Spin Waves for Ferromagnets

To calculate the energy of Pq we evaluate the effect of the different terms of the Hamiltonian separately and then add them up to obtain the energy. 

The zero s in the first two contributions are due to the fact that a spin with maximum M -value cannot climb further on the ladder by From this, we confirm that 4 o is an eigenfunction with eigenvalue 

The action of the ladder operators on such functions is defined in Eq. 1.23 and results in 

By defining 0 = (0i 02)/V2, eigenfunctions of the Heisenberg Hamiltonian are obtained with the following eigenvalues 

E+ is identical to the ground state value and the corresponding wave function has the same spin multiplicity as but the total Ms value is lowered by one. The second energy, E-, is higher than Eq (remember that the Jy are positive for a ferromagnetic system) and describes a state where the total spin moment is no longer equal to the maximum value. 

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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,... 

When speaking of kinematic interaction, it should be noted that the problem of its separation in connection with the transition from Pauli operators to Bose operators is far from new. This problem arises, in particular, for the Heisenberg Hamiltonian, which corresponds, for example, to an isotropic ferromagnet with spin a = 1/2 when spin waves whose creation and annihilation operators obey Bose commutation relations are introduced. This problem was dealt with by many people, including Dyson (6), who obtained the low-temperature expansion for the magnetization. However, even before Dyson s paper, Van Kranendonk (7) proposed to take into account of the kinetic interaction by starting from a picture where one spin wave produces an obstacle for the passage of another spin wave, since two flipped spins cannot be located at the same site (for Frenkel excitons this means that two excitons cannot be localized simultaneously on one and the same molecule). 

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. 

The quadratic dispersion relation of the spin waves should therefore give rise to the usual resistivity for local moment ferromagnetism at low temperature, i.e. 

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