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6.2 Commensurable periodic structures
In the preceding section, we discussed the spin-wave spectra in helical
or conical systems, which are characterized by the important feature
that the magnitude of the ordered moments, and hence of the exchange
field, are constant. This simplification allowed an analytic derivation
of the spin-wave energies, in weakly anisotropic systems. If B 6 only
leads to a slight distortion of the structure, its effects on the spin waves
may be included as a perturbation. If B 6 is large, however, as it is for
instance in Ho, this procedure may not be suciently accurate. Instead
it is necessary to diagonalize the MF Hamiltonian for the different sites,
determine the corresponding MF susceptibilities, and thereafter solve
the site-dependent RPA equation
χ ( ij, ω )= χ i ( ω ) δ ij +
j
χ i ( ω )
( ij ) χ ( j j, ω ) .
J
(6 . 2 . 1)
In uniform para- or ferromagnetic systems, χ i ( ω ) is independent of the
site considered, and the equation may be diagonalized, with respect to
the site dependence, by a Fourier transformation. In an undistorted helix
or cone, the transformation to the rotating coordinate system eliminates
the variation of χ i ( ω ) with respect to the site index, and (6.2.1) may be
solved as in the uniform case. If B 6 is large, the transformation to a (uni-
formly) rotating coordinate system leaves a residual variation in χ i ( ω ),
and in the direction of the moments relative to the z -axis of the rotating
coordinates. This complex situation can usually only be analysed by
numerical methods. A strong hexagonal anisotropy will normally cause
the magnetic structure to be commensurable with the lattice, as dis-
cussed in Section 2.3. We shall assume this condition, and denote the
number of ferromagnetic hexagonal layers in one commensurable period
by m ,with Q along the c -axis. The spatial Fourier transformation of
(6.2.1) then leaves m coupled equations. In order to write down these
equations explicitly, we define the Fourier transforms
χ o ( n ; ω )= 1
N
i
χ i ( ω ) e −in Q · R i
(6 . 2 . 2 a )
and, corresponding to (6.1.28),
χ ( n ; q )= 1
N
ij
χ ( ij, ω ) e −i q · ( R i R j ) e −in Q · R i ,
(6 . 2 . 2 b )
where n is an integer. Equation (6.2.1) then leads to
m− 1
χ ( n ; q )= χ o ( n ; ω )+
χ o ( n
s ; ω )
J
( q + s Q ) χ ( s ; q ) , (6 . 2 . 3)
s =0
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