The following described experimental data points are later used to train a machine learning model described in Section 2.1.3 predicting the intrinsic material properties saturation magnetization and magneto-crystalline anisotropy constant. 119 variations of (Nd,Ce,La,Pr)_13.55-(Fe,Co,Ni)_80.54-B_5.91 (at%) alloys were prepared by arc melting. These alloys were annealed at 1373 K for 24 h in Ar atmosphere. After a homogenizing heat treatment, alloy ingots were pulverized and sorted into particles with diameters of < 20 μ m in an inert atmosphere to make each particle single crystal.

The powder density was determined using a pycnometer (Ulrtapyc1200e, Quantachrome Instruments, United States). The powder compositions were measured by ICP-AES (ICPS8100, Shimazu, Japan) and the main phase (2-14-1 phase) ratio was calculated from the obtained composition. The magnetic properties were measured by using a vibrating sample magnetometer (PPMS EverCool II, QuantumDesign, United States) at a maximum applied field of 9 T.

Sample powder was mixed with an epoxy resin in a Cu container and solidified in a magnetic field of 1 T for vibrating sample magnetometer (VSM) measurements. M(H) curves of magnetically easy and hard directions were measured in the temperature range from 300 K to 453 K. The magnetic anisotropy field (H _A) was estimated by singular point detection (SPD) method (Cabassi, 2020) from the M(H) curve of hard axis. When anisotropy field is less than 9 T (maximum applied magnetic field of our VSM) H _A can be detected by SPD, otherwise the M(H) curves for both easy and hard axis were extrapolated to higher magnetic field to estimate intersection between them as H _A. On the other hand, saturation magnetization (μ ₀ M _s) was estimated by the law of approach to saturation (LAS) (Akulov, 1931; Hadjipanayis et al., 1981; Kronmüller et al., 2003) from the M(H) curve of easy axis. When μ ₀ H _A was lower than the maximum applied field, Eq. 1 was used. μ 0 M = μ 0 M s 1 − b H 2 (1) When μ ₀ H _A was higher than 9 T, Eqs 2, 3 were used. μ 0 M = μ 0 M s 1 − b H 2 + χ 0 H (2) μ 0 d M d H = μ 0 M s 2 b H 3 + χ 0 (3) M _s is the saturation magnetization and b and χ ₀ are constants. The χ ₀ H term is often referred to as the so-called paramagnetic term (Zhang et al., 2010). Eqs 2, 3 dealing with χ ₀ term is more accurate for measuring saturation magnetization, but there is a condition that the applied magnetic field is sufficiently larger than the anisotropy field. The saturation magnetization estimated by the LAS was divided by the main phase ratio of the powder to obtain the saturation magnetization of the 2-14-1 phase in these compositions. For the measurement of the intrinsic properties as described above, grain boundary phases other than the main phase were treated as paramagnetic phases.

The following material database is later used to improve the machine learning model described in Section 2.1.3 predicting the intrinsic material properties saturation magnetization and magneto-crystalline anisotropy constant by appending predicted exchange integral information, Curie temperature and magnetization at 0 K. In this work no attempt to compute the anisotropy constant with first-principles is presented. The material database for the magnetization and Curie temperature (T _C) of ( N d 1 − α − β − γ Pr α L a β C e γ ) 2 - ( F e 1 − δ − ζ C o δ N i ζ ) 14 -B is constructed using the AkaiKKR program package ² (Akai, 1989), where the Korringa-Kohn-Rostoker (KKR) Green’s function method (Korringa, 1947; Kohn and Rostoker, 1954) is implemented within the local density approximation (LDA) of the density functional theory (Hohenberg and Kohn, 1964; Kohn and Sham, 1965). Here, α, β, γ, δ and ζ satisfy the conditions that 0 ≤ α, β, γ, δ, ζ ≤ 1, 0 ≤ α+β+γ ≤ 1 and 0 ≤ δ+ζ ≤ 1. The atomic configurational disorder is treated by the coherent potential approximation (CPA) (Soven, 1970; Shiba, 1971). In the CPA approach, the multiple scattering effects due to random potentials are treated with the single-site approximation. The CPA enables us to calculate the electronic structure and magnetic properties of disordered systems without constructing a large supercell, and hence, is an efficient framework to explore large materials space.

The relativistic effects are included through the scalar relativistic approximation and with spin orbit coupling retaining only the diagonal terms (i.e., l _z s _z terms). LDA, as parametrized by Moruzzi, Janak, and Williams (Moruzzi et al., 2013), is adopted for the exchange-correlation energy functional. The LDA is rather poor for handling strongly localized f states. Therefore, the open core approximation is applied to the f states of the rare earth elements, i.e., Nd, Ce, and Pr, so that the resonances originating from the f states are removed. We assume the electron configurations of Nd³⁺, Ce⁴⁺, and Pr³⁺ in the present calculations.

The T _C is estimated by combining the mean field approximation and the KKR-CPA calculations of magnetic exchange interaction (J _ij ) between sites i and j. Here, the Liechtenstein formula (Liechtenstein et al., 1987) is employed to obtain J _ij . According to this formulation, two magnetic atoms are embedded into the CPA effective medium generated by KKR-CPA calculations, and then we evaluate the J _ij values on the basis of the magnetic force theorem, mapping the changes in the band energy due to the infinitesimal rotations of the two magnetic moments to those of the classical Heisenberg Hamiltonian. Experimental lattice parameters are used for R₂T₁₄B (R = Nd, Pr, La, Ce; T = Fe, Co, Ni) when available in literature (Herbst, 1991). Otherwise, the lattice parameters are estimated by integrating first-principles calculation data and available experimental data, as described in Appendix D in (Harashima et al., 2021). The lattice parameters for non-stoichiometric systems are determined from those for stoichiometric systems according to Vegard’s law.

We constructed a model which uses the chemical composition as the only feature to predict the spontaneous magnetization μ ₀ M _s and the magneto-crystalline anisotropy K ₁ of (Nd,La,Ce,Pr)₂(Fe,Co,Ni)₁₄B magnets combining ab-initio data and experimental measurements with partial least-squares regression (PLS). In this section we cover training data preprocessing, analysis and briefly how partial least-squares regression works.

Partial least-squares regression consists of two major parts, i) the decomposition of a matrix into products of smaller matrices similar to principal components analysis (PCA) and ii) the relation between features X and labels Y . In PLS a matrix X is decomposed into a matrix-matrix product of scores T and loadings P such that X = TP ^T + E . E here is the residuum. The labels matrix or vector Y is decomposed in the same fashion such that Y = UQ ^T + F* . The number of columns in the matrix T is much smaller than the total number of features (the number of columns of the matrix X ). Similarly, the matrix U may have much less columns than the number of labels. In other words, both the features and the labels are projected into a latent space. Similar to PCA, the PLS method reduces the dimensionality. Therefore PLS regression is well suited for problems in which either the number of features or the number of labels is large as compared to the number of samples. The scores T are used to predict the Y -scores U . This in turn gives a linear relation between the features and the labels: Y = XB _PLS + B ₀ . The PLS coefficients B _PLS hold the information of how much each feature contributes to the labels. The latent variables are found through an iterative approach which maximizes the covariance between T and U . In other words the decomposition is built such that i) T best explains the variance in the features, ii) U best explains the variance in the labels, and iii) there is the strongest possible relationship between features and labels (Tobias et al., 1995). There is a lot of well written literature and therefore we may refer to (Geladi and Kowalski, 1986) or (Eriksson et al., 2014) for further reading.

Due to the low amount of training points for heavy rare earth element composites such as with dysprosium (Dy), gadolinium (Gd) and terbium (Tb) it is more difficult to train reliable models and will be covered in future works. These heavy rare earths have been excluded from the training data base as well as entries with holmium (Ho) and manganese (Mn) replacing iron (Fe) and entries where carbon (C) partially replaces boron (B).

Input data for building the machine learning model presented are two data sets: i) Data from ab-initio simulations relating chemical composition to the computed properties such as the spontaneous magnetization, local magnetic moments, the lattice constants, Curie temperature, and the exchange integrals (see Section 2.1.2). ii) Experimental measurements provide the spontaneous magnetization and the anisotropy field for different chemical compositions and temperatures (see Section 2.1.1).

The ab-initio data set and the experimental data do not cover the same combinations of chemical compositions and it is hard to visualize or compute the overlap between two multidimensional sets of partially dependent and independent chemical concentrations of 2-14-1 systems. Our approach is to apply a manifold method exclusively to the ab-initio dataset which transforms each entry to a 2-dimensional projection plane. Each data point receives a repulsive and attractive force between all others, where points close in chemical concentrations attract each other whereas others are pushed away. After relaxation groups of composites will have formed and can be visualized in a 2-dimensional embedding space. In machine learning this is often used to categorize data points in a certain amount of classes, but in our case we use it to gain insight into the overlap of two datasets. After the ab-initio data has been used to calibrate this transformation function, the experimental points are transformed with the same function to verify if those newly added points are far from the ab-initio entries. The result of such a Uniform Manifold Approximation & Projection (UMAP) (McInnes et al., 2018) is shown in Figure 2. One can see that an overlap exists within this projection space which is small but sufficiently encapsulates the ab-initio points. The first principals calculations database consists of many simulations where Co or Ni atoms replace Fe.

Using the ab-initio data, we train a first model that predicts ab-initio data from the chemical composition (see i) in Figure 3). We use this partial least squares regressor to augment the experimental data with predicted values for the ab-initio data (see ii) in Figure 3). In addition, we add estimates for M _s (T) and M s 3 ( T ) estimating the spontaneous magnetization at temperature T with Kuz’min’s equation (Kuz’min, 2005) from the ab-initio-calculated magnetization at 0 K and the Curie temperature. By adding the ab-initio data to the experimental data set, we increase the total number of features which are then used to predict the temperature dependent magnetization and anisotropy with a second partial least squares regressor (see iii) in Figure 3).

To choose the correct number of decomposing components we use nested cross-validation. The data sets shown in Figure 3 are split 5 times into a test set and a training set. With the training set we perform a 5-times repeated 5-fold cross validation. For hyperparameter-optimization we include the polynomial degree of features in the partial least squares regression to leave it to the optimization if higher order contributions should be included or not. The mean squared errors averaged over the test sets are 0.31 MJ/m³ and 0.04 T for K ₁ and μ ₀ M _s , respectively. The feature vector is extended by all combinations of products (2nd degree polynomial) and the final model had 35 components for the partial least squares regression. The forward prediction of this trained model for various chemical compositions is shown in the results section of this manuscript in Section 3.1.

As it is intended to explore and search for new materials, the trained regression is applied to unseen samples of chemical concentration combinations which have no observed or measured output of intrinsic material properties. To gain insight into prediction uncertainty, a rather naive empirical bagging or bootstrapping approach is applied. In our case, not only one but one hundred regressions are calibrated where only slight differences in training data can be found. Each training set is drawn randomly from the full set with the chance of skipping some. The predictions of these hundred regressions provide an average prediction and additionally its prediction variance, referred to as the predictor’s confidence interval (Heskes, 1997). Currently this is our measure of predictor uncertainty but there is potential for improvement here.

Such a trained partial least squares model capable of returning a continuous function of intrinsic material properties along each trained material concentration axis gives an additional opportunity. Combining a multi-objective genetic algorithm framework (Blank and Deb, 2020) with the intrinsic material property predictor can optimize towards a specific target K 1 * and M s * by tuning the chemical concentrations. Similarly, Kim et al. (2018) searched for organic molecules targeting specific molecular designs. Such an inverse design scheme is outlined in Figure 4. There are two objective functions: i) The squared distance between the predicted and the desired K ₁ and ii) the squared distance between the predicted and desired μ ₀ M _s . One major advantage is that we now can define restrictions or constraints to our optimization problem as well e.g., maximum allowed Nd-concentration, only composites of Nd, La and Ce are allowed, and/or the confidence interval has to be below a certain limit.

Models of permanent magnets in the range of 10 nm³ to 1 μm³ can be treated with micromagnetic simulations. Micromagnetism is a continuum theory that describes magnetization processes on length scales large enough to replace discrete atomic spins by a continuous function of position M(x) but also small enough to resolve the transition of the magnetization M(x) between magnetic domains. The first argument allows to replace billions of spins with millions of finite elements and hence the computation of demagnetization curves within a reasonable time. The second argument makes it possible to compute the influence of the microstructure on magnetization processes in contrast to macroscopic simulations based on Maxwell’s equations. The key assumption of micromagnetic theory is that the spin orientation changes only by a small angle from one lattice point to the next in a ferromagnet (Brown, 1959). The resolution of the mesh (meshsize) may have significant influence on the energy necessary to form domain walls and the energy required to move domain walls (Donahue and McMichael, 1997). Domain walls might get pinned on too coarse meshes which is an unwanted effect. When simulating permanent magnets, we usually choose a meshsize close to the Bloch wall parameter δ 0 = A / K 1 which is a measure of the width of the transition region between two magnetic domains. Exchange constant A and anisotropy constant K ₁ of Nd₂Fe₁₄B at room temperature are 8 pJ/M and 4.9 MJ/m³, respectively (Coey, 2010, p. 401). Hence δ ₀ = 1.3 nm defines the meshsize close to microstuctural features where the nucleation of reversed domains is likely to happen. While some growth of the meshsize towards the center of the core phase is wanted to speed-up computation, we limited the maximum meshsize to 2δ ₀ for unsupervised simulations as it is difficult to automatically detect errors due to artificial pinning of domain walls. This fine discretation limits the maximum modelsize even on today’s computing hardware.

We developed an automated simulation process comprising geometry generation, mesh generation, mesh quality checks, computing the hysteresis properties, and storing the results in a database. The simulation runs completely unsupervised without human interaction. Thus, coercivity data can be sampled in a high-throughput like fashion. The sampler explores the design space and selects a point in feature space. A python script was used to create the multi-phase grain geometry (as sketched in Figure 1) using the open-source software Salome ³ . MeshGems ⁴ was used to create a tetrahedral representation of the geometry. The grains have a brick like shape as shown by transmission electron microscopy (Sepehri-Amin et al., 2013). Such multi-phase grains consist of a core and a shell phase enclosed within a ferromagnetic grain boundary phase. Characteristic features of the core/shell geometry were the thickness of a weakly soft magnetic grain boundary phase (2 nm–50 nm), the thickness of the shell (2 nm–50 nm), the extensions of the grain (60 nm–100 nm in c-direction and 100 nm–150 nm in a- and b-direction), and the misorientation of the grain with respect to the field direction (0°–45°). The hysteresis properties were computed by minimizing the Gibbs free energy considering energy terms due to exchange energy, the magneto-crystalline anisotropy energy, the magnetostatic energy and the Zeeman energy (Exl et al., 2019).

Using micromagnetic simulations we created a database which related chemical composition, grain geometry, and hysteresis properties. We implemented different methodologies to sample the design space. Initially, sample points for training were generated using a Bayesian optimizer (Häse et al., 2018) for maximizing coercivity. In a second stage we applied an active learning scheme. The machine learning predictor for coercivity was used to evaluate the objective function (high coercivity and low Nd + Pr content) of a genetic algorithm. The Pareto points were saved and put into a database for recomputation of the hysteresis properties with the micromagnetic solver. Figure 5 shows the active learning scheme to create additional samples to improve the micromagnetic simulation database.

A neural network regressor is trained to predict the coercive field of multi-phase grains. The model was based on a fully dense neural network as implemented in scikit-learn (Géron, 2019). We used one hidden layer with the number of units as hyperparameter. The rectified linear unit is used as activation function. The mean squared error is used as the loss function which is minimized by the limited-memory Broyden–Fletcher–Goldfarb–Shanno (LBFGS) algorithm (Liu and Nocedal, 1989). The feature vector which contains geometry and intrinsic magnetic properties is augmented with the minimum Stoner-Wohlfarth switching field of the core and shell materials.

In Figure 6 we show the prediction performance as a residual plot and true vs. predicted values for the coercive field of single core-shell grains with defect shells. We included the calibration errors of all micromagnetic simulations as final performance measures: mean absolute error (MAE), root mean squared error (RMSE) and the explained variance (R2). These should not be confused with the predictor performance values during fine tuning of the neural network, which are computed via a 5-fold cross validation. The number of units of the single hidden layer was the only parameter optimized, which gave best results with nine units for the multi-layer perceptron (MLP) regressor inside the sklearn package ⁵ . The cross validated explained variance is 0.975 and the mean absolute error is 0.148 T.

The same bootstrapping approach, as reported in Section 2.1.3, is applied to the coercivity predictor. 100 slightly different training sets are prepared to train 100 neural network regressors, predicting the coercivity. The deviations from their averaged prediction are used to estimate a prediction confidence interval. Additionally all features used for the prediction of coercivity, which depend on the previously predicted material properties are propagated to the prediction uncertainty of coercivity. We think that this is a good measure of the distance in feature space between a point in the test set and the nearest point in the training data.

Nd-reduced sintered magnets containing Ce and La were fabricated as base sintered magnets. Hydrogen decrepitation and jet-milling processes were used to crush the strip cast flakes into powders. Magnetically aligned green compacts were sintered at 1338 K for 4 h in a vacuum atmosphere. Grain-boundary diffusion was performed at 1223 K for 165 min in vacuum using Nd-Cu based alloys as infiltration materials. This diffusion process forms the core/shell structure in base sintered magnets. The diffused samples were annealed at 773 K for 1 h in vacuum to enhance coercivity.

Magnetic properties were measured by a vibrating sample magnetometer (PPMS EverCool II, QuantumDesign, United States). Scanning electron microscope (SEM) observations were performed using a field emission SEM (ULTRA55, Carl Zeiss, Germany), equipped with an in-lens type detector. SEM observations were performed on specimens milled with an Ar ion beam at an acceleration voltage of 4 kV using an ion milling system (IM4000, Hitachi, Japan) after flat polishing. Crystal structure analysis was carried out by X-ray diffraction (XRD) (SmartLab SE, Rigaku, Japan).

We apply materials informatics (MI) (Rajan, 2005) to analyze the microstructure of permanent magnets. The following steps were taken to characterize a scanning electron microscope (SEM) image.

1) The two-dimensional power spectrum (Koizumi et al., 2019) obtained by the Fourier transform of the image is converted into a one-dimensional (1D) power spectrum by integrating it in the azimuth direction Figure 7.

We think that the individualization of analysis and the pre-processing of inhomogeneous data are the bottlenecks in MI, and in order to solve these problems, we have built a common platform for data analysis and storage named “WAVEBASE” ⁶ . This system is built on a cloud, and users can upload measured data from a web browser. The uploaded data is uniformly pre-processed, features extracted, and visualized in the system. For example, for image analysis, processing is performed by the function 1)-4) above. Data from other techniques such as XRD, small-angle X-ray scattering (SAXS), infrared (IR) and mass spectroscopy can be analyzed in the same way. By utilizing this system, we try to shorten the time from data acquisition to analysis while sharing data and analysis results within multiple joint research institutes.

A power spectrum as shown in Figure 7C characterizes the microstructure of a magnet. We aim to generate synthetic microstructures for micromagnetic simulations that resemble the grain structure of a given magnet. Instead of matching the SEM image with an image derived from the synthetic microstructure we try to create a synthetic microstructure which gives a power spectrum that is similar to that computed from a SEM image. For this purpose, we combine synthetic microstructure generation, neural-network-based image segmentation and a Bayesian optimizer.

We use the software package Neper ⁷ , which applies Laguerre tessellation for the generation of polycrystalline grains with realistic shape and size distribution (Quey and Renversade, 2018). After slicing the three-dimensional synthetic structure, we create a three-colored image where the different colors represent the main NdFeB phase, the grain boundary phase, and a secondary phase such as for example Nd-oxides. From this artificial microstructure image the power spectrum can be computed. In order to obtain three-colored microstructure images from SEM images we apply automated image segmentation. A convolutional neural network is trained to map a SEM image to a three-color segmented image. We use the well-established unet (Ronneberger et al., 2015; Choudhary et al., 2022) for the network architecture. Instead of manually labeling SEM images we use synthetically generated data for training of the network. Three-colored images which were obtained from synthetic microstructures are randomized by adding noise. Figure 8A shows input output pairs used for training the neural network. Figure 8B shows a SEM image and the segmented three-colored image.

The segmentation of the SEM images ensures close similarity in the input color space for the computation of the power spectra between the experimental and the synthetic images. We then apply a Bayesian optimizer ⁸ to find a set of input parameters for polycrystalline structure generation with the aim to create a synthetic microstructure which is close to the microstructure of a real magnet. The magnet’s microstructure is characterized by the average of power spectra P S D ̄ exp obtained from SEM images as described above. To characterize the three-dimensional synthetic grain structure, we compute the average of the power spectra of nine slices through the magnet P S D ̄ syn . The Euclidean norm between the powerspectra f = | P S D ̄ exp − P S D ̄ syn | will be minimized during optimization.

For micromagnetic modeling to correctly track magnetic behaviour, various features on multiple length scales have to be taken into account. On the smallest length scale of just a few nanometers crystallographic defects can substantially influence the magnetic reversal of much larger systems. To consider these defects, the model needs to operate on a spatial discretization close to δ ₀ in the nanometer regime. Hence, the simulations for large systems require a vast amount of computing resources or are not feasible at all.

To bridge the gap from the nanometer length scale of important microstructural features to a macroscopic length scale, a reduced order model was developed. This model is based on the assumption that nucleation of a sufficiently large reversed domain immediately leads to the magnetic switching of the entire grain in question. Therefore, instead of calculating the state of the magnetic moment on every finite element as in conventional micromagnetism, we assume that each grain is uniformly magnetized and only one macroscopic magnetic moment is assigned per grain. A fast method to approximate the magnetic state of a grain is to employ the embedded Stoner-Wohlfarth (SW) model (Fischbacher et al., 2017; Exl et al., 2018). The embedded approach adapts the original SW method for small ferromagnetic particles to additionally account for long-range interactions of uniformly magnetized grains. Previous research suggests that magnetization reversal of a grain starts close to the edges. Hence, we define evaluation points in all grains at a distance of dL _ex to the grain’s edges. While L ex = A / μ 0 M s 2 is the exchange length (Rave et al., 1998), d is a free parameter which will be tuned by comparison to conventional micromagnetics. At these evaluation points the local switching fields are determined analytically by the SW model as H sw = 2 K 1 / μ 0 M s sin 2 / 3 ⁡ ψ + cos 2 / 3 ⁡ ψ − 3 / 2 . Here ψ is the angle between the total field acting on the respective point and the magneto-crystalline anisotropy direction of the grain. The total field is the sum of the external field H _ext, the magnetostatic field H _mag and the exchange field H _exch. The magnetostatic field in each point is calculated from the surface charge density by using analytical formulas for polyhedral geometries (Guptasarma and Singh, 1999), employing hierarchical matrices as implemented with h2tools ⁹ (Mikhalev and Oseledets, 2016). Hierarchical matrices reduce storage and CPU time, by computing the magnetostatic field from nearby grains exactly, while approximating the field from grains far away from the respective evaluation point. Following the work of Wood, the direction of the magnetization vector of a grain is analytically calculated by the SW model as well (Wood, 2009). This way, in our reduced order model we can also track the reversible part of the demagnetization curve of each grain, and therefore of the entire multigrain system. The exchange field is defined as H _exch = M/l ² with l being a phenomenological distance parameter.

We start from a magnetically saturated state with a large positive external field and gradually reverse H _ext in the negative direction until negative saturation. At each field step for each grain we evaluate the total field and check if it overcomes the local switching field H _sw in each point. If multiple grains would switch, only the magnetization of the one with the smallest difference between the total field and H _sw is reversed and the entire system is recomputed. This routine of reversal and recomputation of the total field is done iteratively until a magnetic equilibrium state is reached for the current applied field. Only then this field is reduced by another step and the iterative process starts again.

The free parameters d and l were tuned by comparing calculated coercive fields of single cubic grains to conventional micromagnetic results. We used efficient global optimization (Huang et al., 2006) as implemented in the Dakota suite ¹⁰ to find the minimal error for grains with sizes between 100 nm and 800 nm. We included grains with a magneto-crystalline anisotropy direction aligned with the applied field and misaligned by 20°. The intrinsic properties for NdFeB are taken from literature as μ ₀ M _s = 1.61 T, K ₁ = 4.9 MJ/m³ and A = 8 pJ/m. With the optimized parameters d = 1.44 and l = 1.24, the reduced order model can reproduce the coercive fields from conventional micromagnetics very well (see Figure 9).

The empirical relation (Kronmüller et al., 1988) for the coercive field H c = α 2 K 1 μ 0 M s − N eff M s (4) is often used to analyze coercivity in permanent magnets. The microstructural parameter α is expected to account for the decrease of the coercive field by soft magnetic defects, misorientation, and intergrain exchange interactions (Kronmüller and Goll, 2002). The microstructural parameter N _eff gives the reduction of the coercive field caused by strong local demagnetizing fields (Grönefeld and Kronmüller, 1989). Experimentally, the microstructural parameters α and N _eff are derived from fitting a straight line to the values of H _c(T)/M _s(T) and 2 K 1 ( T ) / ( μ 0 M s 2 ( T ) ) measured at different temperatures T. In micromagnetic simulations, we can compute the microstructural parameters by separating the effects of the demagnetizing field. Computing the demagnetization curve but neglecting the magnetostatic energy term gives H c * = α 2 K 1 / ( μ 0 M s ) and we can derive α = H c * μ 0 M s / ( 2 K 1 ) . Computing the demagnetizing curve using magnetostatics gives H c = H c * − N eff M s . From the difference of the coercive fields computed with and without magnetostatic effects, we obtain the effective demagnetizing factor N eff = ( H c * − H c ) / M s .