Physics of Magnetism

CHAPTER 15. INVAR ALLOYS 167

The most conventional treatment of magnetovolume effects is based on a model in

which the magnetic moments are assumed to be localized. The magnetic contribution to the

volume change can be represented by the two-spin correlation function via

where the summation is taken over all magnetic sites, and where is the compressibility.

The quantity is the magnetovolume coupling constant. It originates from the volume

dependence of the exchange constant responsible for the magnetic coupling between

two magnetic moments i and j (see, for instance, Eq. 4.4.2). This means that is

proportional to

In the case of alloys or intermetallic compounds based on 3d metals, one has to realize

that the 3d electrons occupy a narrow energy band having a width of a few electron volts,

as has been discussed in Section 7.1. In order to describe magnetovolume effects in these

materials, it is therefore necessary to take the band character of these electrons into con­

sideration. The reason for this is that there is an intimate connection between interatomic

distances, bandwidth and magnetic properties, as will be further discussed below.

It was outlined in Section 7.2 that the spin polarization of the 3d band that causes the

formation of magnetic moments is a trade-off between exchange energy (which is gained)

and the kinetic energy (which is lost). However, the increase in kinetic energy required

for the band polarization can be kept low if this band polarization is accompanied by

volume expansion. This may be seen as follows. If volume expansion occurs, one expects

a concomitant decrease of the bandwidth W on the basis of Eq. (7.1.1). It can be easily

verified by means of Fig. 7.1.1 that the expenditure in kinetic energy required to realize

a given 3d-band polarization (i.e., to realize a given amount of electron transfer from the

minority band to the minority band) will be lower, the smaller the bandwidth (i.e., the higher

the density of states).

To a first approximation, the increase in kinetic energy is proportional to the square of

the magnetization. The volume change due to band polarization can therefore be written as

where represents the magnetovolume coupling constant associated with the band
character of the 3d electrons.
At low temperatures the spin-correlation function in Eq. (15.6) may
be approximated by so that the total volume magnetostriction can now be written as

Well-known materials with Invar properties are alloys of iron and nickel in a concen­

tration range close to the composition It is interesting and instructive to compare

the Invar properties of these alloys with results of calculations of their electronic band

structure. The volume dependence of the total energies of non-magnetic and ferromag­

netic states derived from these calculations (Williams et al., 1983) is shown in Fig. 15.2.

In fcc FeNi (top part), the ferromagnetic state is the ground state, having an energy lower

than the paramagnetic state. The situation for fcc Fe is shown in the bottom part of the