2 Temperature dependence of macroscopic anisotropy in magnetic nanoparticles

To evaluate the temperature dependence of the macroscopic anisotropy constant we use the constrained Monte Carlo method (CMC) introduced in the section 4.2 of the previous chapter. The CMC is an algorithm of the Monte Carlo type which allows to include both thermodynamic fluctuations and entropy into the evaluation of macroscopic quantities such as the temperature dependent magnetic anisotropy. Differently to chapter 2, the internal energy in this case is substituted by the free energy $\mathcal{F}$. We retrieve the anisotropy constants by calculating the average restoring torque in the equilibrium as a function of the constrained magnetization direction $\mathbf{M}_{0}(\theta, \varphi)$, where $(\theta,\varphi)$ are the constrained polar and azimuthal angles. $\theta,\varphi$.

$$\mathcal{T}=\bigl\langle-\sum_i\mathbf{S}_i\times\partial \mathca... ..._i\bigr\rangle \approx -\mathbf{M}\times\partial\mathcal{F}/\partial\mathbf{M}.$$ (76)

where $\mathcal{T}$ is the restoring torque, $\mathbf{S_{i}}$ is the localized spin moment at site i ( $\mid\mathbf{S_{i}}\mid=1$), and $\mathbf{M}$ is the average magnetization $\textbf{M}\equiv \bigl(\sum_{i}\textbf{S}_{i}\bigr)/\Vert\sum_{i}\textbf{S}_{i}\Vert$. The simulated angular dependence of the torque at $\varphi =0$ is then fitted to

$$\mathcal{T}(\theta) =-K_{ua}^{eff}(T) \sin(2 \theta)- \frac{1}{2} K_{ca}^{eff}(T) \sin (4 \theta) .$$ (77)

from which the effective uniaxial ( $K_{ua}^{eff}$) and cubic $K_{ca}^{eff}$ anisotropies are extracted as a function of temperature.