Environmental Engineering Reference
In-Depth Information
The abrupt change in the uniform
α
-strains (Rhyne and Legvold 1965b)
at the transition to the cone phase reduces this energy by 0.19 meV/ion
(Rosen
et al.
1973), corresponding to an increase of
(
0
) by about 0.01
meV. In the cycloidal phase, there is also a longitudinal-strain mode
at wave-vector 2
Q
, which disappears in the cone phase, but the energy
gained by this distortion is estimated to be very small. Since the
c
-axis
moment is substantially squared up in the cycloidal phase just above
the transition, the change of the
α
-strains cannot have its origin in the
single-ion magnetoelastic coupling, which does not distinguish between
positive and negative moments. It must rather be caused by the strain-
dependence of the two-ion interaction
J
I
1
(
ij
)
+
I
3
(
ij
)
33
(
i
)
J
iζ
J
jζ
,
∆
H
me
=
−
{
11
(
i
)+
22
(
i
)
}
(2
.
3
.
3)
ij
which is that part of eqn (2.2.32) which changes at the transition. If
the basal-plane moments and the single-ion magnetoelastic terms are
assumed to be the same immediately above and below
T
C
,∆
H
me
gives
rise to the following changes at the transition:
2
(
c
11
−
c
66
)∆(
11
+
22
)+
c
13
∆
33
=
N
{
I
1
(
0
)
−
I
1
(
Q
)
}|
J
ζ
|
(2
.
3
.
4)
2
,
c
13
∆(
11
+
22
)+
c
33
∆
33
=
N
{
I
3
(
0
)
−
I
3
(
Q
)
}|
J
ζ
|
where the bars denote effective coupling parameters, as in (2.3.1), and
∆
αα
=
αα
(cone)
αα
(cycloid). Since the elastic constants are known,
and the strains are ∆
33
=3
.
1
−
10
−
3
,the
two-ion magnetoelastic-coupling parameters may be determined from
this equation. The nature of this magnetoelastic contribution implies
that it should be possible to suppress the cone phase in Er by apply-
ing hydrostatic pressure. In the zero-temperature limit, the energy dif-
ference between the two phases is estimated to be only about 0.033
meV/ion, so a hydrostatic pressure of about 2.5 kbar, or alternatively a
uniaxial pressure along the
c
-axis of only about half this amount, should
be sucient to quench the cone. The application of this modest pres-
sure should then allow experimental studies of the cycloidal phase in
Er below 18 K, to ascertain, for example, whether the transition to the
phase with an elliptical ordering of the basal-plane moments occurs. We
shall return to this two-ion magnetoelastic interaction when we discuss
Er films and superlattices.
The negative value of
B
2
in Tm is large and
B
6
is also negative, as
may be seen in Table 2.1, so that the moments are firmly anchored to
the
c
-direction, and no ordered basal-plane component appears at any
temperature. A longitudinal-wave structure forms at 56 K, and starts
to square up at about 40 K, as the amplitude approaches the free-ion
10
−
3
and ∆(
11
+
22
)=
·
−
2
.
4
·
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