Bosonic and magnonic magnon dispersions

The analytical crossover events in the magnon dispersion relations are discussed. As the available experimental data show, only a few types of magnon dispersion relations are observed for all spin- and lattice-structures. When the magnon spectrum exhibits a finite energy gap at q = 0 the dispersion consists of two q-sections with different functions of wave vector. Such analytical changes indicate that two mechanisms dominate the dispersion relation alternately. For magnon energies of larger than the gap, the dispersion exhibits over a finite q-range a single q(x) power function of wave vector. Within the experimental error limits, the exponent x is a rational number and is independent of the spin structure, i.e. universal, but depends on whether the spin quantum number is integer or half-integer. As we now know, this is the typical indication that the q(x) function is determined by the bosons of the continuous magnetic medium. The q(x) function holds up to the crossover to a sinefunction of wave vector for antiferromagnets but to a sine-function squared for ferromagnets. One therefore has to distinguish between the bosonic part of the magnon dispersion relation, at small q-values, and the magnonic part at large q-values. As is well known, sine functions of wave vector are the dispersion of the linear spin chain. Since the linear spin chain is not ordered at any finite temperature it follows that the observed magnons are not indicative of a long-range magnetic order. From the fact that no principal change of the magnon dispersions occurs upon crossing the magnetic ordering temperature it follows that the exchange interactions are not involved directly in the magnetic ordering process. The long-range ordered system is the boson field. M the critical temperature, the boson field orders. Ordering of the boson field is associated with the formation of domains and with the emergence of a magnon gap. In the ordered state, propagation of the bosons is restricted to the few different domain axes. The dimensionality of the ordered boson field can be recognized from the number of the in-equivalent domain orientations. For all lattice structures the boson field within each magnetic domain is perfectly one-dimensional, and aligns all spins parallel. This is the origin of the linear chain dispersion. The mass-less bosons are, however, not visible for neutrons. If the magnon-boson interaction is strong, the q-range of the sine functions is small. It then proves necessary to add a phenomenological phase shift in the argument of the sine function. As a consequence, magnon dispersions cannot be understood considering exchange interactions alone.