1.1.3.3 Microscopic model

Now we have to investigate how the exchange interactions play on a larger spatial scale, namely how the elementary magnetic moments $\textbf{M} dV_\textbf{r}$ exchange-interact with one another. We follow the derivation proposed by Landau and Lifshitz in 1935, reported by W.F. Brown Jr. in Ref [12]. In this respect, an energy term which penalizes magnetization disuniformities is introduced in the free energy. This term, in the isotropic case (i.e. cubic cell) is consisted of an expansion in even power series of the gradients of magnetization components [11]. If one stops the expansion to the first term, the disuniformity penalization assumes the form:

$$f_ =A[(\nabla m_x)^2+ (\nabla m_y)^2+ (\nabla m_z)^2] \quad,$$ (1.31)

where the constant $A$, having dimension of [J/m], has to be somehow determined. One way is to identify the exchange constant from experiments, but it is also possible to estimate it with a theoretical approach. In fact, let us consider a cubic lattice of spins, with interaction energy given by the Heisenberg Hamiltonian:

$$W=-2J\sum \textbf{S}_i \cdot \textbf{S}_j \quad,$$ (1.32)

where the sum is extended to the nearest neighbors only and $\textbf{S}_i$, $\textbf{S}_j$ are the spin angular momenta, expressed in units of $\hbar$, associated to sites $i$ and $j$, and $J$ is the nearest neighbor exchange integral. We assume that the forces between spins are sufficiently strong to keep the neighbor spins almost parallel. Thus, if $\textbf{{m}}_i$ is the unit-vector in the direction $-\textbf{S}_i$, such that $\textbf{S}_i=-S\textbf{{m}}_i$ ($S$ is the spin magnitude), and if $\theta_{i,j}$ is the small angle between the directions $\textbf{{m}}_i$ and $\textbf{{m}}_j$, one can rewrite Eq. (1.32) as

$$\begin{split} W=-2JS^2 \sum \cos \theta_{i,j}\simeq -2JS^2\... ...left(\textbf{{m}}_j-\textbf{{m}}_i\right)^2 \quad, \end{split}$$ (1.33)

since for small $\theta_{i,j}$, $\vert\theta_{i,j}\vert=\vert\textbf{{m}}_j-\textbf{{m}}_i\vert$. We now assume that the displacement vector $\textbf{{m}}_j-\textbf{{m}}_i$ can be written in terms of a continuous function $\textbf{{m}}$ such that:

$$\textbf{{m}}_j-\textbf{{m}}_i=\Delta\textbf{r}_j\cdot \nabla\textbf{{m}}\quad,$$ (1.34)

where $\Delta \textbf{r}_j=\textbf{r}_j-\textbf{r}_i$ is the position vector of neighbor $j$ with respect to site $i$. Then, if $\textbf{{m}}=m_x\mathbf{e}_x+m_y\mathbf{e}_y+m_z\mathbf{e}_z$,

$$W = + JS^2\sum(\Delta\textbf{r}_j\cdot\nabla\textbf{{m}})^2=$$ (1.35)

$$= + JS^2\sum[(\Delta\textbf{r}_j\cdot \nabla m_x)^2 + (\Delta\textbf{r}_j\cdot \nabla m_y)^2 + (\Delta\textbf{r}_j\cdot \nabla m_z)^2] \quad.$$

Now we sum over $j$ and multiply by the number of spins per unit volume $n$ in order to obtain the energy per unit volume $f_$ex. It is important to notice that, if $\Delta\textbf{r}_j=x_j\mathbf{e}_x+y_j\mathbf{e}_y+z_j\mathbf{e}_z$, due to the cubic symmetry it happens that $\sum_j x_j y_j=0$, and $\sum_j x_j^2=\frac{1}{3}\sum_j \Delta\textbf{r}_j^2$. By using these properties and neglecting the constant term, one ends up with:

$$f_ =A[(\nabla m_x)^2+ (\nabla m_y)^2+ (\nabla m_z)^2] \quad,$$ (1.36)

where $A$ is the exchange constant:

$$A=\frac{1}{6} n J S^2\sum \Delta\textbf{r}_j^2 \quad,$$ (1.37)

which can be particularized for different lattice geometries (body-centered, face-centered cubic crystals). Typical values of $A$ are in the order of $10^{-11}$ J/m. Finally, one can write the contribution of exchange interactions to the free energy of the whole magnetic body by integrating Eq. (1.36) over the region $\Omega$:

$$F_ =\int_\Omega A[(\nabla m_x)^2+ (\nabla m_y)^2+ (\nabla m_z)^2] dV \quad.$$ (1.38)

It is important to notice that, in this case, the exchange interaction is isotropic in space, meaning that the exchange energy of a given volume $\Delta V$ is the same for any orientation of the magnetization vector, provided that its strength remains the same. In this respect, the expression (1.38) for the exchange energy puts this consideration into evidence.