Spin wave approximation

Longitudinal fluctuations of a Neel-ordered state are usually strongly damped and not included in a spin-wave approximation of its excitation spectrum. Until now similar modes have been only observed in neutron scattering on spin chain systems as broadened maxima [44 46]. The present narrow linewidth is probably related to the small scattering vector involved in light scattering experiments. 

From this spin wave approximation (Wegner, 1967) one can also obtain the correlation function... 

Thus the spin wave approximation predicts a line of critical points at all T > 0, each temperature being characterized by its own (non-universal) critical exponent r). 

For systems containing localized magnetic moments, the thermopower has not been theoretically investigated in such detail as the resistivity. An expression for the thermopower of ferromagnetic materials with localized moments has been obtained by Kasuya (1959) in both the molecular field approximation and the spin wave approximation. In the former case, Kasuya used a molecular field approximation to obtain the energy spectrum of the conduction electrons and the localized magnetic moments. In addition he assumed that the spin-flip transition probabilities for scattering of electrons by local moments dominate the non-spin-flip transition probabilities. 

At low temperatures, B Jiy) becomes exponentially small and pu>ng T) may be neglected, reflecting the insignificance of longitudinal spin fluctuations as the moment approaches saturation at T = 0. The transverse spin fluctuations, however, remain but are more accurately described within the spin wave approximation, because of the inadequacy of the molecular field model in describing the collective excitations. Calculations (Kasuya 1959, Yamada and Takada 1972) of the electron - magnon scattering cross-section yield a resistivity contribution ... 

In order to include the spin of the two electrons in the wave function, it is assumed that the spin and spatial parts of the wave function can be separated so that the total wave function is the product of a spin and a spatial wave function F — iAspace sp n Since our Hamiltonian for the H2 molecule does not contain any spin-dependent terms, this is a good approximation (NB—the complete Hamiltonian does contain spin-dependent terms, but for hydrogen they are rather small and do not appreciably affect the energetics of chemical bonding). For a two-electron system it turns out that there are four possible spin wave functions they are ... 

In the Xa scattered wave approximation, the exchange potential for spin-up electrons may be different from that for spin-down electrons. In particular, when unpaired electrons are present, the exchange potentials, and hence the spin-up and spin-down orbitals and their energy levels, are different. Thus, MO calculations are performed using a spin-unrestricted formalism so that separate orbital energy levels are given for spin-up (a) and spin-down (p) electrons. 

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,... 

For electrons (being fermions of spin 1/2) a scalar wave approximation is often used, whereas for photons (being bosons of spin 1), the full vector field needs to be taken into consideration (see Optics Express, vol. 8, no. 3, 2001, for a focus issue on photonic band gap calculations). 

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

Of course, the approximations made by the spin wave theory are reasonable at very low temperatures only, and thus it is plausible that this line of critical temperatures terminates at a transition point 7 Kt, the Kosterlitz-Thouless (1973) transition, while for T > 7kt one has a correlation function that decays exponentially at large distances. This behavior is recognized when singular spin configurations called vortices (fig. 33 Kawabata and Binder, 1977) are included in the treatment (Berezinskii, 1971, 1972). Because 0(x) is a multivalued function it is possible that a line integral such... 

The application of this approach to spin waves was called by Dyson naive and criticized as incorrect and leading to results different from those obtained by him (see (6), the end of 3). We shall show in what follows, however, on the basis of an exact representation of Pauli operators in terms of Bose operators, that the picture described above does take place for Frenkel excitons. This takes place only because the excitation energy A for Frenkel excitons is large compared with the width of the exciton band. As for the spin waves, where the inequality indicated above is not satisfied, the cross-section for the scattering of long-wavelength spin waves by each other can indeed, in agreement with Dyson, differ substantially from a value that follows from the hard sphere approximation (7). 

Generalizations [71] of the bare anisotropic-planar-rotor model (2.5) include other multipolar interactions such as dipolar and octopolar terms with and without in-plane crystal-field modulations. Several such combinations were analyzed in the mean-field approximation, Landau theory, and spin-" wave expansion [71]. The quadrapole-quadrupole model written in the form... 

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