TC, and 0.69TC. In the case of Dy-substitution,the increase is 144%, 140%, and 132%, respectively, whilein the case of doubly enhanced magnetic anisotropy of Ndatoms, the increase is 123%, 121%, and 111% [22]. Thisindicates that the coercivity enhancement effect is maintainedat high temperatures in the case of Dy substitution.Next, we observed at the change in nucleation dynamicswhen the substituted Dy layer was increased. Figure 16shows the nucleation of n = 1 and n = 5 in system B. Themagnetic field was used near the threshold value.Figure 16(a) shows the case with n = 1 where only onelayer was Dy-substituted. A large part of the soft magnetphase was reversed before the magnetization of the hardmagnet phase was reversed, indicating that depinning occursfrom the surface of the hard magnet phase. On the other hand,Fig. 16(b) has five Dy-substituted layers and nucleationoccurred from the inside of the hard magnet part. Surfacenucleation is suppressed by Dy substitution, and is replacedby bulk nucleation. This change in nucleation mechanismwas also observed in system A, but n boundary for thechange from surface nucleation to internal nucleation wasn ’ 2 in system A and n ’ 3 in system B.There are two possible microscopic origins of thecoercivity enhancement effect of Dy substitution and itsmaintenance at high temperatures. The first is the differencein magnetic interactions. The interaction between Nd andFe atoms is ferromagnetic, while that between Dy and Fe isantiferromagnetic. This antiferromagnetic coupling stabilizesthe Dy moment under the reversed magnetic field andprevents nucleation for reversal. The second is the featureof the temperature dependence of the CEF energy of Dy.Figure 4(b) depicts the temperature dependence of the CEFenergy of Dy atoms in the Nd magnet. Compared withFig. 4(a), the CEF energy of Dy atoms has a minimum at123456780 1 2 3 4 5 6h (T)n106% 121% 132%108%129% 141%116%138%158%123456780 1 2 3 4 5 6h (T)n103%122%140%123456780 1 2 3 4 5 6h (T)n121%151%178%107%123%144%1015202530350 1 2 3 4 5 6h (T)n105%100% 101%150%173%185%(a) (b)(c) (d)0 K 0.34 TC0.46 TC0.69 TCFig. 15 n dependence of the threshold filed in systems A (dotted lines) and B (solid lines) at (a) 0K, (b) 0.34TC, (c) 0.46TC, and (d)0.69TC. Reprinted figure with permission from [M. Nishino et al., Phys. Rev. B 106, 054422 (2022)]. Copyright (2022) by the AmericanPhysical Society. (online color)M. Nishino and S. Miyashita12ª = 0 regardless of temperature, and the potential barrier isrelatively high at all temperatures. Furthermore, when wefocus on the change of the potential barrier between 0.46TC(near room temperature) and 0.69TC, we find that thepotential barrier of Dy is 27% higher than that of Nd at0.46TC, but is 79% higher at 0.69TC. In other words, Dy hasa relatively higher potential barrier at high temperatures. Thismay be the reason why Dy substitution is effective at hightemperatures.8. Magnetic Dipole InteractionSince the atomistic model method takes into account thespins of all atoms, the computable scale is realistically limitedto the order of tens of nm. The magnetic dipole interactionis much smaller than the exchange interaction and is notvery important at that scale. Therefore, the magnetic dipoleinteraction14³®0Xi6¼k1r3ikSi � Sk �3ðrik � SiÞðrik � SkÞr2ik� �ð41Þhas not been taken into account, but it becomes important atthe scale of an assembly of grains (µm) because of theappearance of uniform magnetization in ferromagnets. Thedemagnetization field effect given by this interaction has theeffect of breaking the uniform order, and experiments andmicromagnetic calculations have shown a decrease incoercivity (increase in demagnetization field effect) withrespect to grain size [70–73]. The coercivity of nanograinsshown in this paper (saturation at about 20 nm) is consideredto give an approximation of the upper limit of the coercivityincluding the effect of thermal fluctuations of grain assembly.The exchange interaction is of short range and thecomputational cost of the simulation is proportional to thespin number N, while the magnetic dipole interaction is oflong range and thus it increases with the square of the spinnumber (N2). In Monte Carlo methods, the stochastic cutoff(SCO) and other methods have been proposed to overcomethis difficulty [74, 75]. The SCO method accelerates thecalculation by reducing the selectivity of long-range weakinteractions in a state-update process of the Monte Carlomethod while maintaining the detailed balance condition andensuring that the simulation reaches equilibrium correctly asa steady state. For a 3-dimensional system of magnetic dipoleinteractions, a single Monte Carlo step can be computed inO(N lnN ) time. However, the conventional SCO method hasdifficulties that make the procedure cumbersome for systemswith complex unit cells such as Nd magnets, but a modifiedSCO method (MSCO) using Walker’s algorithm has beendeveloped and the effect of magnetic dipole interaction hasbeen investigated in a thin film system of an atomistic modelof Nd [15]. In addition, magnetic profiles have been studiedusing MSCO for a simple anisotropic Heisenberg model withparameters of anisotropy constant, dipole interaction, andsystem thickness [26].9. Summary and Future PerspectiveThis paper introduces the methodologies for studying theeffects of temperature and thermal fluctuations on themagnetic properties of permanent magnets and describesthe magnetic properties obtained by applying the method-ologies, especially focusing on the properties of neodymiummagnets. The method of the atomistic model introduced hereis an extension of the study of thermal fluctuation effectsinitiated by Brown [76], and correctly treats entropy effectsand temperature effects including local thermal fluctuations inmany-body systems of spins with cooperative interactions. Inthis paper, we first presented a modeling of the interactionfrom the atomic level in Nd2Fe14B. Then, the characteristicsof magnetization process, domain wall, coercivity ofnanograin, nucleation and pinning fields, and dysprosiumsubstitution effect were explained using Monte Carlo andsLLG methods. There, the understanding of thermal(a)(b)n=1n=5Fig. 16 Snapshots of the spin configuration in nucleation for (a) n = 1 and (b) n = 5 at the threshold fields h = 3.70T and h = 5.05T,respectively, at 0:46TC ’ Troom in system B. The yellow-boxed regions correspond to the soft magnetic phase. Red and blue parts denoteup-spin and down-spin ones, respectively. Reprinted figure with permission from [M. Nishino et al., Phys. Rev. B 106, 054422 (2022)].Copyright (2022) by the American Physical Society. (online color)Atomistic Model Study on Magnetic Properties of Permanent Magnets 13fluctuations and temperature effects and their microscopicmechanisms in the magnetic properties of permanentmagnets, which had not been well understood, was advanced,as seen in the significant decrease in coercive force due tothermal fluctuations.Although we did not mention them due to space limitation,the angular dependence of nucleation and pinning fields andthe effect of the orientation distribution of magneticanisotropy of grains are also important for coercivity studies,and studies of their finite temperature properties have beenstarted using the above methodologies [77]. A nontrivialbehavior in the temperature dependence of the ferromagneticresonance of neodymium magnets has also been clarified bythe method of atomistic modeling [18].The magnetization reversal of individual grains has beenobserved experimentally, and the effect of thermal fluctua-tions on the process has become a realistic problem [78].Therefore, the rigorous treatment of thermal fluctuations andtemperature effects by this atomistic model is expected tobecome more and more important in the future. In fact,during the ESICMM project, European and Chinese groupshave also started to study the atomistic model of Nd2Fe14B[79–83], and the study of permanent magnets using thisapproach is expected to be developed hereafter. However,when using the atomistic model, the current computingpower is limited to calculations on the scale of a few tensof nm. This is the range in which a single grain or a fewgrains can be included. Coercivity appears as a property ofan assembly of grains. The assembly of grains requirescalculations on the order of µm, which is done inmicromagnetics calculations.If we start from an atomistic model and obtain macro-scopic A(T ) and K(T ) using the constraint Monte Carlomethod introduced in 4.2, the LLG calculation for micro-magnetics cannot handle thermal activation processes.Adding a noise field for thermal fluctuations leads to adouble temperature effect, which is inappropriate. Therefore,it is necessary to develop a methodology that starts from anatomistic model and connects it to the µm order scale. Forthis purpose, it is necessary to construct a coarse-grainedmodel consisting of local spin variables that incorporates theeffects of thermal fluctuations. For example, the followingmethod can be considered. For a spin set in a certain region,we introduce a coarse-grained magnetization M. Themagnitude of this coarse-grained magnetization is variabledue to temperature effects, and the magnetization distributionP(M ) can be calculated by the Wang-Landau method usingan atomistic model. For this coarse-grained magnetization,we construct a coarse-grained Hamiltonian.H ¼ �Xi;jJi;jMi �Mj þXiEselfðMiÞ; ð42Þwhere Eself (Mi) is the magnetic anisotropy energy and othercontributions. The renormalized parameters (Ji, j, etc.) areadjusted to reproduce the values of physical quantitiesobtained in the original atomistic Hamiltonian. It isconceivable to apply this coarse-grained Hamiltonian tolarge systems, including magnetic dipole interactions. Thedetails are under investigation.AcknowledgementsThe study presented here was supported by the ElementsStrategy Initiative Center for Magnetic Materials (ESICMM)(project number JPMXP0112101004) funded by the Ministryof Education, Culture, Sports, Science and Technology(MEXT) of Japan. This paper is the result of collaborativeresearches with Satoshi Hirosawa, Ismail Enes Uysal, HiroshiHayasaka, Yuta Toga, Taichi Hinokihara, Sasmita Mohakud,Sergio Andraus, Munehisa Matsumoto, Shotaro Doi,Hisazumi Akai, Akimasa Sakuma and Takashi Miyake. Wewould like to take this opportunity to express our deepestgratitude to them. We also thank the other ESICMMmembers for helpful discussions.REFERENCES[1] M. Sagawa and S. Hirosawa: Magnetic hardening mechanism insintered R–Fe–B permanent magnets, J. Mater. Res. 3 (1988) 45–54.[2] S. Hirosawa, Y. Matsuura, H. Yamamoto, S. Fujimura, M. Sagawa andH. Yamauchi: Single Crystal Measurements of Anisotropy Constantsof R2Fe14B (R=Y, Ce, Pr, Nd, Gd, Tb, Dy and Ho), Jpn. J. Appl. Phys.24 (1985) L803.[3] J.F. Herbst: R2Fe14B materials: Intrinsic properties and technologicalaspects, Rev. Mod. Phys. 63 (1991) 819–898.[4] S. Hirosawa, Y. Matsuura, H. Yamamoto, S. Fujimura, M. Sagawa andH. Yamauchi: Magnetization and magnetic anisotropy of R2Fe14Bmeasured on single crystals, J. Appl. Phys. 59 (1986) 873–879.[5] A.V. Andreev, A.V. Deryagin, N.V. Kudrevatykh, N.V. Mushnikov,V.A. Reimer and S.V. Terent’ev: MAGNETIC PROPERTIES OFY2FE14B AND ND2FE14B AND THEIR HYDRIDES, Sov. Phys.JETP 63 (1986) 608–612.[6] O. Yamada, Y. Ohtsu, F. Ono, M. Sagawa and S. Hirosawa:Magnetocrystalline anisotropy in Nd2Fe14B intermetallic compound,J. Magn. Magn. Mater. 70 (1987) 322–324.[7] X.C. Kou, R. Grössinger, G. Hilscher, H.R. Kirchmayr and F.R.de Boer: ac susceptibility study on R2Fe14B single crystals (R = Y, Pr,Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm), Phys. Rev. B 54 (1996) 6421–6429.[8] C. Piqué, R. Burriel and J. Bartolomé: Spin reorientation phasetransitions in R2Fe14B (R = Y, Nd, Ho, Er, Tm) investigated by heatcapacity measurements, J. Magn. Magn. Mater. 154 (1996) 71–82.[9] S. Hirosawa, M. Nishino and S. Miyashita: Perspectives for high-performance permanent magnets: applications, coercivity, and newmaterials, Adv. Nat. Sci.: Nanosci. Nanotechnol. 8 (2017) 013002.[10] S. Hirosawa: Magune 17 (2022) 175–180.[11] M. Nishino and S. Miyashita: Realization of the thermal equilibriumin inhomogeneous magnetic systems by the Landau-Lifshitz-Gilbertequation with stochastic noise, and its dynamical aspects, Phys. Rev. B91 (2015) 134411.[12] S. Mohakud, S. Andraus, M. Nishino, A. Sakuma and S. Miyashita:Temperature dependence of the threshold magnetic field for nucleationand domain wall propagation in an inhomogeneous structure withgrain boundary, Phys. Rev. B 94 (2016) 054430.[13] Y. Toga, M. Matsumoto, S. Miyashita, H. Akai, S. Doi, T. Miyake andA. Sakuma: Monte Carlo analysis for finite-temperature magnetism ofNd2Fe14B permanent magnet, Phys. Rev. B 94 (2016) 174433.[14] M. Nishino, Y. Toga, S. Miyashita, H. Akai, A. Sakuma and S.Hirosawa: Atomistic-model study of temperature-dependent domainwalls in the neodymium permanent magnet Nd2Fe14B, Phys. Rev. B 95(2017) 094429.[15] T. Hinokihara, M. Nishino, Y. Toga and S. Miyashita: Exploration ofthe effects of dipole-dipole interactions in Nd2Fe14B thin films basedon a stochastic cutoff method with a novel efficient algorithm, Phys.Rev. B 97 (2018) 104427.[16] S. Miyashita, M. Nishino, Y. Toga, T. Hinokihara, T. Miyake, S.Hirosawa and A. Sakuma: Perspectives of stochastic micromagnetismof Nd2Fe14B and computation of thermally activated reversal process,Scr. Mater. 154 (2018) 259–265.[17] Y. Toga, M. Nishino, S. Miyashita, T. Miyake and A. Sakuma:M. Nishino and S. Miyashita14https://doi.org/10.1557/JMR.1988.0045https://doi.org/10.1143/JJAP.24.L803https://doi.org/10.1143/JJAP.24.L803https://doi.org/10.1103/RevModPhys.63.819https://doi.org/10.1063/1.336611https://doi.org/10.1016/0304-8853(87)90456-2https://doi.org/10.1103/PhysRevB.54.6421https://doi.org/10.1016/0304-8853(95)00571-4https://doi.org/10.1088/2043-6254/aa597chttps://doi.org/10.1103/PhysRevB.91.134411https://doi.org/10.1103/PhysRevB.91.134411https://doi.org/10.1103/PhysRevB.94.054430https://doi.org/10.1103/PhysRevB.94.174433https://doi.org/10.1103/PhysRevB.95.094429https://doi.org/10.1103/PhysRevB.95.094429https://doi.org/10.1103/PhysRevB.97.104427https://doi.org/10.1103/PhysRevB.97.104427https://doi.org/10.1016/j.scriptamat.2017.11.012Anisotropy of exchange stiffness based on atomic-scale magneticproperties in the rare-earth permanent magnet Nd2Fe14B, Phys. Rev. B98 (2018) 054418.[18] M. Nishino and S. Miyashita: Nontrivial temperature dependence offerromagnetic resonance frequency for spin reorientation transitions,Phys. Rev. B 100 (2019) 020403(R).[19] M. Nishino, I.E. Uysal, T. Hinokihara and S. Miyashita: Dynamicalaspects of magnetization reversal in the neodymium permanent magnetby a stochastic Landau-Lifshitz-Gilbert simulation at finite temper-ature: Real-time dynamics and quantitative estimation of coerciveforce, Phys. Rev. B 102 (2020) 020413(R).[20] I.E. Uysal, M. Nishino and S. Miyashita: Magnetic field threshold fornucleation and depinning of domain walls in the neodymiumpermanent magnet Nd2Fe14B, Phys. Rev. B 101 (2020) 094421.[21] Y. Toga, S. Miyashita, A. Sakuma and T. Miyake: Role of atomic-scalethermal fluctuations in the coercivity, npj Comput. Mater. 6 (2020) 67.[22] M. Nishino, I.E. Uysal and S. Miyashita: Effect of the surface magneticanisotropy of neodymium atoms on the coercivity in neodymiumpermanent magnets, Phys. Rev. B 103 (2021) 014418.[23] M. Nishino, I.E. Uysal, T. Hinokihara and S. Miyashita: Finite-temperature dynamical and static properties of Nd magnets studied byan atomistic modeling, AIP Adv. 11 (2021) 025102.[24] S. Miyashita, M. Nishino, Y. Toga, T. Hinokihara, I.E. Uysal, T.Miyake, H. Akai, S. Hirosawa and A. Sakuma: Atomistic theory ofthermally activated magnetization processes in Nd2Fe14B permanentmagnet, Sci. Technol. Adv. Mater. 22 (2021) 658–682.[25] S. Miyashita, M. Nishino, Y. Toga, T. Hinokihara, I.E. Uysal, T.Miyake, H. Akai, S. Hirosawa and A. Sakuma: Atomistic Theory ofThermally Activated Magnetization Processes in Nd2Fe14B PermanentMagnet, J. Jpn. Soc. Powder Powder Metallurgy 69 (2022) S126–S146.[26] T. Hinokihara and S. Miyashita: Systematic survey of magneticconfigurations in multilayer ferromagnet system with dipole-dipoleinteraction, Phys. Rev. B 103 (2021) 054421.[27] M. Nishino, H. Hayasaka and S. Miyashita: Microscopic origin ofcoercivity enhancement by dysprosium substitution into neodymiumpermanent magnets, Phys. Rev. B 106 (2022) 054422.[28] M. Nishino and S. Miyashita: submitted.[29] H. Kronmüllar and M. Fähnle: Micromagnetism and the Micro-structure of Ferromagnetic Solids, (Cambridge University Press,2003).[30] J.F. Herbst, J.J. Croat, F.E. Pinkerton and W.B. Yelon: Relationshipsbetween crystal structure and magnetic properties in Nd2Fe14B, Phys.Rev. B 29 (1984) 4176–4178.[31] A.I. Liechtenstein, M.I. Katsnelson, V.P. Antropov and V.A. Gubanov:Local spin density functional approach to the theory of exchangeinteractions in ferromagnetic metals and alloys, J. Magn. Magn. Mater.67 (1987) 65–74.[32] Y. Miura, H. Tsuchiura and T. Yoshioka: Magnetocrystallineanisotropy of the Fe-sublattice in Y2Fe14B systems, J. Appl. Phys.115 (2014) 17A765.[33] M. Yamada, H. Kato, H. Yamamoto and Y. Nakagawa: Crystal-fieldanalysis of the magnetization process in a series of Nd2Fe14B-typecompounds, Phys. Rev. B 38 (1988) 620–633.[34] A.J. Freeman and R.E. Watson: Theoretical Investigation of SomeMagnetic and Spectroscopic Properties of Rare-Earth Ions, Phys. Rev.127 (1962) 2058–2075.[35] J.L. García-Palacios and F.J. Lázaro: Langevin-dynamics study of thedynamical properties of small magnetic particles, Phys. Rev. B 58(1998) 14937–14958.[36] H. Suwa and S. Todo: Markov Chain Monte Carlo Method withoutDetailed Balance, Phys. Rev. Lett. 105 (2010) 120603.[37] B. Berg and T. Neuhaus: Multicanonical ensemble: A new approach tosimulate first-order phase transitions, Phys. Rev. Lett. 68 (1992) 9–12.[38] F. Wang and D.P. Landau: Efficient, Multiple-Range Random WalkAlgorithm to Calculate the Density of States, Phys. Rev. Lett. 86(2001) 2050–2053.[39] K. Hukushima and K. Nemoto: Exchange Monte Carlo Method andApplication to Spin Glass Simulations, J. Phys. Soc. 65 (1996) 1604–1608.[40] B.A. Berg, U. Hansmann and T. Neuhaus: Simulation of an ensemblewith varying magnetic field: A numerical determination of the order-order interface tension in the D=2 Ising model, Phys. Rev. B 47(1993) 497–500.[41] K. Watanabe and M. Sasaki: An Efficient Monte-Carlo Method forCalculating Free Energy in Long-Range Interacting Systems, J. Phys.Soc. Jpn. 80 (2011) 093001.[42] K. Binder: Finite size scaling analysis of ising model block distributionfunctions, Z. Phys. B 43 (1981) 119–140.[43] M. Sagawa, S. Fujimura, H. Yamamoto, Y. Matsuura, S. Hirosawa andK. Hiraga: Proceedings of the 4th International Symposium onMagnetic Anisotropy and Coercivity in Rare Earth Transition MetalAlloys, ed. by K.J. Strnat, (University of Dayton, Dayton, OH, 1985)p. 587 and Coercivity in Rare Earth Transition Metal Alloys, ed. byK.J. Strnat, (University of Dayton, Dayton, OH, 1985) p. 587.[44] K. Ono, N. Inami, K. Saito, Y. Takeichi, M. Yano, T. Shoji, A.Manabe, A. Kato, Y. Kaneko, D. Kawana, T. Yokoo and S. Itoh:Observation of spin-wave dispersion in Nd-Fe-B magnets usingneutron Brillouin scattering, J. Appl. Phys. 115 (2014) 17A714.[45] Y. Zhu and M.R. McCartney: Magnetic-domain structure of Nd2Fe14Bpermanent magnets, J. Appl. Phys. 84 (1998) 3267–3272.[46] S.J. Lloyd, J.C. Loudon and P.A. Midgley: Measurement of magneticdomain wall width using energy-filtered Fresnel images, J. Microscopy207 (2002) 118–128.[47] M. Beleggia, M.A. Schofield, Y. Zhu and G. Pozzi: Quantitativedomain wall width measurement with coherent electrons, J. Magn.Magn. Mater. 310 (2007) 2696–2698.[48] P. Asselin, R.F.L. Evans, J. Barker, R.W. Chantrell, R. Yanes, O.Chubykalo-Fesenko, D. Hinzke and U. Nowak: Constrained MonteCarlo method and calculation of the temperature dependence ofmagnetic anisotropy, Phys. Rev. B 82 (2010) 054415.[49] D. Givord, A. Lienard, P. Tenaud and T. Viadieu: Magnetic viscosity inNd-Fe-B sintered magnets, J. Magn. Magn. Mater. 67 (1987) L281–L285.[50] S. Okamoto, R. Goto, N. Kikuchi, O. Kitakami, T. Akiya, H. Sepehri-Amin, T. Ohkubo, K. Hono, K. Hioki and A. Hattori: Temperature-dependent magnetization reversal process and coercivity mechanism inNd-Fe-B hot-deformed magnets, J. Appl. Phys. 118 (2015) 223903.[51] W. Wernsdorfer, E.B. Orozco, K. Hasselbach, A. Benoit, B. Barbara,N. Demoncy, A. Loiseau, H. Pascard and D. Mailly: ExperimentalEvidence of the Néel-Brown Model of Magnetization Reversal, Phys.Rev. Lett. 78 (1997) 1791–1794.[52] R.H. Victora: Predicted time dependence of the switching field formagnetic materials, Phys. Rev. Lett. 63 (1989) 457–460.[53] S. Okamoto: Experimental approaches for micromagnetic coercivityanalysis of advanced permanent magnet materials, Sci. Technol. Adv.Mater. 22 (2021) 124–134.[54] R. Friedberg and D.I. Paul: New Theory of Coercive Force ofFerromagnetic Materials, Phys. Rev. Lett. 34 (1975) 1234–1237.[55] A. Sakuma, S. Tanigawa and M. Tokunaga: Micromagnetic studies ofinhomogeneous nucleation in hard magnets, J. Magn. Magn. Mater. 84(1990) 52–58.[56] A. Sakuma: The theory of inhomogeneous nucleation in uniaxialferromagnets, J. Magn. Magn. Mater. 88 (1990) 369–375.[57] A.L. Wysocki and V.P. Antropov: Micromagnetic simulations withperiodic boundary conditions: Hard-soft nanocomposites, J. Magn.Magn. Mater. 428 (2017) 274–286.[58] T. Pramanik, A. Roy, R. Dey, A. Rai, S. Guchhait, H.C.P. Movva,C.-C. Hsieh and S.K. Banerjee: Angular dependence of magnetizationreversal in epitaxial chromium telluride thin films with perpendicularmagnetic anisotropy, J. Magn. Magn. Mater. 437 (2017) 72–77.[59] Y. Feng, J. Liu, T. Klein, K. Wu and J.-P. Wang: Localized detection ofreversal nucleation generated by high moment magnetic nanoparticlesusing a large-area magnetic sensor, J. Appl. Phys. 122 (2017) 123901.[60] Y. Tatetsu, S. Tsuneyuki and Y. Gohda: First-Principles Study of theRole of Cu in Improving the Coercivity of Nd-Fe-B PermanentMagnets, Phys. Rev. Appl. 6 (2016) 064029.[61] Y. Gohda, Y. Tatetsu and S. Tsuneyuki: Electron Theory on Grain-Boundary Structures and Local Magnetic Properties of NeodymiumMagnets, Mater. Trans. 59 (2018) 332–337.[62] K. Hirota, H. Nakamura, T. Minowa and M. Honshima: CoercivityEnhancement by the Grain Boundary Diffusion Process to Nd–Fe–BSintered Magnets, IEEE Trans. Magn. 42 (2006) 2909–2911.[63] F. Xu, J. Wang, X. Dong, L. Zhang and J. Wu: Grain boundaryAtomistic Model Study on Magnetic Properties of Permanent Magnets 15https://doi.org/10.1103/PhysRevB.98.054418https://doi.org/10.1103/PhysRevB.98.054418https://doi.org/10.1103/PhysRevB.100.020403https://doi.org/10.1103/PhysRevB.102.020413https://doi.org/10.1103/PhysRevB.101.094421https://doi.org/10.1038/s41524-020-0325-6https://doi.org/10.1103/PhysRevB.103.014418https://doi.org/10.1063/9.0000070https://doi.org/10.1080/14686996.2021.1942197https://doi.org/10.2497/jjspm.69.S126https://doi.org/10.2497/jjspm.69.S126https://doi.org/10.1103/PhysRevB.103.054421https://doi.org/10.1103/PhysRevB.106.054422https://doi.org/10.1103/PhysRevB.29.4176https://doi.org/10.1103/PhysRevB.29.4176https://doi.org/10.1016/0304-8853(87)90721-9https://doi.org/10.1016/0304-8853(87)90721-9https://doi.org/10.1063/1.4869061https://doi.org/10.1063/1.4869061https://doi.org/10.1103/PhysRevB.38.620https://doi.org/10.1103/PhysRev.127.2058https://doi.org/10.1103/PhysRev.127.2058https://doi.org/10.1103/PhysRevB.58.14937https://doi.org/10.1103/PhysRevB.58.14937https://doi.org/10.1103/PhysRevLett.105.120603https://doi.org/10.1103/PhysRevLett.68.9https://doi.org/10.1103/PhysRevLett.86.2050https://doi.org/10.1103/PhysRevLett.86.2050https://doi.org/10.1143/JPSJ.65.1604https://doi.org/10.1143/JPSJ.65.1604https://doi.org/10.1103/PhysRevB.47.497https://doi.org/10.1103/PhysRevB.47.497https://doi.org/10.1143/JPSJ.80.093001https://doi.org/10.1143/JPSJ.80.093001https://doi.org/10.1007/BF01293604https://doi.org/10.1063/1.4863380https://doi.org/10.1063/1.368515https://doi.org/10.1046/j.1365-2818.2002.01048.xhttps://doi.org/10.1046/j.1365-2818.2002.01048.xhttps://doi.org/10.1016/j.jmmm.2006.10.995https://doi.org/10.1016/j.jmmm.2006.10.995https://doi.org/10.1103/PhysRevB.82.054415https://doi.org/10.1016/0304-8853(87)90185-5https://doi.org/10.1016/0304-8853(87)90185-5https://doi.org/10.1063/1.4937274https://doi.org/10.1103/PhysRevLett.78.1791https://doi.org/10.1103/PhysRevLett.78.1791https://doi.org/10.1103/PhysRevLett.63.457https://doi.org/10.1080/14686996.2021.1874836https://doi.org/10.1080/14686996.2021.1874836https://doi.org/10.1103/PhysRevLett.34.1234https://doi.org/10.1016/0304-8853(90)90162-Jhttps://doi.org/10.1016/0304-8853(90)90162-Jhttps://doi.org/10.1016/0304-8853(90)90660-Ihttps://doi.org/10.1016/j.jmmm.2016.11.128https://doi.org/10.1016/j.jmmm.2016.11.128https://doi.org/10.1016/j.jmmm.2017.04.039https://doi.org/10.1063/1.5001919https://doi.org/10.1103/PhysRevApplied.6.064029https://doi.org/10.2320/matertrans.M2017258https://doi.org/10.1109/TMAG.2006.879906microstructure in DyF3-diffusion processed Nd–Fe–B sinteredmagnets, J. Alloy. Compd. 509 (2011) 7909–7914.[64] K. Löewe, C. Brombacher, M. Katter and O. Gutfleisch: Temperature-dependent Dy diffusion processes in Nd–Fe–B permanent magnets,Acta Mater. 83 (2015) 248–255.[65] W. Chen, J.M. Luo, Y.W. Guan, Y.L. Huang, M. Chen and Y.H. Hou:Grain boundary diffusion of Dy films prepared by magnetronsputtering for sintered Nd–Fe–B magnets, J. Phys. D 51 (2018)185001.[66] T.-H. Kim, T. Sasaki, T. Ohkubo, Y. Takada, A. Kato, Y. Kaneko andK. Hono: Microstructure and coercivity of grain boundary diffusionprocessed Dy-free and Dy-containing Nd–Fe–B sintered magnets,Acta Mater. 172 (2019) 139–149.[67] S. Bance, J. Fischbacher, A. Kovacs, H. Oezelt, F. Reichel and T.Schrefl: Thermal Activation in Permanent Magnets, JOM 67 (2015)1350–1356.[68] J. Fischbacher, A. Kovacs, L. Exl, J. Kuhnel, E. Mehofer, H. Sepehri-Amin, T. Ohkubo, K. Hono and T. Schrefl: Searching the weakest link:Demagnetizing fields and magnetization reversal in permanentmagnets, Scr. Mater. 154 (2018) 253–258.[69] C. Mitsumata, H. Tsuchiura and A. Sakuma: Model Calculation ofMagnetization Reversal Process of Hard Magnet in Nd2Fe14B System,Appl. Phys. Express 4 (2011) 113002.[70] S. Bance, B. Seebacher, T. Schrefl, L. Exl, M. Winklhofer, G. Hrkac,G. Zimanyi, T. Shoji, M. Yano, N. Sakuma, M. Ito, A. Kato and A.Manabe: Grain-size dependent demagnetizing factors in permanentmagnets, Appl. Phys. 116 (2014) 233903.[71] R. Ramesh, G. Thomas and B.M. Ma: Magnetization reversal innucleation controlled magnets. II. Effect of grain size and sizedistribution on intrinsic coercivity of Fe-Nd-B magnets, J. Appl. Phys.64 (1988) 6416–6423.[72] K. Uestuener, M. Katter and W. Rodewald: Dependence of the MeanGrain Size and Coercivity of Sintered Nd–Fe–B Magnets on the InitialPowder Particle Size, IEEE Trans. Magn. 42 (2006) 2897–2899.[73] T. Fukada, M. Matsuura, R. Goto, N. Tezuka, S. Sugimoto, Y. Uneand M. Sagawa: Evaluation of the Microstructural Contribution to theCoercivity of Fine-Grained Nd–Fe–B Sintered Magnets, Mater. Trans.53 (2012) 1967–1971.[74] M. Sasaki and F. Matsubara: Stochastic Cutoff Method for Long-Range Interacting Systems, J. Phys. Soc. Jpn. 77 (2008) 024004.[75] K. Fukui and S. Todo: Order-N cluster Monte Carlo method for spinsystems with long-range interactions, J. Comput. Phys. 228 (2009)2629–2642.[76] W.F. Brown, Jr.: Thermal Fluctuations of a Single-Domain Particle,Phys. Rev. 130 (1963) 1677–1686.[77] H. Hayasaka, M. Nishino and S. Miyashita: Microscopic study on theangular dependence of coercivity at zero and finite temperatures, Phys.Rev. B 105 (2022) 224414.[78] T. Yomogita, S. Okamoto, N. Kikuchi, O. Kitakami, H. Sepehri-Amin,Y.K. Takahashi, T. Ohkubo, K. Hono, K. Hioki and A. Hattori: Directdetection and stochastic analysis on thermally activated domain-walldepinning events in micropatterned Nd-Fe-B hot-deformed magnets,Acta Mater. 201 (2020) 7–13.[79] Q. Gong, M. Yi, R.F.L. Evans, B.-X. Xu and O. Gutfleisch:Calculating temperature-dependent properties of Nd2Fe14B permanentmagnets by atomistic spin model simulations, Phys. Rev. B 99 (2019)214409.[80] Q. Gong, M. Yi and B.-X. Xu: Multiscale simulations towardcalculating coercivity of Nd-Fe-B permanent magnets at hightemperatures, Phys. Rev. Mater. 3 (2019) 084406.[81] Q. Gong, M. Yi, R.F.L. Evans, O. Gutfleisch and B.-X. Xu:Anisotropic exchange in Nd–Fe–B permanent magnets, Mater. Res.Lett. 8 (2020) 89–96.[82] S.C. Westmoreland, R.F.L. Evans, G. Hrkac, T. Schrel, G.T. Zimanyi,M. Winklhofer, N. Sakuma, M. Yano, A. Kato, T. Shoji, A. Manabe,M. Ito and R.W. Chantrell: Multiscale model approaches to the designof advanced permanent magnets, Scr. Mater. 148 (2018) 56–62.[83] S.C. Westmoreland, C. Skelland, T. Shoji, M. Yano, A. Kato, M. Ito,G. Hrkac, T. Schrefl, R.F.L. Evans and R.W. Chantrell: Atomisticsimulations of ¡-Fe/Nd2Fe14B magnetic core/shell nanocompositeswith enhanced energy product for high temperature permanent magnetapplications, J. Appl. Phys. 127 (2020) 133901.M. Nishino and S. Miyashita16https://doi.org/10.1016/j.jallcom.2011.05.023https://doi.org/10.1016/j.actamat.2014.09.039https://doi.org/10.1088/1361-6463/aab912https://doi.org/10.1088/1361-6463/aab912https://doi.org/10.1016/j.actamat.2019.04.032https://doi.org/10.1007/s11837-015-1415-7https://doi.org/10.1007/s11837-015-1415-7https://doi.org/10.1016/j.scriptamat.2017.11.020https://doi.org/10.1143/APEX.4.113002https://doi.org/10.1063/1.4904854https://doi.org/10.1063/1.342055https://doi.org/10.1063/1.342055https://doi.org/10.1109/TMAG.2006.879889https://doi.org/10.2320/matertrans.MAW201207https://doi.org/10.2320/matertrans.MAW201207https://doi.org/10.1143/JPSJ.77.024004https://doi.org/10.1016/j.jcp.2008.12.022https://doi.org/10.1016/j.jcp.2008.12.022https://doi.org/10.1103/PhysRev.130.1677https://doi.org/10.1103/PhysRevB.105.224414https://doi.org/10.1103/PhysRevB.105.224414https://doi.org/10.1016/j.actamat.2020.09.074https://doi.org/10.1103/PhysRevB.99.214409https://doi.org/10.1103/PhysRevB.99.214409https://doi.org/10.1103/PhysRevMaterials.3.084406https://doi.org/10.1080/21663831.2019.1702116https://doi.org/10.1080/21663831.2019.1702116https://doi.org/10.1016/j.scriptamat.2018.01.019https://doi.org/10.1063/1.5126327