Environmental Engineering Reference
In-Depth Information
A helix is formed at the Neel temperature in Tb, Dy, and Ho, while
the longitudinal-wave structure is preferred in Er and Tm. If the
Q
-
vectors are zero, a
ferromagnetic structure
results, with the ordered mo-
ment along some specified direction. In Tb and Dy at low temperatures,
the easy direction of magnetization lies in the plane, while in Gd, which
has a very small magnetic anisotropy, it is along the
c
-axis just below
the Curie temperature, but is tilted about 30
◦
towards the
b
-axis at low
temperatures. If a ferromagnetic component in the
c
-direction is added
to the helix, the moments rotate on the surface of a cone with its axis
in the
c
-direction. This
conical structure
is stable in both Ho and Er
at the lowest temperatures, but the cone angle between the
c
-axis and
the moments at 4 K is large (about 80
◦
) in the former, and small (about
30
◦
) in the latter. If the plane of the moments in the helix is rotated
about an axis in the hexagonal plane, so that its normal makes a non-
zero angle with
Q
, the structure becomes the
tilted helix
,whichmaybe
regarded as a combination of a helix and a longitudinal wave, with the
same
Q
-vectors. This structure has not been definitively identified in
the elements in zero field. The moments in the hexagonal plane of Er
do order below 52 K, with the same period as the
c
-axis modulation, but
they are most probably confined to the
a
-
c
plane, in an elliptically polar-
ized
cycloidal structure
(Miwa and Yosida 1961; Nagamiya 1967) in the
whole temperature interval between 52 K and the transition to the cone
(Jensen 1976b). As the temperature is reduced, in the modulated
c
-axis
phases, the moments on the individual sites approach their saturation
values, resulting in a squaring of the longitudinal wave which manifests
itself in higher odd harmonics. This phenomenon is observed in both Er
and Tm and, in the latter, results in a low-temperature ferrimagnetic
square-wave structure
in which alternately four layers of moments point
up and three layers point down.
The hexagonal anisotropy
B
6
tends to distort the helical structure,
by deflecting the moments towards the nearest easy axis. In a helix
which is
incommensurable
with the lattice periodicity, this effect may be
treated by perturbation theory, which predicts a change of the energy
in second order. However, in Ho at low temperatures,
B
6
is so large
that the magnetic structure is forced to be
commensurable
with the lat-
tice, so that
Q
has the magnitude
π/
3
c
, and the turn angle between the
moments in successive planes averages 30
◦
. It was verified experimen-
tally by Koehler
et al.
(1966) that, under these circumstances, the large
hexagonal anisotropy causes the helix to distort so that the moments in
the plane bunch about the
b
-directions, as illustrated in Fig. 1.20. This
bunched helix
is described by
J
iξ
J
⊥
(
u
sin
Q
·
R
i
−
v
sin 5
Q
·
R
i
)
=
(1
.
5
.
3
a
)
J
iη
=
J
⊥
(
u
cos
Q
·
R
i
+
v
cos 5
Q
·
R
i
)
,
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