Spin Hamiltonians solving

Once a hyperfine pattern has been recognized, the line position information can be summarized by the spin Hamiltonian parameters, g and at. These parameters can be extracted from spectra by a linear least-squares fit of experimental line positions to eqn (2.3). However, for high-spin nuclei and/or large couplings, one soon finds that the lines are not evenly spaced as predicted by eqn (2.3) and second-order corrections must be made. Solving the spin Hamiltonian, eqn (2.1), to second order in perturbation theory, eqn (2.3) becomes 4... 

This spin Hamiltonian is solved in Appendix D for the S= spin system. Comparing the solution of Eq. (48) to Eqs. (41) and (42) we find that the behavior of the two ground-state functions in the presence of a magnetic field can be represented by the solution of the spin Hamiltonian of Eq. (48) in which g, and gL are simply constants to be evaluated by experiment. 

The spin Hamiltonian is an artificial but useful concept. It is possible that more than one spin Hamiltonian will fit the data. Further, we should note that in solving Eq. (48), we start with pure + and — spin functions and talk about the upper and lower states as being pure spin states. This is not the true case for the ion, as has already been noted in Eq. (38). As regards the Zeeman interaction, however, the final state behaves as a pure spin state, except that we must assign g values different from that of the free electron. 

General Form. From Sec. III.D and III.E, it becomes apparent that most ESR spectra can be accounted for by solving a spin Hamiltonian of the form... 

Special Hamiltonians. For some ions in which the zero-field interaction is very large, it is possible to see only a few of the allowed transitions. In these cases the absorption lines have sometimes been fitted to spin Hamiltonians of somewhat different character than that of Eq. (78). For example, in the 5= 1 system it may be that only the (— l ->+ 1) transition can be observed when D is very large. In this case it will be shown in Sec. IV that this absorption can be represented fairly well by solving the spin Hamiltonian... 

A very common situation encountered is axial symmetry where gz=g and gx=gy=g - The spin Hamiltonian for axial symmetry has been solved for S=i in Appendix D, giving... 

When dealing with transitions between these two levels we can drop the since this only adds a constant energy to each level. The same behavior for two energy levels can be obtained by solving the spin Hamiltonian... 

The same result can be obtained by solving the spin Hamiltonian... 

In tetrahedral and cubic symmetry, the crystal-field levels are inverted, giving a single orbital state lowest. For dl in an octahedral field, the 4F state also has the single orbital state lowest, so we would expect the d1 configuration in a tetrahedral or cubic field to behave in a similar fashion and fit the spin Hamiltonian given in Eq. (158) solved for S=f. For most of the examples listed in Table XV, the tetrahedral symmetry is not distorted so that D = E=0 and no fine structure is reported. The 5=f character of the spin state is revealed in these cases by the fact that Eq. (80) must be added to the spin Hamiltonian to explain the ESR results on d1 in tetrahedral and cubic fields (131). For Co2+ in Cs3CoCl5, D= —4.5 cm-1 (222) and in CdS, D > 2 cm-1 (223). 

In real systems, the number of spin centers N is, of course, too large to deterministically solve the corresponding eigenvalue problem of a corresponding spin Hamiltonian. Thus, no exact solution exists. Several approximate expressions were developed, such as the Bonner Fisher finite-chain model for equidistant antiferro-magnetically coupled S = 1/2-based chains.20 Here, the susceptibility is calculated for finite chain segments (ca. N 10 spin centers) and extrapolated to an infinite chain (N > oo). The extrapolated expression for the susceptibility is as follows ... 

One of the limiting problems of numerically solving the eigenvalue problem of a given spin Hamiltonian for a system with a finite number of spin centers is that the Hilbert space dimension Q that translates into the dimension of matrices that need to be diagonalized increases with... 

Jansen and van der Avoird (1985) have also made spin-wave calculations as described earlier. The RPA equations with the effective spin Hamiltonian (140), averaged over the translations and librations, could be solved analytically for any wave vector q. The optical (q = 0) magnon frequencies emerging from these calculations are 6.3 and 20.9 cm-1, in reasonable agreement with the experimental values 6.4 and 27.5 cm-1. This agreement is very satisfactory if we realize that the spin Hamiltonian has been obtained from first principles, with none of its parameters fitted to the magnetic data. We conclude that the RPA model, both for the lattice modes and the spin waves, when based on a complete crystal Hamiltonian from first principles, yields a realistic description of several properties of solid O2 that were not well understood before. 

In all single-crystal studies, the variation in resonance frequency or magnetic field is studied as a function of the orientation of the crystal in the magnetic field. A spin Hamiltonian of appropriate form is then solved and the parameters adjusted to fit the calculated variation with the experimental data. Most errors in doing this occur because approximate solutions of spin Hamiltonians are used for systems for which the approximations are not justified. Second-order effects are often very important in analyzing single-crystal ESR and ENDOR measurements. 

For the simulation of ESR spectra one has to solve the spin Hamiltonian of Eq. (10). The easiest way to do this is to regard all the different terms in the spin Hamiltonian as small compared with the electron Zeeman interaction and to use perturbation theory of the first order. The Zeeman term can easily be solved within the eigensystem of the Sz operator (in the main axis system of the g-tensor or S 2=5 for isotropic cases), for instance in the isotropic case ... 

In order to avoid any assumptions about the size of D and relative to gPB > the characteristic secular determinant may be solved numerically. The secular determinant is set up within the manifold of S=f states using the spin Hamiltonian of Eq. (16). The g tensor may be taken equal to 2.0 since for d systems the principal... 

We now turn to the spin-dependent terms listed in Section 11.3, and discuss some examples of their observable effects. Such effects are most commonly interpreted in terms of a phenomenological Hamiltonian, which usually contains only spin operators (for the various nuclei and for the total electron spin) and applied fields, together with numerical parameters that serve as coupling constants . This spin Hamiltonian H describes a model spin system whose behaviour may be determined by solving ... 

The usual way of solving eqn (7) requires its transformation into the interaction representation (Dirac picture) that is often called rotating frame for a particular case, when static part of the spin Hamiltonian is restricted to the electron Zeeman interaction. In the Dirac picture only the stochastic dipolar interaction is left in the spin Hamiltonian, its matrix elements get additional oscillatory factors due to the static Hamiltonian transitions. The integral on each matrix element of the double commutator in eqn (7) thus evolves into the Fourier transform /(co ) of the correlation function for the corresponding stochastic process. This Fourier transform is often called spectral density of the stochastic process and it is to be taken at a frequency co of a particular transition of the static Hamiltonian operator, driven by a single transition operator ki ... 

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