Spin Hamiltonian hyperfine coupling

The similarity of the preceding first-order ESR treatment to the first-order NMR treatment of two coupled protons is evident. For an unpaired electron interacting with n equivalent nuclei of spin the hyperfine coupling term in the spin Hamiltonian is... 

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 ... 

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... 

The spin Hamiltonian for a biradical consists of terms representing the electron Zeeman interaction, the exchange coupling of the two electron spins, and hyperfine interaction of each electron with the nuclear spins. We assume that there are two equivalent nuclei, each strongly coupled to one electron and essentially uncoupled to the other. The spin Hamiltonian is ... 

The method presented here for evaluating energy levels from the spin Hamiltonian and then determining the allowed transitions is quite general and can be applied to more complex systems by using the appropriate spin Hamiltonian. Of particular interest in surface studies are molecules for which the g values, as well as the hyperfine coupling constants, are not isotropic. These cases will be discussed in the next two sections. 

The hyperfine constant a in Eq. (1) was also taken to be a scalar quantity for the hydrogen atom however, it is in general a tensor because of the various directional interactions in a paramagnetic species. The hyperfine term in the spin Hamiltonian is more correctly written as S-a-I, where a is the hyperfine coupling tensor. 

Although simple /rSR spectra that do not depend on the nuclear terms in the spin Hamiltonian are the easiest to observe, one loses valuable information on the electronic structure. Under certain circumstances it is possible to use conventional /rSR to obtain a limited amount of information on the largest nuclear hyperfine parameters. The trick is to find an intermediate field for which the muon is selectively coupled to only the nuclei with the largest nuclear hyperfine parameters. Then a relatively simple structure is observed that gives approximate nuclear hyperfine parameters. A good example of this is shown in Fig. 3a for one of the /xSR... 

We now come back to the simplest possible nuclear spin system, containing only one kind of nuclei 7, hyperfine-coupled to electron spin S. In the Solomon-Bloembergen-Morgan theory, both spins constitute the spin system with the unperturbed Hamiltonian containing the two Zeeman interactions. The dipole-dipole interaction and the interactions leading to the electron spin relaxation constitute the perturbation, treated by means of the Redfield theory. In this section, we deal with a situation where the electron spin is allowed to be so strongly coupled to the other degrees of freedom that the Redfield treatment of the combined IS spin system is not possible. In Section V, we will be faced with a situation where the electron spin is in... 

In general, fluctuations in any electron Hamiltonian terms, due to Brownian motions, can induce relaxation. Fluctuations of anisotropic g, ZFS, or anisotropic A tensors may provide relaxation mechanisms. The g tensor is in fact introduced to describe the interaction energy between the magnetic field and the electron spin, in the presence of spin orbit coupling, which also causes static ZFS in S > 1/2 systems. The A tensor describes the hyperfine coupling of the unpaired electron(s) with the metal nuclear-spin. Stochastic fluctuations can arise from molecular reorientation (with correlation time Tji) and/or from molecular distortions, e.g., due to collisions (with correlation time t ) (18), the latter mechanism being usually dominant. The electron relaxation time is obtained (15) as a function of the squared anisotropies of the tensors and of the correlation time, with a field dependence due to the term x /(l + x ). 

The calculation of magnetic parameters such as the hyperfine coupling constants and g-factors for oligonuclear clusters is of fundamental importance as a tool for the evaluation of spectroscopic data from EPR and ENDOR experiments. The hyperfine interaction is experimentally interpreted with the spin Hamiltonian (SH) H = S - A-1, where S is the fictitious, electron spin operator related to the ground state of the cluster, A is the hyperfine tensor, and I is the nuclear spin operator. Consequently, it is... 

The EPR spectra have always been interpreted2994 using an effective S = 2 spin Hamiltonian including the Zeeman term, /iBB g-S, and the hyperfine term, ICa-A-S, which describes the interaction of the unpaired electrons with the copper nucleus (7Cu = I). The spectra are very sensitive to the ratio between the isotropic coupling constant J and the local zero field splitting of nickel(II), Z)Ni.2982 In the limit J DNi it can easily be shown that the following relations hold ... 

The most important examples of 2S states to be described in this book are CO+, where there is no nuclear hyperfine coupling in the main isotopomer, CN, which has 14N hyperfine interaction, and the Hj ion. A number of different 3E states are described, with and without hyperfine coupling. A particularly important and interesting example is N2 in its A 3ZU excited state, studied by De Santis, Lurio, Miller and Freund [19] using molecular beam magnetic resonance. The details are described in chapter 8 the only aspect to be mentioned here is that in a homonuclear molecule like N2, the individual nuclear spins (1 = 1 for 14N) are coupled to form a total spin, It, which in this case takes the values 2, 1 and 0. The hyperfine Hamiltonian terms are then written in terms of the appropriate value of h As we have already mentioned, the presence of one or more quadrupolar nuclei will give rise to electric quadrupole hyperfine interaction the theory is essentially the same as that already presented for1 + states. 

We continue to use the case (b) hyperfine-coupled basis set used earlier. The matrix elements of the first three terms in the effective Hamiltonian (8.251) do not involve the nuclear spins and are therefore independent of, and diagonal in, the quantum number F. The required matrix elements are now tabulated. 

The CN radical in its 21 ground state shows fine and hyperfine structure of the rotational levels which is more conventional than that of CO+, in that the largest interaction is the electron spin rotation coupling../ is once more a good quantum number, and the effective Hamiltonian is that given in equation (10.45), with the addition of the nuclear electric quadrupole term given in chapter 9. The matrix elements in the conventional hyperfine-coupled case (b) basis set were derived in detail in chapter 9,... 

The theory of the magnetic hyperfine interactions in NCI is essentially the same as that already described for the PF radical in the previous section, except that the nuclear spins / are 1 for 14N and 3/2 for 35C1. The form of the effective Hamiltonian for the quadrupole interaction and its matrix elements for two different quadrupolar nuclei was described in some detail in chapter 8 when we discussed the electric resonance spectra of CsF and LiBr. We now use the same case (b) hyperfine-coupled basis set as was used for PF. The quadrupole Hamiltonian for the two nuclei can be written as the sum of two independent terms as follows ... 

We have chosen to use the hyperfine-coupled representation, where for 12CH, F is equal to J 1 /2. An appropriate basis set is therefore t], A N, A S, J, /, F), with MF also important when discussing Zeeman effects. As usual the effective zero-field Hamiltonian will be, at the least, a sum of terms representing the spin-orbit coupling, rigid body rotation, electron spin-rotation coupling and nuclear hyperfine interactions, i.e. 

If one or more spin-coupling terms also occur (e.g. hyperfine or nuclear spin-spin), then each such spin-Hamiltonian parameter matrix Y occurs in the transition energy and intensity expressions within combinations of type... 

Anisotropic proton hyperfine couplings are measured by rotating the crystals in the external magnetic field. From hundreds of angular measurements various proton hyperfine couplings are obtained as follows. EPR data are analyzed using the spin Hamiltonian (A in MHz)... 

For radicals with magnetic nuclei, the hyperfine structure of ESR spectra is produced by the interaction of the electron magnetic moment with the nuclear spin of those nuclei covered by the molecular orbital of the unpaired electron. This interaction splits further the two spin levels in a magnetic field. The hyperfine coupling is often given by the Hamiltonian HgN ... 

---