Exchange energy

The phenomenon whereby individual atomic magnetic moments will attempt to align all other atomic magnetic moments within a material with itself is known as the exchange interaction (Aharoni, 2000). If the magnetic moments align in a parallel fashion, the material is ferromagnetic; if the magnetic moments align antiparallel, the material is antiferromagnetic.

Table 2.1: Magnetic moments of important transition metals (Kittel, 1996)

name	symbol	configuration	lattice type	moment (A$\cdot$m$^2$)
iron	Fe	3d$^6$	bcc	2.22$\times$10$^{-23}$
cobalt	Co	3d$^8$	hcp	1.72$\times$10$^{-23}$
nickel	Ni	3d$^7$	fcc	0.61$\times$10$^{-23}$

The exchange energy between two neighbouring magnetic moments $\mu$$_i$ and $\mu$$_j$ is usually described by:

$$\mathcal{E}^{i,j}_{\mathrm{ex}} = -\mathcal{J} \ensuremath{\mathbf{S}}_i\cdot\ensuremath{\mathbf{S}}_j$$ (2.2)

$\mathcal{E}^{i,j}_{\mathrm{ex}}$The exchange energy between two neighbouring magnetic moments $\mu$$_i$ and $\mu$$_j$ $\mathcal{J}$The exchange integral, originating from the wave function for two electrons $\Psi(\ensuremath{\mathbf{r}}_1,\ensuremath{\mathbf{r}}_2)$ being antisymmetric $\ensuremath{\mathbf{S}}$A normalised magnetic moment, $\mu$ |$\mu$|

where $\ensuremath{\mathbf{S}}$ is the unit vector of the direction of the magnetic moment:

$$\ensuremath{\mathbf{S}} = {\mbox{\boldmath {$\mu$}} \over \vert\mbox{\boldmath {$\mu$}}\vert}$$ (2.3)

and $\mathcal{J}$ is the exchange integral which originates from the wave function overlap of two electrons.

Consequently, the exchange energy for a system of particles, under the assumption that the exchange energy is short-ranging and subsequently only acts on direct neighbours, is:

$$\mathcal{E}_{\mathrm{ex}} = {1 \over 2} \sum_i \sum_{j\in {\mathcal{N}_{i}}} \mathcal{E}^{i,j}_{\mathrm{ex}}$$ (2.4)

$\mathcal{N}$Used to represent nearest neighbours in summations

where $\mathcal{N}_i$ represents the nearest neighbours $i$. The value of $\mathcal{J}$ is derived experimentally and expressed as a function of $A$ (see equation 2.25).

The sign of $\mathcal{J}$ is important -- if $\mathcal{J}$ is positive, it indicates the material exhibits ferromagnetic behaviour and the exchange energy is at a minimum when two neighbouring moments are in parallel alignment.

Antiferromagnetic materials have a negative $\mathcal{J}$, and as such have a minimum exchange energy when aligned antiparallel.

If a ferromagnet is heated above a critical point known as the Curie temperature (Curie, 1895), when the applied field is zero, the average magnetisation also becomes zero.

Typical values of exchange energy between two parallel ferromagnetic magnetic moments for iron, cobalt and nickel are given in table 2.2.

Table 2.2: Exchange energy between parallel ferromagnetic magnetic moments of important transition metals. Reversing the sign gives the energy between antiparallel moments

name	symbol	energy between parallel neighbours (J)
iron	Fe	-1.21$\times$10$^{-21}$
cobalt	Co	-5.15$\times$10$^{-21}$
nickel	Ni	-4.46$\times$10$^{-21}$