Physics of Magnetism

15

Invar Alloys

The origin of thermal expansion is the presence of anharmonic terms in the potential energy

expression describing the mutual separation of a pair of atoms at a temperature T. If x

represents the displacement of the atoms from their equilibrium position, the potential

energy may be written as

The term in is a measure of the asymmetry of the mutual repulsion of the atoms, and

the term in can be regarded as describing the general softening of the vibrations at large

amplitudes.

In order to calculate the average displacement, we will follow Kittel (1953) and use

the Boltzmann distribution function (analogous to Eqs. 3.1.3 and 3.1.4), which weights the

possible values of x with a factor representing their thermodynamic probability.

For small displacements, the anharmonic contribution to the potential energy is relatively
small. In this case, the integrands may be expanded as

and

so that

This result shows that the temperature coefficient of the thermal expansion is a constant.

In classical mechanics, the mean value of the energy E of an oscillator in the harmonic

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