SPIN WAVES IN PERIODIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 310

is close to its maximum value, this mixing is unimportant, but it
has significant effects on the excitations at higher temperatures. In the
RPA, the pure longitudinal response contains an elastic component, and
the (mixed) excitation spectrum in the long-wavelength limit therefore
comprises an elastic and an inelastic branch. The inelastic mode is cal-
culated to lie around 1 meV in the temperature interval 50-80 K. In the
RPA, this feature is independent of whether the magnetic periodicity
is commensurable with the lattice. In the incommensurable structure,
the free energy is invariant to a rotation of the helix around the c -axis,
implying that χ t ( q ,ω ) diverges in the limit ( q ,ω )

J z

( 0 , 0). However,
the corresponding generator of rotations no longer commutes with the
Hamiltonian, as in the regular helix, because B 6 is now non-zero. The
divergence of χ t ( q ,ω ) is therefore not reflected in a conventional Gold-
stone mode, but is rather manifested in the elastic, zero-energy pha-
son mode, which coexists with the inelastic mode. Beyond the RPA,
the elastic response is smeared out into a diffusive mode of non-zero
width. This broadening may essentially eliminate the inelastic phason
mode, leaving only a diffusive peak centred at zero energy in the long-
wavelength limit. The intensity of this peak diverges, and its nominal
width goes to zero, when the magnetic Bragg reflection is approached.
However, a diffusive-like inelastic response is still present at q = 0 ,and
a true inelastic mode only appears some distance away. In the calcula-
tions, the elastic single-site response was assumed to be broadened by
about 6 meV, corresponding to the spin-wave bandwidth. This assump-
tion gives a reasonable account of the excitations in the long-wavelength
limit, suggesting that they become overdamped if the wave-vector is
less than about 0.1 times 2 π/c . Although the inelastic phason mode is
largely eliminated, the calculations suggest that a residue may be ob-
servable. The most favourable conditions for detecting it would occur
in a neutron-scattering scan with a large component of the scattering
vector in the basal plane at about 50 K.
Another example to which the above theory has been applied is
Tm (McEwen et al. 1991), where the c -axis moments order below 57.5
K in a longitudinally polarized structure, which becomes commensu-
rable around 32 K. Below this temperature, as described in Section 2.3.1,
the structure is ferrimagnetic, comprising four layers with the moments
parallel to the c -axis, followed by three layers with the moments in
the opposite direction. Although Tm belongs to the heavy end of the
rare earth series, the scaling factor for the RKKY-exchange interaction,
( g

→

1) 2 =1 / 36, is small, and the Neel temperature is low compared to
the crystal-field energy-splittings. Thecrystal-fieldeffects are therefore
more important in this element than in the other heavy rare earths. The
energy difference between the MF ground state and the dipolar excited

−