1.3.4 Normalized equations

It is very useful to write the micromagnetic equations in dimensionless units. This is helpful as soon as one wants to investigate which terms are prevalent in given situations and moreover, the normalization considerably simplifies the expressions. We start our discussion from the expression of the free energy (1.53). By dividing both sides of Eq. (1.53) by $\mu_0 M_s^2 V_0$ ($V_0$ is the volume of the body) one obtains:

$$g(\textbf{{m}},\textbf{h}_ )=\frac{G(\textbf{M},\mathbf{H}_\text{a})}{\mu_0 M_s^2 V_0}=\int_... ...cdot\textbf{h}_\text{m}- \textbf{{m}}\cdot\textbf{h}_\text{a}\bigg] dv \quad,$$ (1.90)

where the normalized volume $v$ is measured in units of $V_0$. In this framework, we can obtain the normalized effective field $\textbf{h}_{\text{eff}}=\textbf{H}_{\text{eff}}/M_s$ by taking the variational derivative $\delta g/\delta \textbf{{m}}$ of the normalized free energy:

$$\textbf{h}_{\text{eff}}=\frac{2}{\mu_0 M_s^2}\nabla\cdot(A\nabla\... ...xt{an}}{\partial \textbf{{m}}} + \textbf{h}_\text{m}+ \textbf{h}_\text{a}\quad.$$ (1.91)

It is important to focus on the following quantity with the dimension of a length in Eq. (1.90):

$$l_ =\sqrt{\frac{2A}{\mu_0 M_s^2}} \quad,$$ (1.92)

which is commonly referred to as exchange length. The exchange length gives an estimation of the characteristic dimension on which the exchange interaction is prevalent. For typical magnetic recording materials $l_$ex is in the order of $5\div 10$ nm. Therefore, one expects that on a spatial scale in the order of $l_$ex the magnetization is spatially uniform. This is very important when spatial discretization of micromagnetic equations has to be preformed. In fact, one should be sure that the mesh characteristic dimension is smaller than $l_$ex. Now let us consider the Landau-Lifshitz-Gilbert equation (1.83). By dividing both sides by $\gamma M_s^2$ one obtains:

$$\frac{1}{\gamma M_s^2}\frac{\partial \textbf{M}}{\partial t}=-\f... ... M_s^2 M_s} \textbf{M}\times\frac{\partial \textbf{M}}{\partial t} \quad.$$ (1.93)

Now, remembering that

$$\textbf{{m}}=\frac{\textbf{M}}{M_s} \quad,\quad \textbf{h}_{\text{eff}}= \frac{\textbf{H}_{\text{eff}}}{M_s}$$ (1.94)

and by measuring the time in units of $(\gamma M_s)^{-1}$, Eq. (1.93) can be rewritten in the following dimensionless form:

$$\frac{\partial \textbf{{m}}}{\partial t} = - \textbf{{m}}\times ... ...ff}}+ \alpha\textbf{{m}}\times \frac{\partial \textbf{{m}}}{\partial t} \quad .$$ (1.95)

In the case of $M_s\simeq 796$ kA/m ( $\mu_0 M_s=1$ T), the dimensionless time unit corresponds to $(\gamma M_s)^{-1}\simeq 5$.7 ps.