Exchange energy

Taking the atomic representation for exchange energy between two moments (equation 2.2), we can assume that the angle between two neighbouring spins is $\phi_{i,j}$. The sum of all the exchange energies based on equation 2.4 can be rewritten as:

$$\mathcal{E}_{\mathrm{ex}} = - \mathcal{J}S^2 \sum_i \sum_{j\in \mathcal{N}_i} \cos{\phi}_{i,j}$$ (2.18)

where $S=1$ since $\ensuremath{\mathbf{S}}$ is a unit vector (equation 2.3) and for small values of $\phi_{i,j}$ we use the leading terms in the Taylor expansion of $\cos{\phi}_{i,j}$ (figure 2.6):

$$\cos{\phi}_{i,j} \approx 1-{\phi^2_{i,j} \over 2}$$ (2.19)

With this assumption, equation 2.18 can be rewritten:

$$\mathcal{E}_{\mathrm{ex}} = \mathcal{K} + {\mathcal{J}S^2 \over 2} \sum_i \sum_{j\neq i}^{\mathcal{N}} {\phi}_{i,j}^{2}$$ (2.20)

where $\mathcal{K}$ is a constant. Since $\ensuremath{\mathbf{S}}_i = {\ensuremath{\mathbf{M}}(\ensuremath{\mathbf{r}}_i) \over M_s}$ and $\vert\ensuremath{\mathbf{S}}_i\vert = \vert\ensuremath{\mathbf{S}}_j\vert = 1$ (figure 2.5):

$$\vert\phi_{i,j}\vert \approx \vert\ensuremath{\mathbf{S}}_i-\ensuremath{\mathbf{S}}_j\vert$$ (2.21)

$$= a { \vert\ensuremath{\mathbf{S}}_i-\ensuremath{\mathbf{S}}_j\vert \over a }$$ (2.22)

and ${ \vert\ensuremath{\mathbf{S}}_i-\ensuremath{\mathbf{S}}_j\vert \over a }$ approximates the spatial derivative of $\ensuremath{\mathbf{S}}$ over the lattice spacing $a$.

If we take $\ensuremath{\mathbf{r}}_{i,j}$ to be a lattice translation vector of magnitude $a$ as in figure 2.5, the directional derivative $\nabla_{\ensuremath{\mathbf{r}}_{i,j}}\ensuremath{\mathbf{S}}$ can be used to express $\vert\ensuremath{\mathbf{S}}_i-\ensuremath{\mathbf{S}}_j\vert$.

Inserting this into equation 2.18, the exchange energy can now be represented as (Blundell, 2001):

$$\mathcal{E}_{\mathrm{ex}} = - \mathcal{J}S^2 \sum_i \sum_{j\neq i}^{\mathcal{N}} \left [ (\ensuremath{\mathbf{r}}_{i,j} \cdot \nabla)\ensuremath{\mathbf{S}} \right ] ^2$$ (2.23)

$$= - \mathcal{J}S^2a^2 \sum_i \sum_{j\neq i}^{\mathcal{N}} \left [(\nabla m_x)^2 + (\nabla m_y)^2 + (\nabla m_z)^2 \right ]$$ (2.24)

if we take $\ensuremath{\mathbf{r}}_{i,j}$ outside the summations and redefine this as $a$ (the nearest neighbour distance). Since we will integrate over volume to obtain the continuous representation, if we consider a unit cell site number $z=1$, $2$ or $4$ (for simple cubic, body-centred cubic and face-centred cubic respectively), we can define the exchange coupling constant (Aharoni, 2000):

$$A = {\mathcal{J}S^2z \over a}$$ (2.25)

$A$The exchange coupling constant $z$The cell site number; $z=1$, $2$ or $4$ for simple cubic, body-centred cubic and face-centred cubic respectively $a$The distance between nearest neighbours in a crystalline lattice

We can now ignore the discrete lattice, yielding the continuous form:

$$\mathcal{E}_{\mathrm{ex}} = A \int_V \left [ (\nabla S_x)^2 + (\nabla S_y)^2 + (\nabla S_z)^2 \right ] \mathrm{d}^3r$$ (2.26)

Figure: The functions $\cos{\phi}$ (solid black) and $1-{\phi ^2 \over 2}$ (dashed red). The dotted green line represents the difference between the two functions