SPIN WAVES IN PERIODIC STRUCTURES - Rare Earth Magnetism: Structures and Excitations - page 285

6
SPIN WAVES IN
PERIODIC STRUCTURES
Because of the modification of the translational symmetry, the spin
waves in modulated magnetic structures display certain interesting as-
pects which are not shared by the simple ferromagnetic structure. How-
ever, this same feature makes their experimental study considerably
more dicult, and the results which have been obtained on such sys-
tems are still relatively sparse. This chapter is correspondingly short,
andinitstwosectionswedistinguish between structures incommensu-
rable with the lattice periodicity, when the translational symmetry in
the direction of the wave-vector is, in principle, destroyed, and com-
mensurable structures, in which this symmetry is only modified, though
possibly quite drastically.
The stringent mathematical definition of an incommensurable struc-
ture is straightforward, but it presupposes that the coherence lengths of
the lattice and of the magnetic system are both infinite. In this idealized
case, an irrational ratio between the periodicities of the two subsystems
breaks the translational invariance, the wave-vector q is consequently no
longer a good quantum number, and neutron-diffraction peaks acquire
a non-zero width. Furthermore, the energy eigenvalues which deter-
mine the excitation spectrum also have a certain width when projected
on to q -space, which is the appropriate representation for interpreting
constant- q neutron-scattering experiments. The alternative method of
distinguishing experimentally between the two types of structure is to
follow the ratio between the two periodicities as a function of temper-
ature or external field. If this ratio changes discontinuously between
constant steps, the structure is commensurable. On the other hand,
if the variation is observed to be continuous , the structure is usually
classified as being incommensurable. As examples of transverse incom-
mensurable structures, we shall accordingly take the helix in Tb, which
exists only over a small range of temperature below T N ,andthelow-
temperature cone in Er. It is generally questionable whether any partic-
ular structure can strictly be classified as incommensurable but, as we
shall see, the distinction in these cases is unimportant. The magnetic
high-temperature phase of Er is treated as an example of an incom-
mensurable longitudinal-wave structure , although it may only be truly
incommensurable at the highest temperatures in the ordered phase. In