Environmental Engineering Reference
In-Depth Information
knowledge of the equilibrium strains may also be used for a reasonable
estimate of the magnetoelastic modifications of the second derivatives,
provided that the additional assumption is made that the couplings of
lowest rank are dominant. For example, the higher-rank
γ
-strains in the
basal-plane magnet make contributions to the axial anisotropy which
cannotbewrittenintermsof
C
and
A
in eqn (2.2.27). A more direct
estimate of the contributions to the second derivatives requires an exper-
imental determination of how the strains behave when the direction of
the magnetization is changed. In basal-plane ferromagnets, such as Tb
and Dy, it may be possible to observe the
φ
-dependence of the strains
(Rhyne and Legvold 1965a), whereas if the axial anisotropy is large, it
may be very dicult to determine the variation of the strains with
θ
.
Inthecaseofthe
α
-strains, the argument that the (
l
= 2) couplings are
dominant is not sucient for a determination of their effect on the axial
anisotropy. The reason is that the
two-ion magnetoelastic couplings
of
lowest rank, i.e. the dipolar interactions
D
10
(
ij
)
α
1
+
D
20
(
ij
)
α
2
J
i
·
J
j
+
D
13
(
ij
)
α
1
+
D
23
(
ij
)
α
2
J
iζ
J
jζ
,
α
me
=
∆
H
−
ij
(2
.
2
.
32)
may be important. This is the case in Tb and Dy, as shown by the
analysis of the stress-dependence of the Neel temperatures (Bartholin
et
al.
1971). These interactions affect the
α
-strains, but they contribute
differently to the axial anisotropy from the (
l
= 2)-terms in the single-ion
magnetoelastic Hamiltonian (1.4.10).
The simplifications introduced in the above discussion of the ferro-
magnet may also be utilized in non-uniform systems, because the MF
approximation allows the individual ions to be treated separately. How-
ever, the isotropic two-ion contributions no longer cancel in
δF
(
θ, φ
)in
(2.2.13), since the direction of the exchange field depends on the site
considered. We consider as an example the helically ordered phase. If
we neglect the bunching effect due to the hexagonal anisotropy, the axial
anisotropy is independent of the site considered. Treating the ions as
isolated, but subject to a constant exchange-field, we may calculate
F
θθ
,
corresponding to 1
/χ
xx
, and then use (2.1.19) to account for the induced
exchange-field due to an applied field in the
x
-or
c
-direction, modulated
with a wave-vector
q
along the
c
-axis. If the two-ion coupling between
the moments is allowed to be anisotropic, the leading order result is
−J
(
q
)+
3
κ
2
−
1
2
κ
6
/
(
σJ
)
2
.
κ
4
+
10
8
1
/χ
xx
(
q
)=
J
⊥
(
Q
)
(2
.
2
.
33)
This is the anisotropy parameter which determines the plane in which
the moments spiral, and it vanishes at the temperature
T
N
at which
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