166 CHAPTER 15. INVAR ALLOYS
approximation is equal to For this reason, one may also write Eq. (15.3) in the form
which suggests that the approximate quantum-mechanical result would be obtained by
substituting for the corresponding quantum-mechanical expression for the energy of
a harmonic oscillator with frequency This then leads to
It is useful to bear in mind that the specific heat as well as the thermal-expansion coefficient
are temperature derivatives of E, which means that the thermal-expansion coefficient is
proportional to the specific heat.
On the basis of Eq. (15.5), one would furthermore expect the thermal-expansion
coefficient to decrease rather abruptly when the temperature falls below the characteris
tic temperature of the oscillator and to go to zero if the temperature goes to zero kelvin.
This is what is commonly observed. The third law of thermodynamics requires that the
thermal-expansion coefficient vanishes if the temperature goes to zero.
A schematic representation of the thermal-expansion behavior expected, if only the
lattice anharmonicity contributes, is shown in Fig. 15.1 (dashed-dotted curve). In many
magnetic materials, the thermal expansion takes quite a different form, as shown for instance
by the full curve in the same figure. The total thermal expansion can be subdivided into a
lattice contribution and a contribution due to magnetic effects. The latter contribution
is called the spontaneous volume magnetostriction and is indicated by the broken curve
in Fig. 15.1.