SPIN WAVES IN PERIODIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 306

where χ o ( n + m ; ω )= χ o ( n ; ω ). The m matrix equations may be solved
by replacing ω by ω + i . Instead of taking the limit 0 + ,asrequired

by the definition of the response function, is considered as non-zero
but small, corresponding to a Lorentzian broadening of the excitations.
Equation (6.2.3) may then be solved by a simple iterative procedure,
after the diagonal term χ ( n ; q ,ω ) has been isolated on the left-hand
side of the equation. If m is not too large, and if is not chosen to
be too small, this procedure is found to converge rapidly, requiring only
10-20 iterations at each ( q ,ω ). The energies of the magnetic excitations
at the wave-vector q are then derived from the position of the peaks, of

→

width 2 h , in the calculated response function Im χ (0; q ,ω ) .
The use of numerical methods, which is unavoidable in systems with
complex moment-configurations, leads to less transparent results than
those obtained analytically. However, compared with the linear spin-
wave theory, they have the advantage that anisotropy effects may be
included, even when they are large, without diculty. The introduc-
tion of a non-zero value for means that the response function is only
determined with a finite resolution, but this is not a serious drawback.
The experimental results are themselves subject to a finite resolution,
because of instrumental effects. Moreover, intrinsic linewidth phenom-
ena, neglected within the RPA, provide a justification for adopting a
non-zero .
The numerical method summarized above has been used for calcu-
lating the spin-wave energies in the various structures of Ho discussed
in Section 2.3. In Fig. 5.9, we presented the dispersion relations in the
c -direction of Ho containing 10% of Tb, in its ferromagnetic and helical
phases (Larsen et al. 1987). The Tb content has the desirable effects
of confining the moments to the basal plane, and inducing the simple
bunched helix or zero-spin-slip structure (1.5.3) in the range 20-30 K,
and ferromagnetism below 20 K. The commensurability of the 12-layer
structure implies that the energy of the helix is no longer invariant un-
der a uniform rotation, and an energy gap appears at long wavelengths,
reflecting the force necessary to change the angle φ which the bunched
moments make with the nearest easy axis. The excitations in this rel-
atively straightforward structure can be treated by spin-wave theory,
and the energies in the c -direction may be written in the form of eqn
(6 . 1 . 10 b ):

E q = A q − B q 1 / 2 ,

where now

A q + B q = A + B + J J ( 0 ) −J ( q ) , (6 . 2 . 4 a )