SECTION 14.3. THE RANDOM-ANISOTROPY MODEL 157
As the local magnetocrystalline anisotropies are averaged out this way, the scale on which
the exchange interaction dominates expands at the same time. Thus, the exchange length,
has to be renormalized by substituting for in Eq. (14.2.1), that is, is
self-consistently related to the averaged anisotropy by
After combining Eqs. (14.2.1) and (14.3.2), one finds for grain sizes smaller than the
exchange length that the averaged anisotropy is given by
It should be borne in mind that this result is essentially based on statistical and scaling
arguments. This implies that it is not limited to uniaxial anisotropies, but also applies to
cubic or other symmetries.
The most prominent feature of the random-anisotropy model is that it predicts a strong
dependence of on the grain size. Because it varies with the sixth power of the grain size,
one finds for (grain sizes in the order of 10–15 nm) that the magnetocrystalline
anisotropy is reduced by three orders of magnitude (toward a few It is this very
property, that is, the small grain size and the concomitant strongly lowered anisotropy that
gives the nanocrystalline alloys their superior soft-magnetic behavior. Correspondingly,
the renormalized exchange length as given by Eq. (14.3.2) reaches values that fall into
the This is almost two orders of magnitude larger than the natural exchange
length as given by Eq. (14.2.1). This has as a further consequence that the domain-wall
width, discussed in Section 12.3, can become fairly large in these nanocrystallrne materials.
It has already been mentioned briefly in Section 13.2 that magnetic domains of different