Magnetization spin wave theory

De Jongh showed that in 2-D systems, spin-reduction can also be deduced from the perpendicular magnetic susceptibility7,269. The experimental value of xi extrapoled to 0 K is in good agreement with the theoretical value obtained by means of the spin-wave theory (Xi (0). The difference between the latter value and that calculated via the molecular field approach Xkmf) is essentially due to the zero-point spin reduction. 

Those curves that do not approach T = 0°K with zero slope are not realized in nature. The N6el model is a molecular field model, and is subject to the same criticisms as the Weiss field model for ferromagnets. Kaplan (325) has applied spin wave theory to ferri-magnets and worked out a Bloch Tz/2 law, similar to equation 98, for low temperatures. In this approximation M /M% remains constant,... 

The heat capacity of EuS was measured to test the predictions of spin-wave theory from 1° to 38°K. by McCollum and Callaway (137) and independently from 10° to 35°K. by Moruzzi and Teaney (145). A sharp Neel peak was found at 16.2 °K. Magnetic and lattice contributions to the heat capacities were resolved on the assumption of a dependence for the lattice and a T dependence for the magnetic contribution at temperatures above the Neel point. A plot of CT vs. yields a straight line between 21° and 31 °K. and a Debye temperature of 208 °K. 

The FMR technique can also be used as a convenient means of studying the temperature dependence of the magnetization. In spin wave theory this temperature dependence is given by... 

Deviations from spin wave theory at low temperatures in various amorphous alloys were found by Bhagat et al. (1980), and explained in terms of a model where proper account is taken of the presence of magnetic clusters. A spin-cluster model was also used by Bhagat and Paul (1975) to explain the FMR data obtained by these authors in several amorphous RFe alloys. Weissenberger et al. (1984) investigated FMR in amorphous Y-Co alloys and did not observe deviations from normal spin wave behaviour below T. ... 

The chief source of all this confusion and apparent indecision lies in our inability to ascribe a definite form to the magnetic spin wave contribution to the heat capacity. A number of theories were introduced in the early 1950 s, for a review of which reference should be made to Mackintosh and Mbller (1972) or, for a review in brief, one may consult Lounasmaa and Sundstrom (1%6). For the sake of completeness, we present in table 5.1 the temperature dependences of the various models. 

Figure 7.17 Reduced magnetization as a function of reduced temperature. Calculations were performed within the framework of the spin wave theory applied to thin Fe films with various numbers of atomic layers [82).

For an XT-model at dimensionalities d > dt this coefficient vanishes at Tc with a power law S a ]f 2jff T,v. Although in d = 2 the magnetization (cos0(x)) = 0, the stiffness S is non-zero in the spin wave regime in fact, comparison of eqs. (160) and (168) suggests S = J, independent of T. The Kosterlitz-Thouless (1973) theory implies that S is reduced from J at non-zero temperatures due to vortex-antivortex pairs, and that the equation that yields the transition temperature rather is... 

In conclusion, it may be said that the stumbling blocks in analysing heat capacity measurements on the rare earth metals are the difficulty of obtaining satisfactorily high purity samples and the current state of the theory for spin waves. There are too many unknown variables for unambiguous analysis. Nevertheless, in nearly all cases the nuclear contribution has been determined accurately, and for the majority some indication of the most likely form for the magnetic term exists. 

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