Page 297 - Electrical Properties of Materials

Microscopic theory (quantum-mechanical) 279

average magnetic moment is analogous to the one we used for electric dipoles
but must now be applied to a discrete distribution.
The possible magnetic moments are
M J gμ mB , where M J = J, J –1, ... ,–J +1, –J.

Hence, their energies are

E mag =–M J gμ mB μ 0 H, (11.41)
and the average magnetic moment may be obtained in the form,

J M J gμ mB exp(M J gμ mB μ 0 H/k B T)
μ m = –J . (11.42)
J
–J exp(M J gμ mB μ 0 H/k B T)
The macroscopic magnetic moment may now be calculated by multiplying
μ m by the number of atoms per unit volume.
Equation (11.42) turns out to be a very accurate formula for describing ∗ We need not be too impressed by these
∗
the average magnetic moment as shown in Fig. 11.18, where it is compared close agreements between theory and ex-
periments. The theoretical curve was not
with the experimental results of Henry on potassium chromium alum. The ver-
calculated from first principles, in the
tical scale is in Bohr magnetons per ion. Note that experimental results for sense that the value of J was arrived
paramagnetic properties are often given for ions embedded in some salt. The at by semi-empirical considerations. The
reason being that in these compounds the ions responsible for magnetism (Cr 3+ problem is far too difficult to solve ex-
actly. The usual approach is to set up
in the case of potassium chromium alum) are sufficiently far from each other
a simple model and modify it (e.g. by
for their interaction to be disregarded. taking account of the effect of neigh-
If the exponent is small enough, the exponential function may be expanded bouring atoms) until theory and experi-
to give ment agree. It is advisable to stop rather
abruptly at that point because further re-
J finement of the model might increase the
–J M J (1 – M J gμ mB μ 0 H/k B T) discrepancy.
μ m =–gμ mB
J
–J (1 – M J gμ mB μ 0 H/k B T)
J
2 2
g μ mB 0 H 2
μ
= M , (11.43)
J
(2J +1)k B T
–J
because
J

M J = 0. (11.44)
–J
3
Bohr magnetons/ion 2

1
Fig. 11.18
The magnetic moment as a function
of H/T for potassium chromium
0
0 1 2 3 4 H/T alum (after Henry).