Chemistry Reference
In-Depth Information
J
+
1
2
-
1
2
J
Figure 7.2
Energy level diagram due to the exchange interaction between two elec-
trons on the same atom: the exchange energy,
E
ex
1
=−
2
J
when the
1
electron spins are parallel, while
E
ex
=+
2
J
when the spins are anti-parallel.
to occupy states of opposite spin. We can represent the exchange interaction
energy,
E
ex
, for two electrons on the same atom by
E
ex
=−
2
J
s
1
·
s
2
1
2
J
1
2
=−
when
s
1
z
=
s
2
z
=±
1
2
J
=+
when
s
1
z
=−
s
2
z
(7.3)
as illustrated in fig. 7.2.
The situation is more complicated in molecules and solids, where the
exchange interactionalsoplays amajor role indetermining the ground state
energy. Consider a diatomic molecule with two electrons: the total energy
now includes interactions not only between the electrons but also between
the electrons and the two nuclei. We saw in Chapter 2 using the indepen-
dent electron approximation how the ground state wavefunction favours
a build-up of charge between the two nuclei. This is best achieved with
anti-parallel spins (see fig. 7.3), and indeed explains why most molecules
and covalent solids are diamagnetic.
Individual ions can retain a net magnetic moment in some molecules
and solids, in particular transition metal ions (e.g. Fe, Co) or rare earth
ions (e.g. Nd, Sm, Gd). The direct exchange interaction between elec-
trons on two such ions decreases very rapidly with increasing distance.
Indirect exchange interactions are, however, also possible, for example,
mediated through the electrons on a shared neighbouring atom or through
the spin of conduction electrons in a metal (fig. 7.4). Because competing
effects are present, these exchange interactions favour parallel spins and
ferromagnetism in some cases, while favouring anti-parallel spins and
antiferromagnetism in others.
The exact treatment of the spin interactions in magnetic solids, and their
temperature dependence is very complex and difficult, because we are
dealing essentially with a many-body effect, which can depend on both
local and non-local interactions. Much insight can, however, be obtained
from the Heisenberg Hamiltonian,
H
, which is widely used to describe the
