Environmental Engineering Reference
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where
φ
i
=
Q
·
R
i
+
ϕ
. Carrying out this transformation, we find that
J
i
·
J
j
becomes
(
J
iz
J
jz
+
J
iy
J
jy
)cos(
φ
i
−
φ
j
)+(
J
iy
J
jz
−
J
iz
J
jy
)sin(
φ
i
−
φ
j
)+
J
ix
J
jx
,
so that the Hamiltonian (3.5.1) may be written, in the (
xyz
)-coordinate
system,
H
=
2
i
=
j
1
i
H
J
(
J
ix
)
−
J
iα
J
αβ
(
ij
)
J
jβ
,
(6
.
1
.
4)
αβ
where
α
and
β
signify the Cartesian coordinates
x
,
y
,and
z
.Herewe
have assumed that the dependence of the single-ion anisotropy on
J
iξ
and
J
iη
can be neglected, and that only even powers of
J
iζ
=
J
ix
occur,
since otherwise the helical structure becomes distorted and (6.1.1) is no
longer the equilibrium configuration. The ordering wave-vector
Q
is de-
termined by the minimum-energy condition that
(
q
) has its maximum
value at
q
=
Q
. After this transformation, the MF Hamiltonian is the
same for all sites:
J
)
j
(
J
iz
−
2
H
J
(
J
ix
)
H
MF
(
i
)=
−
J
⊥
J
⊥
J
(
ij
)cos(
φ
i
−
φ
j
)
(
J
iz
−
2
H
J
(
J
ix
)
,
(6
.
1
.
5)
as is the corresponding MF susceptibility
χ
o
(
ω
). The price we have paid
is that the two-ion coupling
=
−
J
⊥
)
J
⊥
J
(
Q
);
J
⊥
=
J
z
J
(
ij
) is now anisotropic, and its non-zero
Fourier components are
J
zz
(
q
)=
1
J
xx
(
q
)=
J
(
q
)
;
J
yy
(
q
)=
2
{J
(
q
+
Q
)+
J
(
q
−
Q
)
}
−J
zy
(
q
)=
i
J
yz
(
q
)=
2
{J
(
q
+
Q
)
−J
(
q
−
Q
)
}
.
(6
.
1
.
6)
However, it is straightforward to take account of this complication in
the RPA, and the result is the same as (3.5.8) or (3.5.21), with
J
(
q
)
replaced by
J
(
q
),
χ
t
(
q
,ω
)=
1
(
q
)
−
1
χ
o
(
ω
)
,
χ
o
(
ω
)
−
J
(6
.
1
.
7)
where the index
t
indicates that this is the (
xyz
)-susceptibility, and not
the (
ξηζ
)-susceptibility
χ
(
q
,ω
) which determines the scattering cross-
section. From the transformation (6.1.3), it is straightforward, but
somewhat cumbersome, to find the relation between the two suscep-
tibility tensors.
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