ELEMENTS OF RARE EARTH MAGNETISM - Rare Earth Magnetism: Structures and Excitations - page 55

anisotropy and magnetoelastic forces, which are averaged out or ineffec-
tive in the helical structure, are strong enough to overcome the relatively
weak tendency to periodic ordering. In Tm, on the other hand, a com-
promise obtains, by which the moments take their maximum value along
the c -axis, but alternate in direction so as to take advantage of the large
peak in ( q ). In Ho, the balance is so delicate that the weak classical

dipolar interaction plays a crucial role, as we shall discuss in Section 2.3.
In order to explain the temperature dependence of the structures,
it is necessary to determine the configuration of the moments which
minimizes the free energy , taking into account the influence of increasing
temperature and magnetic disorder on the interactions. Provided that
the magnitude

J

of the ordered moment is the same on all sites, the
entropy term is independent of the details of the ordering (Elliott 1961),
so the stable structure has the minimum energy . In exchange-dominated
systems, like the heavy rare earths, the ordered moment approaches its
saturation value at low temperature. As the temperature is increased,
the structure which has the lowest energy may change as the effective
interactions renormalize . This may occur either through a second-order
transition, in which some order-parameter goes continuously to zero or,
more commonly, discontinuously through a first-order transition. At
elevated temperatures, the entropy may favour a structure, such as the
longitudinal wave, in which the degree of order varies from site to site.
A conceptually simple but powerful means of calculating magnetic
properties, and their dependence on the temperature, is provided by the
molecular-field approximation or mean-field theory . We shall describe
this method in some detail in the next chapter, but it is convenient to
introduce it here in order to establish a few elementary results. The
essential feature of the theory is the approximation of the two-ion in-
teractions by effective single-ion terms, by replacing the instantaneous
values of the J operators on the surroundings of any particular ion by
their thermal averages. The effect of the exchange interaction (1.4.21)
with the surrounding ions on the moment at R i may then be written

| J i |

( J i − 2 J i H ff ( i ) − ) · J ( ij ) J j , (1 . 5 . 6)

j

which in turn may be written in terms of an effective magnetic field

H eff ( i )=( gµ B ) − 1
j

J ( ij ) J j , (1 . 5 . 7)

plus a constant contribution to the energy. If the sum of the applied and
effective fields is small, which will generally be true in the paramagnetic