MAGNETIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 100

the system as long as σ is greater than about 0.8. The same picture holds
true for other combinations of Stevens operators, but the discrepancies
between the different theories are accentuated as the rank increases.
Figure 2.3 shows the example of Q 6 . The absolute magnitude of this

quantity is reduced by nearly 40% in the zero-temperature limit, as
compared with the Callen-Callen theory, and the slope of the numerical
calculation, in the semi-logarithmic plot, changes with σ , leading to an
effective power-law depending on the interval over which it is measured.
In the zero-temperature limit, Q 6

is proportional to approximately
σ 26 , instead of the Callen-Callen result σ 21 .

Fig. 2.3. The dependence on the relative magnetization of the expec-
tation value of the Stevens operator Q 6 , which determines the hexag-
onal magnetic anisotropy, in Tb. The numerical calculations and the
spin-wave theory both predict a large reduction in this quantity at low
temperatures, compared with the Callen-Callen theory.

The numerical results are expected to be sensitive to the magnitude
of the anisotropy, rather than to the actual parameters which determine
the anisotropy, and the spin-wave theory indicates that this expectation
is fulfilled, at least at low temperatures. However, in order to obtain
the right variation of the anisotropy fields with temperature, i.e. of b
compared to σ , it is necessary to select appropriate linear combinations
of Stevens operators of various ranks for the modelling of the different
anisotropy terms. At high temperatures, for instance, b is determined