Chemistry Reference
In-Depth Information
E
F
B
B
B
B
Spin
Spin
Density of states
Figure 6.7
Density of states for spin up (LHS) and spin down (RHS) electrons of
a free electron metal in an applied field
B
. When
B
=
0, the bottom
of the spin up and spin down bands are at the same energy, with equal
number of spin up and down states filled to the Fermi energy,
E
F
. In field
B
, the electrons which occupied the cross-hatched states on the LHS
must be transferred into previously empty states on the RHS, giving a
net paramagnetic magnetisation.
1
2
g
0
by
mB
=
µ
B
B
while states with
m
against the field direction shift up
by
2
g
0
µ
B
B
, as illustrated in fig. 6.7. The Fermi energy must be the same
for both sets of states in thermal equilibrium. Considering the shaded area
in fig. 6.7, this occurs by moving
1
2
g
0
1
2
g
electrons per
unit volume from states with moment aligned against the field (above the
Fermi energy) to states aligned with moment parallel to the field (below
the Fermi energy). The net magnetisation
M
is then given by
N
=
(
µ
B
B
)
·
(
(
E
F
))
1
2
g
0
1
4
g
0
µ
B
g
M
=
µ
·
2
N
=
(
E
F
)
B
(6.42)
B
so that the paramagnetic susceptibility,
χ
P
, is then given by
2
B
g
χ
∼
µ
µ
(
E
F
)
(6.43)
P
0
Because thermal energies
kT
are much smaller than the Fermi energy
E
F
,
the same argument can be applied at finite temperature, and the paramag-
netic susceptibility of a metal, referred to as the Pauli spin susceptibility,
is then approximately independent of temperature. Because electrons near
