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Taking the atomic representation for exchange energy between two
moments (equation 2.2), we can assume that the angle
between two neighbouring spins is
. The sum of all the
exchange energies based on equation 2.4 can be rewritten as:
where since
is a unit vector (equation 2.3) and for small values of
we use the leading terms in the Taylor expansion of
(figure 2.6):
With this assumption, equation 2.18 can be rewritten:
where
is a constant. Since
and
(figure 2.5):
and
approximates the spatial
derivative of
over the lattice spacing .
If we take
to be a lattice translation vector of magnitude as in
figure 2.5, the directional derivative
can be used
to express
.
Inserting this into equation 2.18, the exchange energy can
now be represented as (Blundell, 2001):
if we take
outside the summations and redefine this
as (the nearest neighbour distance). Since we will integrate over
volume to obtain the continuous representation, if we consider a unit
cell site number , or (for simple cubic, bodycentred
cubic and facecentred cubic respectively), we can define the exchange
coupling constant (Aharoni, 2000):
The exchange coupling constant
The cell site number; , or for simple cubic, bodycentred
cubic and facecentred cubic respectively
The distance between nearest neighbours in a crystalline lattice
We can now ignore the discrete lattice, yielding the continuous form:
Figure:
The functions
(solid black) and
(dashed red). The dotted green line represents the difference between the two functions

Next: Anisotropy energy
Up: Micromagnetic description
Previous: Micromagnetic description
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Richard Boardman
20061128