Magnon dispersion

Not any feature was observed in the single-magnon dispersion that would correlate with the lower-energy two-magnon peak in the inelastic neutron scattering measurements of the stripe-ordered nickelate Lai.69Sr0.3iNiC>4 also.311... 

Fig. 31. Ratio of magnon dispersion constants D(x)/D(0) as a function of composition t for the system Pt3Mn [Cr1 l . Filled circles, inelastic neutron scattering open circles, bulk magnetization both from Williams et al. (1981) Paul and Stirling (1979).

With a few exceptions practical implementations of Eq. (27) treat the exchange and correlation (XC) field in a collinear approximation with Bxc = Bxc z aligned along the (arbitrary) i-axis. Non-collinear structures play an important role for the groxond state of some magnetic systems as well as for excited state properties like magnon dispersions. A discussion of the technical details of non-collinear schemes and their applications is beyond the scope of this text. Reviews on this subject can be found in Refs. [23] and [24]. 

Fig. 9. Magnon dispersion for Tb in the ferromagnetic phase along the three principal axis directions. Symbols along the top of the diagram label the directions in the conventional notation shown in fig. 4. Dashed curves labelled PH are [longitudinal acoustic (LA), transverse acoustic (TA), and transverse optic (TO)] phonon branches which interact with the magnons. (After Mackintosh and Bjerrum-Moller 1972.)...

Spin wave theory for a simple ferromagnet gives a quadratic magnon dispersion relation at very low temperatures in the long wavelength limit ... 

For an antiferromagnet spin wave theory gives a linear magnon dispersion relation ... 

Magnetic anisotropy arising from the crystalline Held introduces a finite gap in the magnon dispersion curve even in the long wave-length limit. Thus where... 

Finally, it appears that Cm for gadolinium does not follow one simple temperature dependence if we allow Cm T" then theoretically (Sedaghat and Cracknell 1971, Stevens and Krukewich 1973) n varies from 1.5 at low temperatures to 2.3 above 10 K. The above theoretical calculations are based on experimentally determined magnon dispersion relations. Lounasmaa and Sund-strom (1966) tentatively proposed n 2.7 from analysis of heat capacity results. Wells et al. (1974), on the other hand, are able with more confidence to propose n 1.75, and that if magnetic anisotropy were allowed then the introduced energy gap Eg will be very small, in the region of 0.05 K. If there were no anisotropy at all, then a behaviour would have been expected, as gadolinium, with L = 0, is a special case (see e.g. Lounasmaa and Sundstrom 1966). 

Fig. 7.35. Magnon dispersion relations for Tb along symmetry lines in the zone at 4.2 K. The effects of interactions with certain phonon branches (dashed lines) are also shown (after Mackintosh and Bjerrum-Mpller, 1972).

Fig. 7.40. Magnon dispersion in ErFej along <110) and (111) symmetry directions. The squares denote points measured on the lower optic branch the circles denote points on the acoustic and upper optic branches. The solid line is the result of the nearest neighbor exchange interaction fitting calculation described in the text (after Rhyne et al., 1976).

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. 

The magnetic structures of these compounds are often complex (Rossat-Mignod et al. 1984, Lander 1993). Although the magnetic structure defines the position of the lowest cncigy in the excitation spectrum, the shape of the magnon dispersion curves are not usually sensitive to the exact magnetic structure. 

Measurements of the magnon dispersion relationship on NdAU and TbAl2 and also crystal field and molecular field calculations agree well with the values calculated using the above-mentioned hamiltonian (Houmann et al., 1974). 

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