Spin-wave dispersion

There has been a considerable effort in the physics and chemistry communities to use INS methods to study magnetic dynamics, which can often be described as spin waves. Measurements of spin wave dispersion curves can provide information about the interactions between atomic magnetic moments, the so-called exchange interactions. There have been comparatively few INS measurements on magnetic minerals. INS methods have been used to produce spin wave dispersion curves for hematite. Crystal field magnetic transitions in cobalt bearing cordierite, and spinel phases have also been studied by INS. ... 

In order to obtain spin-wave dispersions, we use the following approximations ... 

The spin-wave dispersion is calculated using the zero temperature structure factor given by... 

Since crystals of Fe3Al have been widely examined, and magnetization and thermal expansion (Cook and Pavlovic 1984), anisotropy constants (Takahashi 1975), resistivity (Thomas et al. 1973), Hall effect (Puzey et al. 1975) and spin wave dispersion (Frikee 1978) have been reported. 

The magnetic term in the heat capacity is what would be expected from the ferromagnetic spiral (i.e. the cone) structure of erbium (Kaplan 1961) and has been supported by the discovery of a linear spin-wave dispersion law along the c-axis in neutron scattering experiments (Nicklow et al. 1971). 

Fig. 16. Spin wave dispersion along [110] direction for ErBa2Cu307. The solid curve is a guide to the eye (Skanthakumar et al. 1997a).

The anisotropy will also affect the spin waves (Mills 1991). Measurements of spin wave relaxation along the lines of the work by Vaterlaus, Beutler and Meier (1992) are indicated. There is also a need for mapping the spin wave dispersion, as has been undertaken for Tm (McEwan et al. 1995), but for thin films. We would expect that the lanthanide metals will exhibit spin waves at temperatures above the Curie temperature, much in the same way as has already been observed for Ni (Lynn and Mook 1981, Mook et al. 1973, Mook and Paul 1985, 1988, Mook and Lynn 1986, Steinsvoll et al. 1983, Uemura et al. 1983). Plasmon and other collective electron effects, as well as magnon dispersion, also need to be investigated and such studies are currently only at a preliminary stage for surfaces. 

Recent detailed study of the spin-wave dispersion by Bohn et al. (1980) in a single crystal of EuS (enriched with Eu) by inelastic neutron scattering technique (fig. 6a) confirms previous assumptions that the range of the exchange interactions is essentially limited to the second nearest neighbors = 0.220 K and 2/ 6 "... 

Fig. 6. (a) Spin-wave dispersion in EuS, measured by inelastic neutron scattering at 7 =1.3K (T = 16.6 K). The solid lines represent the best fit using up to fifth neighbors exchange interactions. The arrows indicate the boundary of the first Brillouin zone in the various symmetry directions (from Bohn et al. 1980). (b) Dependence of the exchange interactions, /, and /j, on the Eu-Eu distance in the Eu-chal-cogenides. 

The measurements on Euq yoSrp 30S (in powder form) which orders ferromag-netically below = 8.5 K show well-defined spin-wave excitations at 4.5 K (fig. 40a), and a quadratic form of the spin-wave dispersion at low q (eq. 61) similar to pure EuS. The stiffness constant D, related to the exchange constants in the non-diluted ferromagnet by the relation... 

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