Spin-lattice Hamiltonian

As discussed in Sect. 6.2, the electronic states of a paramagnetic ion are determined by the spin Hamiltonian, (6.1). At finite temperamres, the crystal field is modulated because of thermal oscillations of the ligands. This results in spin-lattice relaxation, i.e. transitions between the electronic eigenstates induced by interactions between the ionic spin and the phonons [10, 11, 31, 32]. The spin-lattice relaxation frequency increases with increasing temperature because of the temperature dependence of the population of the phonon states. For high-spin Fe ", the coupling between the spin and the lattice is weak because of the spherical symmetry of the ground state. This... 

Now we consider thermodynamic properties of the system described by the Hamiltonian (2.4.5) it is a generalized Hamiltonian of the isotropic Ashkin-Teller model100,101 expressed in terms of interactions between pairs of spins lattice site nm of a square lattice. Hamiltonian (2.4.5) differs from the known one in that it includes not only the contribution from the four-spin interaction (the term with the coefficient J3), but also the anisotropic contribution (the term with the coefficient J2) which accounts for cross interactions of spins a m and s m between neighboring lattice sites. This term is so structured that it vanishes if there are no fluctuation interactions between cr- and s-subsystems. As a result, with sufficiently small coefficients J2, we arrive at a typical phase diagram of the isotropic Ashkin-Teller model,101 102 limited by the plausible values of coefficients in Eq. (2.4.6). At J, > J3, the phase transition line... 

Consider a spin system whose spin Hamiltonian consists of a time-independent Hamiltonian H0 and a stochastic perturbation Hamiltonian H,(t) due to a small spin-lattice coupling,... 

Substituting Eq. (14) into Eq. (12), and neglecting any correlation between the density matrix and the spin-lattice coupling Hamiltonian, one obtains to first order... 

Lso is the commutator with the electron spin Zeeman Hamiltonian (assuming isotropic g tensor, Hso = gS- Bo), Lrs = Lzfs (the sub-script RS stands for coupling of the rotational and spin parts of the composite lattice) is the commutator with the ZFS Hamiltonian and Lr = —ir, where is a stationary Markov operator describing the conditional probability distribution, P(QolQ, t), of the orientational degrees of freedom through ... 

A more general theory for outer-sphere paramagnetic relaxation enhancement, valid for an arbitrary relation between the Zeeman coupling and the axial static ZFS, has been developed by Kruk and co-workers (96 in the same paper which dealt with the inner-sphere case. The static ZFS was included, along with the Zeeman interaction in the unperturbed Hamiltonian. The general expression for the nuclear spin-lattice relaxation rate of the outer-sphere nuclei was written in terms of electron spin spectral densities, as ... 

According to standard NMR theory, the spin-lattice relaxation is proportional to the spectral density of the relevant spin Hamiltonian fluctuations at the transition frequencies coi. The spectral density is given by the Fourier transform of the auto-correlation fimction of the single particle fluctuations. For an exponentially decaying auto-correlation function with auto-correlation time Tc, the well-known formula for the spectral density reads as ... 

Electron spin resonance (ESR) measures the absorption spectra associated with the energy states produced from the ground state by interaction with the magnetic field. This review deals with the theory of these states, their description by a spin Hamiltonian and the transitions between these states induced by electromagnetic radiation. The dynamics of these transitions (spin-lattice relaxation times, etc.) are not considered. Also omitted are discussions of other methods of measuring spin Hamiltonian parameters such as nuclear magnetic resonance (NMR) and electron nuclear double resonance (ENDOR), although results obtained by these methods are included in Sec. VI. 

Values of the spin Hamiltonian reported for d4 ions are given in Table XI. The difficulty in detecting the ESR is due most likely to short spin-lattice-relaxation times and large zero-field splittings. In both octahedral and tetrahedral fields the 5 D state of d4 gives degenerate orbital states which,... 

Values for the spin Hamiltonian are given in Table XIV. The 5D state of d6 has three orbital states for the ground state in octahedral symmetry. Since these three states are connected by the spin-orbit coupling, the spin-lattice-relaxation time is quite short and the zero-field splitting very large. In a distorted octahedral field the large zero-field distortion makes detection of ESR difficult. In the case of ZnF2 the forbidden AM = 4 transition was measured and fitted to Eq. (164). 

The Kondo-lattice Hamiltonian conserves total spin and being an interacting model is nontrivial to solve. However, as with the conjugated systems, it is possible to solve finite Kondo chains efficiently by employing the VB method. The VB... 

When 1/2spin permutations in the first and second order in t2 we can write the lattice Hamiltonian for this filling in the form... 

So, to use the spin permutation technique we constructed the symmetry adapted lattice Hamiltonian in a compact operator form and essentially reduced the dimensionality of the corresponding eigenvalue problem. The effects of tpp 0 and the additional superexchange of copper holes are considered in [48]. 

In principle, very good distance measurements can be made with arguments based on the C and D terms of the dipolar Hamiltonian, which predict how spin-lattice relaxation of one species can affect spin-lattice relaxation of the other species. 

In summary, then, all relaxation processes can ultimately be described as some linear combination of spectral density functions of the form shown in Eq. (11). We have here only considered explicitly the case of longitudinal or spin lattice relaxation in the laboratory frame (the so-called spin-lattice relaxation in the rotating frame being a different process), but a similar case can be made for transverse relaxation, relaxation processes in the rotating frame and crossrelaxation processes. The spectral densities involved in each case are J f( Mo> ) where co is the frequency of rotating frame transformation required to remove the stationary part of the total spin Hamiltonian in each case. This will be the Larmor frequency, co0, for any relaxation process taking place in the laboratory frame. For relaxation processes taking place... 

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