Environmental Engineering Reference
In-Depth Information
There is no change in the axial susceptibility in the axial ferromag-
net, for which
θ
= 0, but the higher derivatives are affected by the
modifications
κ
2
(0) =
κ
2
(1
4
6
35
bκ
2
. The correction
to the Callen-Callen theory is proportional to
b
,whichisoftheorder
1
/J
times the ratio between the anisotropy and the exchange energy
(
7
b
)and
κ
4
(0) =
−
B
2
J
(2)
/J
2
(
0
)), and hence becomes smaller for larger values of
J
.
This calculation may be extended to higher order and to non-zero tem-
peratures, but the complications are much reduced by the application of
the
Holstein-Primakoff transformation
which utilizes directly the factor
1
/J
in the expansion, as we shall see in the discussion of the spin-wave
theory in Chapter 5.
In the ferromagnetic phase, the ordered moments may distort the
lattice, due to the magnetoelastic couplings, and this gives rise to addi-
tional contributions to
F
(
θ, φ
). We shall first consider the effects of the
γ
-strains by including the magnetoelastic Hamiltonian, incorporating
(1.4.8) and (1.4.11),
H
γ
=
i
∝
J
2
B
γ
2
Q
2
(
J
i
)
γ
1
+
Q
−
2
(
J
i
)
γ
2
c
γ
(
γ
1
+
γ
2
)
−
2
(2
.
2
.
23)
B
γ
4
Q
4
(
J
i
)
γ
1
−
(
J
i
)
γ
2
,
Q
−
4
4
−
retaining only the lowest-rank couplings (
l
= 2 and 4 of respectively the
γ
2and
γ
4 terms). The equilibrium condition
∂F/∂
γ
1
=
∂
H
γ
/∂
γ
1
=0
,
(2
.
2
.
24)
and similarly for
γ
2
, leads to the equilibrium strains
γ
1
=
B
γ
2
Q
4
/c
γ
Q
2
+
B
γ
4
(2
.
2
.
25)
γ
2
=
B
γ
2
/c
γ
.
Q
−
2
2
Q
−
4
4
−
B
γ
4
The conventional magnetostriction parameters
C
and
A
are introduced
via the equations
−
2
γ
1
=
C
sin
2
θ
cos 2
φ
A
sin
4
θ
cos 4
φ
(2
.
2
.
26
a
)
γ
2
=
C
sin
2
θ
sin 2
φ
+
2
A
sin
4
θ
sin 4
φ
(Mason 1954). Expressing
Q
l
in terms of
O
l
, and retaining only the
terms with
m
= 0, we may derive these parameters from (2.2.25), ob-
taining
1
c
γ
B
γ
2
J
(2)
I
5
/
2
[
σ
]
C
=
(2
.
2
.
26
b
)
2
c
γ
B
γ
4
J
(4)
I
9
/
2
[
σ
]
.
A
=
−
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