MAGNETIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 94

exponents , but it is nevertheless decisive for which universality classes
the phase transitions belong to. The transitions which are predicted to
be continuous by the MF theory, i.e. all those considered above which
are not accompanied by a change of Q to a commensurable value, may
be driven into (weak) first-order behaviour by the fluctuations. An im-
portant parameter for determining the nature of the phase transition is
the product ( n ) of the number of components of the order parameter,
and of the star of the wave-vector (Mukamel and Krinsky 1976; Bak
and Mukamel 1976), the latter being two, corresponding to

± Q ,forthe

periodically-ordered heavy rare earths. If n 3, the transition is ex-

pected to remain continuous, which is in accord with the observation by
Habenschuss et al. (1974) of a second-order transition in Er, since n =2
for the transition between the paramagnetic and the longitudinally or-
dered phase. The transition from the paramagnet to the helix is less
clear-cut, since it belongs to the class n = 4, and a theoretical analysis
by Barak and Walker (1982) suggested that it might be discontinuous.
The bulk of the experimental evidence points towards a continuous tran-
sition (Brits and du Plessis 1988) but some measurements, especially by
Zochowski et al. (1986) on pure Dy, indicate a very weak discontinuity.
In the case of the multiple- Q structures, the fluctuations may drive the
transition to the single- Q structure to be discontinuous, whereas that to
the triple- Q structure, if it is stable, should stay continuous (Bak and
Lebech 1978). In Nd, for example, a single- Q state is formed at T N
and the transition is found to be weakly discontinuous (Zochowski and
McEwen 1986). In accordance with the MF analysis above, a first-order
transition leads to a double- Q structure less than a degree below T N
(McEwen et al. 1985).

≤

2.2 The magnetic anisotropy
In this section, we shall discuss the thermal expectation-values of the
Stevens operators of the single ions when their moments are non-zero,
so that = σJ . We shall then consider the contribution which

the single-ion terms in the Hamiltonian make to the free energy, and
thereby derive the relationship between the microscopic parameters and
the macroscopic magnetic-anisotropy and magnetoelastic coecients.

| J i |

2.2.1 Temperature dependence of the Stevens operators
In a ferromagnet, the Zener power-law (1.5.15) for the expectation values
of the Stevens operators is valid only at the lowest temperatures. Callen
and Callen (1960, 1965) have derived O l in exchange-dominated sys-

tems and obtained results which are useful over a much wider temper-
ature range than the Zener expression. They begin with the density
matrix for a single site in the MF approximation, including only the