SPIN WAVES IN PERIODIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 286

this case, the incommensurability has significant qualitative effects on
the excitation spectrum.
In a commensurable structure, where m = mQ c is p times the

length 4 π/c of the (effective) reciprocal lattice vector, the number m
of layers between magnetically-identical lattice planes will have an im-
portant influence on the character of the excitations. If m is small,
they will be well-defined, long-lived at low temperatures, and relatively
easy to study experimentally. As examples of such ideal commensurable
structures, we shall consider the low-temperature bunched helix in Ho,
and the longitudinal ferrimagnetic structure in Tm, where the crystal
fields make a decisive contribution. These relatively simple structures
are stable at low temperatures but, in both cases, the configuration of
the moments becomes more complicated as the temperature is increased.
In Ho, for example, spin-slip structures of reduced symmetry and gener-
ally increasing m evolve, and the excitations, of which only preliminary
studies have so far been made, become correspondingly complex. Al-
though these excitations may be just as well-defined as when m is small,
the extension of the magnetic Brillouin zone, in the c -direction, is re-
duced by the factor 1 /m , while the number of branches in the dispersion
relation is multiplied by m . The different branches are separated from
each other by energy gaps at the boundaries of the magnetic Brillouin
zone, and the corresponding excitations scatter neutrons with different
weight, depending on the scattering vector and described in terms of the
dynamical structure factor .If m is large, however, it may be extremely
dicult to resolve the different branches experimentally. As m increases
towards the values of the order of 50 which characterize some of the
commensurable structures presented in Section 2.3, imperfections in the
lattice, or boundaries between different magnetic domains, become more
important. We should also expect that disordering phase-slips would be-
come relatively more frequent, leading to a less well-defined structure,
and disturbing the very long-range periodicity. It is unlikely that a spe-
cific pattern involving 100 or more layers could be repeated many times
in the crystal, without significant errors, and we might rather anticipate
a somewhat chaotic arrangement, where the phase factor characterizing
the moments changes systematically from layer to layer, but with an
occasional minor phase slip, so that the structure never repeats itself
exactly. This kind of structure may frequently in practice be described
as incommensurable.

| Q |

6.1 Incommensurable periodic structures
In this section, we shall first discuss the spin waves in the regular helix or
cone, including the hexagonal anisotropy only as a minor perturbation.
On account of the infinitely larger number of irrational than rational