# Calculation of Three-dimensional Energy Product for Isotropic Nd\({}_{2}\)Fe\({}_{14}\)B Magnet
## 3 Simulation Results  
The calculated magnetic hysteresis loops are represented in Figure 2a. The black solid line follows the magnetization (\(\mu_{0}M\)) with respect to the applied magnetic field (\(H_{\text{app}}\)) from the simulation results. One of the main points is the difference between the methods used to obtain the values of the energy product (\(BH\)), which are expressed in Figure 2b. The orange line and squares indicate the energy products calculated in the conventional way, in which the values are from the product between the \(B\)-field and the \(H\)-field. The \(B\)-field is the sum of the magnetization along the direction of the applied magnetic field (\(+z\) in this simulation), while the \(H\)-field consists of both the applied field and the demagnetizing field along the \(+z\) direction. To reflect the experimentally measured values, the average values for the system were used. The violet line with triangle symbols represents the energy product calculated by integrating the energy product of each cell, which considers only a one-dimensional component along the magnetic field direction. On the other hand, the olive line and circle symbols in Figure 2b are the energy products calculated by the integration of the energy products of each cell; these energy products were obtained from the inner product between the \(B\)-field and the \(H\)-field. The three-dimensional energy product of each cell was calculated by  
\[B\cdot H=B_{x}H_{x}+B_{y}H_{y}+B_{z}H_{z}, \tag{3}\]  
where the \(B\)-field and the \(H\)-field follow the indicated direction. There are definite gaps between the energy product from the conventional method and the integration method, which can be explained by the simple mathematical relation expressed below. \[\int_{i}B\cdot HdV\ \
eq\ \overline{B}\cdot\overline{H}. \tag{4}\]  
Figure 1: Schematic image of the Nd\({}_{2}\)Fe\({}_{14}\)B magnet consisting of 4871 grains with randomly distributed anisotropy directions. The magnetizations and the demagnetizing fields during the reversal process were also investigated. Figure 3a represents the magnetization of each cell in color, while Figure 3b shows that of the demagnetizing field. It must be noted that the color indicates the magnitude along the +\(z\) direction. With the magnetic field of 10 T, all the cells were saturated along the +\(z\) direction, which is represented by the image filled in with the color red. With the reduction of the applied magnetic field, the magnitude of the magnetization was reduced and changed into the color blue. At the remanent state without the application of the magnetic field, the magnetizations were distributed to certain values on a grain-by-grain basis, which still have a positive value averaging to remanent magnetization (\(M_{\rm r}\)). What we are focusing on is the difference of tendency between the magnetization and the demagnetizing field, and it can be easily understood from the close-up image of a cube envelopted in dashed lines. While the magnetizations in the same grain have similar values due to exchange interaction, the magnitudes of the demagnetizing fields are differently distributed in a grain. The edge of the grain has stronger magnitudes of the demagnetizing fields. To clearly compare this difference, the histograms for cells with certain values of magnetizations and demagnetizing fields were investigated and represented in Figure 4. At the saturation state, the values of the magnetizations are distributed across a narrow range with high intensity, whereas the values of the demagnetizing fields follow a different distribution from that of the magnetization. The magnetizations were determined by the combination of the easy axis direction of the grain and the strength of the applied magnetic field, while the demagnetizing fields are associated with the magnetizations and the geometry of the whole system.