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Effects of trace elements on the crystal field parameters of Nd ions at the surface of Nd$_2$Fe$_{14}$B grains

Using first-principles calculations, we investigate the positional dependence of trace elements such as O and Cu on the crystal field parameter $A_2^0$, proportional to the anisotropy constant $K_u$ of Nd ions placed at the surface of Nd$_2$Fe$_{14}$B grains. The results suggest the possibility that the $A_2^0$ parameter of Nd ions at the (001) surface of Nd$_2$Fe$_{14}$B grains exhibits a negative value when the O or Cu atom is located near the surface, closer than its equilibrium position. At the (110) surface, however, O atoms located at the equilibrium position provide a negative $A_2^0$, while for Cu additions $A_2^0$ remains positive regardless of Cu's position. Thus, Cu atoms are expected to maintain a positive local $K_u$ of surface Nd ions more frequently than O atoms when they approach the grain surfaces in the Nd-Fe-B grains.