Tensor electron-nuclear hyperfine

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

A major advantage of studying pure compounds is that single crystals can be used, and hence e.s.r. parameters, which are generally anisotropic, can be accurately extracted. Furthermore, if the crystal structure is known, and if, as is frequently the case, the paramagnetic centres retain the orientation of the parent species, the directions of the g- and electron-nuclear hyperfine tensor components can be identified relative to the radical frame. 

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 ). 

Copper porphyrin is one of the best-characterized of the metalloporphyrins, and its electron spin resonance (ESR) spectrum has been known for a quarter of a century.(17) More recently, electron nuclear double resonance (ENDOR) investigations have provided the complete hyperfine tensors for the metal, the nitrogens and the pyrrole protons.(18) We have used this detailed knowledge earlier(, ) to assess the quality of scattered-wave calculations. 

Here /, is the 13C nuclear spin, S is the unpaired electronic spin, and A j- is the Fermi contact hyperfine coupling tensor. This coupling is identical for all 13C nuclei as long as the C60 ion is spherical, but becomes different for different nuclei after the Jahn-Teller distortion leading to an inhomogeneous frequency distribution. The homogeneous width of the 13C NMR lines is, on the other hand, mainly determined by the electron-nuclear dipolar interaction... 

H. Thomann, L. R. Dalton, Y. Tomkiewicz, N. S. Shiren, and T. C. Clarke, Electron-nuclear double-resonance determination of the 13C and 1H hyperfine tensors for polyacetylene, Phys. Rev. Lett. 50 533-536 (1983). 

Inequations (7.155) and (7.158) we have taken the diagonal = 0 component of the second-rank spherical tensors T2(/ . S) and T2(/a, Ia). In general, these interactions and others like them will have off-diagonal terms also, with q = 1 and 2. The q = 2 components are particularly interesting because, for a molecule in a Id electronic state, they connect the A = +1) and A = — 1) components directly. They therefore make additional hyperfine contributions to the /I-doubling of molecules in Id electronic states. As a result, the nuclear hyperfine splitting of one component of a A-doublet is different from that of the other component. The two contributions are ... 

The nuclear hyperfine interaction splits the paramagnetic states of an electron when it is close to a nucleus with a magnetic moment. For a random orientation of spins and nuclei, the tensor quantities in Eq. (4.11) are replaced by scalar distributions, and the resonance magnetic field is shifted from the Zeeman field // by... 

Lunsford [3b] and Hoffman and Nelson [23] first reported the ESR spectra for adsorbed NO molecules. Then, Kasai [4b] revealed that ESR spectra of NO probe molecules are very sensitive to the interaction with metal ions and Lewis acid sites in zeolites. The earlier ESR studies of the NO/zeolite system have been summarized in several review papers [3a, 4a, 8]. A number of ESR studies have been also carried out for NO adsorbed on metal oxides such as MgO and ZnO as reviewed by Che and Giamello [5]. Modern ESR techniques such as pulsed ESR [25-27], ENDOR (Electron Nuclear Double Resonance) [26], and multi-frequency (X-, Q-, and W-band) ESR [28] are especially useful for an unambiguous identification of the ESR magnetic parameters (g, hyperfine A, and quadrupole tensors, etc.) and, consequently, for a detailed characterization of structural changes and motional dynamics involved. Some recent advancements in ESR studies on NO adsorbed on zeolites are presented in this section. 

In Equation (2), D and E are the axial and rhombic zero-field splitting (ZFS) parameters, respectively, and g is the electronic g tensor. The magnetic hyperfine interactions of the electronic system with the Fe nucleus are described by S-a-I, and —is the nuclear Zeeman term. The quadrupole interaction involves the traceless EFG tensor. The EFG tensor has principal components Vyy, and The asymmetry parameter t] = Vxx- VyyyiV can be confined to 0 < 7 < 1 if the convention V zl > I Vyy > V xl is adopted. A quadrupole doublet... 

Later [38, 39], oxygen vacancies (Fig. 2.2) and E point defects present in glassy Si02 could be studied in great detail, including also full ab-initio calculations of the hyperfine parameters experimentally detected by electron-nuclear double resonance (ENDOR) experiments. Indeed, these types of measurements are nowadays routinely done to identify this class of paramagnetic defects. In the ENDOR technique, some Si atoms are substituted with their isotopes Si. This confers anon-zero nuclear spin I to the atomic nucleus that couples to the electron spin S via a tensor A. On the theoretical front, the calculation from first principles DFT approaches does not pose particular problems since the hyperfine interaction is still a ground state property which can be expressed in terms of the electronic density p x). The interaction between an electron spin (S) and a nuclear (I) spin is in fact described by the Hamiltonian... 

Another source of inaccuracy for most of these nuclei is represented by the large, non-negligible contribution of the diamagnetic CS A, which may pollute the experimental measurements, yielding a combined interaction rather than the electron-nuclear dipolar anisotropy alone. Simulations where the CSAs are added to the hyperfine tensors indeed provide a better approximation to the observed spectra. However, as described in the case of lanthanide acetates by Brough et al. [23], this correction is experimentally demanding, as it requires the measurement of CS A... 

In Eq. (2) the summations are taken over all the nuclei in the molecular species. The new symbols in Eq. (2) are defined as follows g is the nuclear g factor, which is dimensionless is the nuclear magneton, having units of joules per gauss or per tesla the nuclear spin angular momentum operator I the electron-nucleus hyperfine tensor A the quadrupole interaction tensor Q and Planck s constant h. 

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 ... 

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