The version derived from the partition function is ``correct''
in that it holds for all temperature including
0 where
M
Nm.
However, the exact formula does reduce to the Curie law in the appropriate
high-temperature limit: as mB/
0,
exp(
mB/
)
1
mB/
and
tanh(mB/
)
mB/
.
Thus M
Nm2B/
[Curie law].
Magnetization as a function of temperature: dashed line represents the Curie law
(valid in the high temperature limit). At low
the spin polarization levels off at 100%.
This approaches unity as
,
indicating that new ``degrees of freedom'' (higher energy levels)
are thermally activated at a constant rate as the temperature increases.
This is known as the Law of Dulong & Petit
and describes the heat capacity of many solids quite well -
namely, those crystals which are built up from a basis
of dissimilar atoms which vibrate against each other at a well-defined
frequency . (Such vibrations are known as
optical phonons because their frequencies are high, comparable
with infrared photons.) In simpler crystals
the lowest energy vibrational modes involve many atoms moving together,
the so-called ``acoustic phonon modes;''
we shall treat that case in some detail later on.