Neodymium (Nd) magnets (${\mathrm{Nd}}_2{\mathrm{Fe}}_{14}\mathrm{B}$) are important permanent magnets due to their strong coercive force, which contributes to high-efficiency energy conversion technologies. This coercivity is often enhanced by substituting dysprosium (Dy). Therefore, understanding the magnetic properties of Dy-substituted systems, ${\left({\mathrm{Nd}}_{1-x}{\mathrm{Dy}}_x\right)}_2{\mathrm{Fe}}_{14}\mathrm{B}$, is essential. We investigate the spin reorientation transition in ${\left({\mathrm{Nd}}_{1-x}{\mathrm{Dy}}_x\right)}_2{\mathrm{Fe}}_{14}\mathrm{B}$ using a recently developed atomistic modeling approach. This modeling method captures the microscopic mechanisms of magnetic interactions and temperature effects, including thermal fluctuations. We study the $x$ dependence of the spin reorientation transition temperature ($T_{\mathrm{SR}}$) and the canting angle ($\theta$) of the total magnetization using an importance-sampling Monte Carlo method, based on a model with microscopic parameters derived primarily from first-principles calculations. Our estimates of $T_{\mathrm{SR}}$ and $\theta$ are consistent with experimental results. We also compare our results with those obtained from a previous mean-field-like study and show significant differences, particularly at higher Dy concentrations. Our model more accurately captures experimental trends in this regime. We attribute this improvement to more accurate representations of canting angles and anisotropy energies of the constituent atoms. Additionally, we investigate an anomalous behavior in the total magnetization of ${\left({\mathrm{Nd}}_{1-x}{\mathrm{Dy}}_x\right)}_2{\mathrm{Fe}}_{14}\mathrm{B}$. We find a nondifferentiable point in the magnetization at $T_{\mathrm{SR}}$, which becomes more pronounced with increasing $x$, peaking at $x=0.5$. We discuss the origin of this anomaly in detail through an analysis of the magnetic properties of the constituent atoms.

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• Magnetic phase transitions
• Spin reorientation transition
• Permanent magnets
• Transition-metal rare-earth alloys
• Crystal-field theory
• Monte Carlo methods