# Calculation of Three-dimensional Energy Product for Isotropic Nd\({}_{2}\)Fe\({}_{14}\)B Magnet
###### Abstract  
A conventional energy product calculated by the product of the \(B\)-field and the \(H\)-field is not sufficient for representing the performance of a magnet because it considers the homogeneous and only the uniaxial magnetic properties of the magnet. The conventional energy product has been compared with another energy product obtained by integrating the scalar product of the \(B\)-field and the \(H\)-field of each cell composed of the three-dimensional components. We investigated a model system by micromagnetic simulation using finite differential method (FDM) and calculated the full hysteresis of the magnet. The model system of a Nd\({}_{2}\)Fe\({}_{14}\)B magnet composed of grains with a diameter of about 100 nm was assumed. In the case of the isotropic multi-grain magnet, the energy product calculated by the integration method was 28% larger than the energy product obtained by the conventional way, although a discrepancy between the distribution of the magnetizations and the demagnetizing fields at the reversal process resulted in the decrease of the energy product. permanent magnet; energy product; micromagnetic simulation; multidomain reversal +
Footnote †: journal: Applied Sciences  
between the magnetic flux density \(B\) and the magnetic field \(H\), and the maximum energy product (\(BH_{\text{max}}\)) is widely used as a figure of merit for evaluating the performance of hard magnetic materials. However, there are some missing points for the usage of the \(BH_{\text{max}}\). Firstly, \(BH_{\text{max}}\) is the highest value of the energy products among various magnitudes of the magnetic field. The energy products obtained from the magnetic hysteresis of a magnet loop can supply the expectation value of the energy product of magnets for different shapes, although it does not perfectly match the energy product of the magnets [5,6]. The only meaningful value for a system is an energy product at the remanent state, which is determined from the demagnetizing field by the shape of the magnet. Another missing point comes from how the energy product is calculated. Although the energy product is usually given as the area of a rectangle in a \(B\)-\(H\) hysteresis loop, there is a clear difference between the energy product and the real value of the energy product calculated from its original physical meaning. One difference is that the conventional energy product assumes the magnetization and the demagnetizing field as uniaxial components in spite of the three-dimensional emission of the magnetic field. Another difference comes from the discrepancy between the distribution of the magnetizations and the demagnetizing fields in the magnet. Because the permanent magnets are not used in the saturation state, the influence by the internal magnetic state during magnetic switching is inevitable for determining the energy product of the magnet. Most studies on the switching process cover the dynamic reversal process of soft magnets [7, 8, 9, 10, 11, 12], and there are some research works that investigate the reversal mechanism of permanent magnets [13, 14, 15]. The reversal process is studied in order to estimate the effect of temperature [15], exchange spring [13], and grain boundary diffusion [14]. However, the effect of the reversal mode on the energy product has scarcely been studied despite its importance in estimating the practical magnetic energy of the permanent magnets. One of the reasons why there is a lack of studies devoted to estimating the energy product based on the theory is the difficulty in measuring the internal state of the magnet. Finally, an easy way to estimate the energy product, which multiplies the average values of \(B\) and \(H\), is widely used. Recent advances in computing technology have made it possible to investigate the complex magnetic properties inside magnets.