Environmental Engineering Reference
In-Depth Information
The aforementioned magnetic structures may readily be understood
as the result of the co-operation and competition between the oscillatory
indirect exchange, which is relatively strong in the heavy rare earths, be-
cause ( g
1) J is generally large, and the crystal-field and magnetoelastic
anisotropy forces. The origin of the periodic structures can be explained
by writing (1.4.21) in the form
2
N
H ff =
q J
( q ) J ( q )
· J (
q ) ,
(1 . 5 . 4)
where the Fourier transform of the magnetic structure is
N
i
J ( q )= 1
J i e −i q · R i .
(1 . 5 . 5)
In order to minimize the energy of the magnetic system, this term will
favour a Q vector which corresponds to the maximum in
( q ). The
maxima shown in Fig. 1.17 thus reflect the observed Q values in the
heavy rare earths through their position, and the relative stability of the
periodic structures through their magnitude. The isotropic exchange
does not in itself specify any orientation of the moments relative to
the crystal axes. The normal to a planar helix can, for example, be
rotated into an arbitrary direction without altering the exchange energy.
This flexibility is realized in Eu, where the crystal-field anisotropy is
very small because, like Gd, it has no ionic orbital moment. Neutron-
diffraction studies of a single crystal by Millhouse and McEwen (1973)
showed a first-order transition to a helical structure, and magnetization
measurements indicate that the plane of the helical structure is always
normal to the direction of a moderate applied field, even though Q
remains along a four-fold axis of the bcc structure.
It is the magnetic anisotropy which fixes the magnetic structure rel-
ative to the crystal axes. As may be seen from eqn (1.4.4), the two-fold
axial anisotropy (proportional to J ζ ) is also proportional to the Stevens
factor α .If A 2 is negative throughout the heavy rare earths, as we shall
see is the case, the values in Table 1.4 immediately explain why Tb and
Dy have easy axes in the hexagonal plane, while the moments in Tm
are strongly bound to the c -axis. In Ho and Er the higher-order axial
anisotropy is important, but the values of α are consistent with the re-
spectively large and small cone angles. Similarly, the alternation in the
sign of γ in the series of the heavy elements is reflected in the easy direc-
tions of magnetization in the hexagonal plane. The competition between
the exchange and the anisotropy is manifested in the low-temperature
magnetic structures.
J
In the ferromagnetic phases of Tb and Dy, the
Search WWH ::




Custom Search