Page 298 - Electrical Properties of Materials

280 Magnetic materials

The summation in eqn (11.43) is one of the simpler ones to perform,
yielding
1 J(J + 1)(2J +1),
3
which gives finally
2 2
μ m = g μ mB J(J +1)μ 0 H/3k B T. (11.45)
We may now express the above equation in terms of the total angular
momentum,
1/2
= {J(J +1)} (11.46)
and total magnetic momentum
μ m = ge /2m (11.47)
to get
2
μ m = μ μ 0 H/3 k B T, (11.48)
m
∗ This perhaps shows the power of hu- in agreement with the classical result. ∗
man imagination. If one has a fair idea
how the final conclusion should look,
one can get a reasonable answer in spite 11.7.3 Paramagnetic solids
of following a false track.
As we have seen, the magnetic properties of electrons combine to produce the
magnetic properties of atoms. These properties can be measured in a Stern–
Gerlach apparatus, where each atom may be regarded as a separate entity.
This is because the atoms in the vapour are far enough from each other not
to interact. However, when the atoms aggregate in a solid, the individual mag-
netic properties of atoms combine to produce a resultant magnetic moment.
The electrons that are responsible for chemical bonding are usually respons-
ible for the magnetic properties as well. When, for example, sodium atoms and
chlorine atoms combine to make up the ionic solid, NaCl, then the valence
electron of the sodium atom moves over to the chlorine atom and fills up
the shell. Hence, both the sodium and the chlorine ions have filled shells,
and consequently, solid NaCl is non-magnetic. A similar phenomenon oc-
curs in the covalent bond, where electrons of opposite spin strike up a durable
companionship, and as a result, the magnetic moments cancel again.
How then can solids have magnetic properties at all? Well, there is first the
metallic bond, which does not destroy the magnetic properties of its constitu-
ents. It is true that the immobile lattice ions have closed shells and hence no
magnetic properties, but the pool of electrons does contribute to magnetism,
owing to their spin. Some spins will be ‘up’ (in the direction of the magnetic
field); others will be ‘down’. Since there will be more up than down, the sus-
–5
ceptibility of all metals has a paramagnetic component, of the order of 10 .
This is about the same magnitude as that of the diamagnetic component; hence
some metals are diamagnetic.
Another possibility is offered by salts of which potassium chromium alum
is a typical example. There again, as mentioned above, the atoms responsible
for the magnetic properties, being far away from each other, do not interact.
In these compounds, however, the atoms lose their valence electrons; they
are needed for the chemical bond. Hence, the compound will have magnetic