Analytic Expression for Magnetic Activation Energy

We theoretically investigate the magnetic activation energy of permanent magnets. Practically, it is widely used in a phenomenological form as $\mathcal{F}_\mathrm{B}(H_\mathrm{ext})=\mathcal{F}_\mathrm{B}^0\left(1-H_\mathrm{ext}/H_0\right)^n,$ where $\mathcal{F}_\mathrm{B}^0$ is the activation energy in the absence of an external magnetic field $H_\mathrm{ext}$, $n$ is a real parameter, and $H_0$ is defined by the equation $\mathcal{F}_\mathrm{B}(H_0)=0$. We derive the general and direct expressions for these phenomenological parameters under the restriction of uniform rotation of magnetization and on the basis of the perturbative theory with respect to $H_\mathrm{ext}$. Further,we apply our results to Nd$_2$Fe$_{14}$B magnets and confirm the validity of the proposed method by comparing with the Monte Carlo calculations.