Total spin Hamiltonian

The spin Hamiltonian is thus generated. In particular it can be used to examine the Tq-S mixing of electron spin states and its relationship to the distributions of populations of nuclear spin states. The total spin Hamiltonian is given in equation (15) which contains both electron and nuclear terms. 

The leading term in T nuc is usually the magnetic hyperfine coupling IAS which connects the electron spin S and the nuclear spin 1. It is parameterized by the hyperfine coupling tensor A. The /-dependent nuclear Zeeman interaction and the electric quadrupole interaction are included as 2nd and 3rd terms. Their detailed description for Fe is provided in Sects. 4.3 and 4.4. The total spin Hamiltonian for electronic and nuclear spin variables is then ... 

The interactions of an electron and a magnetic nucleus with an applied magnetic field and the interaction between the electron and the nucleus may be represented by the total spin Hamiltonian ... 

In the presence of dipolar couplings the total spin Hamiltonian in the rotating reference frame is given by... 

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... 

Summarizing, foiu different magnetic interactions may occiu, which influence the behavior of electrons in a magnetic field (a) the Zeeman interaction, Hu (b) the nuclear hyperfine interaction. Hup, (c) the dectrostatic quadrupole interaction, Hq and (d) the zero-field spHtting if S > V2> Hps- The sum of these interactions results in the total spin Hamiltonian, Hf. 

The spin hamiltonian The total spin hamiltonian may be written as... 

Origin of EPR spectra from dinuclear iron centers. EPR spectra of dinuclear iron species can be described by a total spin Hamiltonian involving the sum of the individual spin Hamiltonians and the interaction Hamiltonian [64—66] ... 

The simple argument to describe the coupling is identical to the derivation of the Lande interval rule for atomic spectra and has been discussed many times (Gibson et al, 1966 Griffith, 1972 Orme-Johnson and Sands, 1873 Palmer, 1973). The total spin Hamiltonian for the system is taken as... 

In this work, relativistic effects are included in the no-pah or large component only approximation [13]. The total electronic Hamiltonian is H (r R) = H (r R) + H (r R), where H (r R) is the nom-elativistic Coulomb Hamiltonian and R) is a spin-orbit Hamiltonian. The relativistic (nomelativistic) eigenstates, are eigenfunctions of R)(H (r R)). Lower (upper)... 

E. Quantitative Aspects of Tq-S Mixing 1. The spin Hamiltonian and Tq-S mixing A basic problem in quantum mechanics is to relate the probability of an ensemble of particles being in one particular state at a particular time to the probability of their being in another state at some time later. The ensemble in this case is the population distribution of nuclear spin states. The time-dependent Schrodinger equation (14) allows such a calculation to be carried out. In equation (14) i/ (S,i) denotes the total... 

The singlet function corresponds to zero total electron spin angular momentum, S = 0 the triplet functions correspond to S = 1. Operating on these functions with the spin Hamiltonian, we get ... 

In the above derivation, we have made no explicit assumption about the total electron spin quantum number S so that the results should be correct for S = 1 /2 as well as higher values. However, the fine structure term is not usually included in spin Hamiltonians for 5=1/2 systems. The fine structure term can be ignored since in that case the results of operating on a spin-1/2 wave function is always zero ... 

If two or three unpaired electrons are present so that the total spin is greater than one-half, additional terms must be added to the spin Hamiltonian of Eq. (6). The new terms may be written as... 

Hence, for a given total spin eigenvalue S there exist 2S + 1 states that all yield the same energy but may split when magnetic fields described as spin interactions are important in the Hamiltonian. The individual spin states are referred to as the S = 0 singlet state with 2S + 1 = 1, as doublet S — j with 2S + 1 = 2, as triplet S = 1 with 2S + 1 = 3, and so on. 

The spin Hamiltonian can be obtained from the MO s in a manner similar to that used in Sec. III. In this case the parameters of the spin Hamiltonian are determined by the Cy/ s of the ground and excited state MO s as well as by the values of (E0 — E ), f, and av. In a complete calculation the values of cjt and (E0 — En) would be found by minimizing the total energy, but this is a difficult computation and has been attempted only infrequently. The most notable attempt in this direction is the calculation by Shulman and Sugano (24,25) on KNiF3. The general practice has been to determine values of (E0 — En) from optical spectra, from atomic spectra, and <>-3>av from free-ion wave functions and to use these values plus the experimental values of the spin Hamiltonian parameters to determine the values of the Cy/ s. 

The total spin-free Hamiltonian including internal motion is taken to be... 

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