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contribution from the crystal-field anisotropy, just balances the differ-
ence in exchange energy between the helical and ferromagnetic phases:
−
1
2
NJ
2
σ
2
∆
U
ff
=
{J
h
(
Q
)
−J
f
(
0
)
}
.
(1
.
5
.
35)
There has been some discussion about the relative importance of the
two terms in stabilizing the ferromagnetic phase. From an analysis of
the field required to induce the transition above
T
C
, Cooper (1968a)
concluded that the magnetoelastic energy plays the dominant role. This
conclusion was, however, based on the implicit assumption that the ex-
change energy changes little between the phases, and later measurements
of the spin waves by Nicklow
et al.
(1971b) demonstrated that this is
not the case. The energy difference
1
2
J
2
σ
2
is about
2 K/ion in the helical phase, but the corresponding quantity is substan-
tially smaller in the ferromagnetic phase. Del Moral and Lee (1975)
reanalysed the data and concluded that the change (1.5.35) in the ex-
change energy makes the major contribution to driving the transition.
Any statement about what drives a
first-order
,asdistinctfroma
second-
order
transition must necessarily be imprecise, since all contributions to
the energy change discontinuously at the transition. Immediately below
T
N
, the exchange dominates and the anisotropy forces are small. As
the temperature is lowered, the peak in
−
{J
h
(
Q
)
−J
h
(
0
)
}
(
Q
) decreases and moves, as
was shown explicitly for the analogous case of Tb by the spin-wave mea-
surements of Bjerrum Møller
et al.
(1967), illustrated in Fig. 6.1. The
magnetoelastic forces therefore increase in relative importance, until a
balance is reached and the transition to the ferromagnetic phase takes
place. At the transition, a large change occurs in the exchange. With-
out the magnetoelastic term,
T
C
would be determined by the hexagonal
crystal-field anisotropy, and would therefore be much lower. In this
sense, the cylindrically-symmetric magnetoelastic forces drive the tran-
sition.
J
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