Chemistry Reference
In-Depth Information
to consider some of the features which enable the diversity of properties.
Finally, we consider some specific applications.
7.2 The exchange interaction
Magnetic ordering in a ferromagnet is
not
due to magnetic interactions
between neighbouring dipoles. Consider two neighbouring atoms, each
with a magnetic moment equal to the Bohr magneton,
B
. The mag-
nitude of the magnetic field at one atom due to the other is given
approximately by
µ
∼
µ
µ
0
B
B
(7.1)
r
3
4
π
where
r
is the separation between the two atoms,
∼
3Å for nearest neigh-
µ
bours. The interaction energy
E
is then of order
B
B
, and substituting
for
B
in eq. (7.1) we can estimate the direct magnetic interac-
tion between neighbouring atoms as
µ
0
and
µ
10
−
6
eV, which is considerably
less than thermal energies. Direct magnetic interactions are, therefore, too
weak to overcome thermal disordering effects, and we must seek an alter-
native explanation for magnetic ordering. This is provided by the
exchange
interaction
.
The exchange interaction is a quantummechanical effect - it has no clas-
sical analogue - and arises due to the electrostatic interaction between
electrons, as discussed below. We have already encountered the Pauli
Exclusion Principle, which states that no two electrons can occupy the
same energy state. An alternative expression of the exclusion principle
states that the wavefunction
E
∼
describing two electrons with coordinates
r
1
and
r
2
and spins
s
1
and
s
2
must be anti-symmetric when all the coor-
dinates of the two electrons are exchanged, including their position and
spin:
ψ
ψ(
r
1
,
s
1
;
r
2
,
s
2
)
=−
ψ(
r
2
,
s
2
;
r
1
,
s
1
)
(7.2)
This immediately requires that
s
2
,sothat
there is zero probability of finding two electrons of the same spin at the
same point in space. By contrast, electrons with opposite spin
can
be at
the same point. We therefore expect that for two electrons on the same
atom, their average separation
ψ
=
0 when
r
1
=
r
2
and
s
1
=
|
−
|
r
1
r
2
will be larger for parallel spins
(
s
1
=
s
2
)
than for anti-parallel spins
(
s
1
=−
s
2
)
. Hence the inter-electron
e
2
Coulomb repulsion energy
is smaller for parallel than
for anti-parallel spins. This effect is referred to as the exchange interaction.
It immediately explains Hund's first rule in Section 6.6, whereby electrons
first occupy all states of the same spin in an isolated atom before they start
/(
4
πε
|
r
1
−
r
2
|
)
0
