SPIN WAVES IN PERIODIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 292

there might be a diffusive mode in the excitation spectrum of the diverg-
ing susceptibility component, with an intensity ( χ ( ω ) /ω )whichgoes

to infinity as the critical q is approached. Outside the critical region, the
inelastic excitation-energies must approach zero, in the absence of a dif-
fusive mode, as a consequence of the Kramers-Kronig relation, but the
excitations may be overdamped, i.e. still become diffusive, suciently
close to the critical q . In this case, the generator 1

∝

iδϕ i J xi of an
infinitesimal rotation δϕ of the helix commutes with the Hamiltonian,
and the Goldstone theorem applies, predicting that the spin waves are
perfectly well-defined excitations in the limit of q → 0 .
If

−

H J can be neglected, the system contains one more Goldstone

mode, since i J iξ or i J iη now also commute with H . The transfor-

iθ i J iξ ) generates a tilting of the plane spanned by the
moments, relative to the ξ - η plane perpendicular to Q , giving rise to the
tilted helix structure. In this configuration, the J iζ = J ix -component is
non-zero and oscillates with the phase Q · R i . The magnitude of the mod-
ulation is determined by the susceptibility component χ xx ( q = Q , 0),
which diverges in the limit where

mation exp( −

H J or 2 A vanishes. The situation is
very similar to the rotation of the helix considered above. The Gold-
stone mode is the spin-wave excitation at q = Q , and the spin-wave
energy vanishes linearly with . The Heisenberg ferromagnet may

be considered to be a helix with Q = 0 , and in this case the two Gold-
stone modes collapse into one at q = 0 , where the spin-wave dispersion
now becomes quadratic in q .
The first study of the spin waves in a periodic magnetic structure
was performed by Bjerrum Møller et al. (1967) on a Tb crystal, to which
10% Ho had been added to stabilize the helix over a wider temperature
range. The results of these measurements are shown in Fig. 6.1. The
hexagonal anisotropy in Tb is small, and

| q − Q |

O 6

has renormalized to neg-
ligible values in the helical phase, so the theory for the incommensurable
structure would be expected to apply. The dispersion relations do in-
deed have the form of eqn (6.1.10), rising linearly from zero at small q ,
and with a non-zero value of E Q , due to the axial anisotropy B 2 .An
analysis of the experimental results in terms of this expression gives the
exchange functions illustrated in Fig. 6.1. The decrease in the size of
the peak in

( q ) with decreasing temperature contributes towards the
destabilization of the helix, as discussed in Section 2.3. The effects of the
change in this function with temperature can be seen fairly directly in
the dispersion relations since, from (6.1.10), the initial slope is propor-
tional to the square root of the curvature

J

J ( Q ), and E Q is proportional

1
2 J

1
2 J

1 / 2 . Similar results have been obtained for
Dy by Nicklow et al. (1971b) and analysed in the same way, even though

to {J ( Q ) − ( 0 ) − (2 Q ) }