Physics of Magnetism

92 CHAPTER 10. CALORIC EFFECTS IN MAGNETIC MATERIALS

In writing down these equations, one has to realize that M depends on temperature and
that varies strongly with temperature. When inspecting Fig. 4.2. 1c, one sees that
M = 0 in a ferromagnet material above whereas M is almost temperature-independent
at temperatures much below However, M varies strongly just below In terms of
Eq. (10.1.4), this means that vanishes at very low temperatures and above

Just below
the specific heat will be large. Infact, shows a discontinuity at
The size of this discontinuity can be calculated as follows. The molecular field constant
can be expressed in terms of by rewriting Eq. (4.2.5) as:

In Section 4.2, it has already been shown that the reduced magnetization M(T)/M( 0 ) if
plotted as a function of the reduced temperature has the same shape for all ferromagnetic
materials characterized by the same quantum number J. By substituting M(T)/M(0) of
Eq. (4.2.11) into Eq. (10.1.5), one can calculate exactly over the whole temperature
range from to by means of a simple computational procedure.
If one is only interested in the magnitude of the specific-heat discontinuity at one
may write down a series expansion for of Eq. (4.2.1) and retain only the first two terms
(Eq. 3.2.1). After some algebra, one finally finds for the magnitude of the discontinuity
at

It is useful to keep in mind that for the simple case the specific heat jump at equals
for a mole of magnetic material. The temperature dependence of
for the case is shown in Fig. 10.1.1.
It is instructive to compare the molecular field results shown in Fig. 10.1.1 with the
experimental results obtained for nickel, shown in Fig. 10.1.2. The upper curve in Fig. 10.1.2
is the total specific heat In order to compare this quantity with the molecular field
prediction, one has to subtract the non-magnetic contributions due to lattice vibrations,
thermal expansion, and the electronic specific heat. These non-magnetic contributions may