# Calculation of Three-dimensional Energy Product for Isotropic Nd\({}_{2}\)Fe\({}_{14}\)B Magnet
## 3 Simulation Results
Due to the exchange coupling, the magnetizations (Figure 4a) have almost the same value within the same grain, but they have different values depending on the grain because the anisotropy direction of each grain is different. On the other hand, the demagnetizing fields (Figure 4b) follow a distribution similar to the Gaussian distribution regardless of the magnitude of the applied magnetic field [25]. This discrepancy between the magnetizations and the demagnetizing fields results in the gap between the energy products calculated from the average values of the \(B\)-field and the \(H\)-field and from the integral of each cell. The distribution of the demagnetizing fields is speculated by cumulative distribution function (CDF) in Figure 4c. Empirical CDF from the simulation is plotted as a purple line, and for comparison, the standard normal CDF is plotted as black scatter. The empirical CDF almost coincides with the standard normal CDF. Figure 2: (a) Hysteresis loops of magnetization \(\mu_{0}\)M with the applied magnetic field \(H_{\rm app}\), and the magnetic flux density \(B\) versus the magnetic field strength \(H\) (=\(H_{\rm app}\) + \(H_{\rm d}\)). (b) Energy product \(BH\) calculated in three different ways. CDF. Therefore, it is justified to approximate the demagnetizing field distribution with a Gaussian. The demagnetizing factor (\(N=-H_{\mathrm{d}}/M_{\mathrm{z}}\)) is a geometry-dependent value representing the magnet, which refers to the proportionality of the demagnetizing field with the magnetization [26]. To clearly show the mismatch between the magnetization and the demagnetizing field, the demagnetizing factor of each cell was investigated; a histogram for the distribution of the demagnetizing factor is given in Figure 5. In the saturation state, the demagnetizing factor is affected by the distribution of the demagnetizing field and follows a narrow Gaussian distribution because the magnetizations in the magnet are aligned along the applied field direction. On the other hand, in the case of the reman  
Figure 4: Histograms for cells with certain values for (**a**) magnetizations and (**b**) demagnetizing fields at 10 T (saturation), 0 T (remanent state), and \(-3\) T (during reversal). (**c**) Cumulative distribution function (CDF) for the computed demagnetizing field at the remanent state, and the standard normal CDF by the mean and the standard deviation of the demagnetizing field in each cell. Figure 3: (**a**) Magnetizations and (**b**) demagnetizing fields of the model magnet at 10 T (saturation), 0 T (remanent state), \(-3\) T (during reversal). [PERSON] and blue colors correspond to \(M_{\mathrm{s}}\) and \(-M_{\mathrm{s}}\), respectively. state, the demagnetizing factor spreads over a wide range due to the mismatch between the magnetizations and the demagnetizing fields. For these differences in distribution between the magnetization and the demagnetizing field during the magnetic reversal, the energy product calculated from the integration of each element is inevitably different from the energy product given by the product of the mean \(B\) and the mean \(H\).