Chemistry Reference
In-Depth Information
the progress shown in fig. 7.10. The heavy rare-earths Dy and Ho have the
largest atomic moment
of any element in the periodic table and
so could in theory show energy products as high as 3000 kJ/m
3
, enabling
a flux density approaching 4 T. However, prospects are poor for realising
room temperature ferromagnetic alloys based on these elements. They are
likely to be restricted to cryogenic applications, if they are ever used at all.
Nevertheless, the progress shown in fig. 7.10 has enabled the development
of a multi-billion dollar industry and the widespread use of permanent
magnets in applications such as motors and generators, actuators and
printers, loudspeakers and magnetic resonance imaging devices, as well
as a diverse range of scientific instruments from large-scale colliders to
tiny ammeters. A considerably more detailed discussion of the science
and application of high-performance magnets is given in Coey (1996).
(
10
µ
)
B
7.10 Itinerant ferromagnetism
We have assumed throughout this chapter that the magnetisation in a fer-
romagnet is due to interactions between localised magnetic moments on
neighbouring ions. There are, in addition, several metals, in particular iron
and nickel, whose ferromagnetism is due to interactions between delo-
calised, so-called itinerant conduction electrons. This is entirely analogous
to the situation considered in the previous chapter, where we saw that
paramagnetism can be associated not just with isolated spins but also with
delocalised electrons in a metal (see Section 6.8).
We can divide the conduction band in a metal into spin-up and spin-
down sub-bands. Potential energy is gained due to the exchange interac-
tion if one sub-band is preferentially occupied at the expense of the other.
This preferential occupation, however, costs kinetic energy. For most met-
als the cost in kinetic energy exceeds the gain in potential energy, so that
they remain paramagnetic. There is, however, a narrow partly-filled band
associatedwith the 3d electrons inNi and Fe, leading to a very large density
of states near the Fermi energy,
g
. The exchange interaction can domi-
nate over kinetic energy effects in this case, leading to a net magnetisation,
as illustrated in fig. 7.12.
The dependence of band ferromagnetismon the density of states near the
Fermi energy can be understood as follows. Suppose a d band possesses
equal numbers of spin-up and spin-down states. Application of an exter-
nal magnetic field
H
can lead to a relative shift,
(
E
F
)
E
, in the two sub-bands,
with electrons transfering from spin-up to spin-down states, as in fig. 6.7.
If we now allow an exchange interaction between electrons of the same
spin, then the shift in the two sub-bands,
E
=
µ
µ
B
H
is increased by an
0
=
(
N
+
−
N
−
)
amount
J
is
the difference in occupation of the two sub-bands, and
J
is an effective
exchange energy between parallel spins. We can, therefore, write the
N
relative to the purely paramagnetic case.
N
