Environmental Engineering Reference
In-Depth Information
These parameters determine the ( xy )-components of χ o ( ω ) in (6.1.9),
and are valid, at least, to first order in 1 /J . From (6.1.8), we finally
obtain
A q
B q
χ xx ( q )=
J z
A q
B q
(
C q ) 2
A q + B q
χ yy ( q )=
J z
(6 . 1 . 18)
A q
B q
(
C q ) 2
i (
C q )
χ xy ( q )=
J z
C q ) 2 ,
A q
B q
(
where the parameters are now
A q
2 J
2 J
}
A q + B q =( A q − B q )cos 2 θ 0 + L + J z {J ( 0 ) −J ( q ) } sin 2 θ 0
C q = 2
B q =
J z {J
( Q )
( q + Q )
( q Q )
J z {J
( q Q )
−J
( q + Q )
}
cos θ 0 ,
(6 . 1 . 19)
and the axial anisotropy constant is
L =
+ f ( u 0 ) /
J z {J
( Q )
−J
( 0 )
}
J z
,
with
f ( u 0 )=3 κ 2 ( T )+ 15
κ 4 ( T )(7 cos 2 θ 0
1)
(6 . 1 . 20)
2
+ 10 8
κ 6 ( T )(33 cos 4 θ 0
18 cos 2 θ 0 +1) .
This constant, to order 1 /J , is that determined by the c -axis bulk sus-
ceptibility: χ ζζ ( 0 , 0) =
/L . The dispersion relation, derived from
the pole at positive energies, is
E q = C q + A q
J z
B q 1 / 2 , (6 . 1 . 21)
which is no longer even with respect to q , because the parameter C q
changes sign, whereas A q and B q are unaffected, if q is replaced by
q .
The other pole, with a minus before the square root, lies at negative
energies. If the two energies for q were both positive, the two poles at
q would both lie at negative energies, indicating an instability of the
magnetic system. Hence in a stable cone C q <A q
B q (Cooper et al.
1962).
The scattering cross-section of the spin waves is still determined by
(6 . 1 . 12 a ), but (6 . 1 . 12 b ) is replaced by
χ ζζ (
)sin 2 θ 0
κ
)= χ xx (
κ
4 χ xx (
+ Q ) cos 2 θ 0
)= 1
χ ξξ (
κ
)= χ ηη (
κ
κ Q )+ χ xx (
κ
4 χ yy (
+ Q )
+ 1
κ Q )+ χ yy (
κ
2 χ xy (
+ Q ) cos θ 0 .
(6 . 1 . 22)
i
κ Q )
χ xy (
κ
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