MAGNETIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 116

the form of (2.1.1), augmented by the magnetic dipole-dipole interaction
(1.4.26) which, as we shall see, is of crucial importance. The crystal-field
parameters B l were determined primarily from a fit to the magnetic
structures and magnetization curves at low temperatures, shown in Fig.
1.20, and the temperature dependence of these parameters was assumed
to be negligible. The initial values for the isotropic Heisenberg exchange
were taken from an analysis of the spin waves in Ho (Jensen 1988a),
and depend explicitly on the temperature, as shown in Fig. 2.4. They
were adjusted slightly (Mackintosh and Jensen 1990) to reproduce cor-
rectly the transition fields from the helical phase, but remain consistent
with the spin-wave data, within the experimental error. The magnetic
properties are calculated by means of the method described in Section
2.1.2, assuming an initial distribution

J i

of the moments at a given
temperature. The structure is taken to be commensurable, with a re-
peat distance, deduced from experimental data, which may be as high as
50-100 atomic layers for the more complex configurations. The assumed
values of

are inserted into the Hamiltonian and a new set of moments
calculated, using the mean-field method to reduce the two-ion term to
the single-ion form. This procedure is repeated until self-consistency is
attained. The free energy and the moments on the different sites can
then readily be calculated for the self-consistent structure.
The results of such self-consistent calculations for different temper-
atures and commensurable periodicities are shown in Fig. 2.5. The data
indicate that B 4 is zero, to within the experimental error, whereas B 6
has the opposite sign to B 2 . As the temperature is reduced in the helical
phase and B 6

J i

O 6 increases, this term tends to pull the moments out

of the plane. If the only two-ion coupling were the isotropic exchange,
this would give rise to a continuous transition to a tilted helix, which re-
duces the exchange energy more effectively than the cone (Elliott 1971,
Sherrington 1972). However, the dipolar interaction strongly favours
a ferromagnetic orientation of the c -axis moments, because the dipolar
energy associated with a longitudinal wave is very high, as we discuss in
detail in Section 5.5.1. Consequently, the dipolar contribution shifts the
position of the maximum in ( q )from q = Q to zero wave-vector, as

illustrated in Fig. 5.7, and the vanishing of the axial anisotropy (2.2.33)
at q = 0 leads to a second-order transition at T N to the cone phase. In
this special case, we can therefore conclude that it is the temperature
dependence of B 6

J

O 6

which drives the helix into instability, and that
the dipolar interaction chooses the cone, rather than the tilted helix, as
the stable low-temperature phase.
At 4 K, in the cone phase, the large hexagonal anisotropy causes the
helical component of the moments to bunch around the easy directions
of magnetization, in the twelve-layer structure described by eqn (1.5.3),