Hamiltonians isotropic spin

Therefore, the effective isotropic spin Hamiltonian for the radical pair Hrp is given by equation (21). 

In liquid solution the orientation of the ions is rapidly changing relative to the magnetic field. If the reorientation frequency is greater than the frequency width of the powder absorption line, the spectrum collapses to narrow lines and fits the isotropic spin Hamiltonian... 

The time independent isotropic spin-spin scalar coupling Hamiltonian is... 

Let us investigate the situation of having an orbitally non-degenerate atomic term 2S+ S or a molecular state 2S+ A and 25+1B, respectively. In the absence of thermally accessible excited states the only contribution to the spin Hamiltonian is an isotropic (spin) Zeeman term... 

The direct observation of hyperfine resonances of the T7 ground state of isotopically enriched Er in a powdered gold matrix was reported by Sjostrand and Seidel (1975). In weak external fields for 5=2 and / =, two hyperfine multiplets, F=3 and 4, are expected. The appropriate isotropic spin Hamiltonian is given by... 

The radical rotates at a shorter correlation time than the reciprocal of spectrum in frequency. Therefore, in Fig. 15 the spectrum can be interpreted in terms of the following "isotropic" spin Hamiltonian ... 

This spectrum was analyzed to determine the types of free radicals manifested in this experiment. In order to investigate only resolved hfs lines, a single imresolved line (Fig. 22a) was subtracted from the spectrum (Fig. 22b). The spectrum resulting the subtraction (Fig. 22c) can be interpreted in terms of the isotropic spin Hamiltonian (eq. 6). 

The special case of isotropic g and hyperfine interaction will now be considered. This simplification is valid when the anisotropic interactions are averaged by rapid tumbling. The quadrupole interaction will be omitted because it is purely anisotropic. The resulting simplified spin Hamiltonian is given in Equation (9). 

This is a simplified Hamiltonian that ignores the direct interaction of any nuclear spins with the applied field, B. Because of the larger coupling, Ah to most transition metal nuclei, however, it is often necessary to use second-order perturbation theory to accurately determine the isotropic parameters g and A. Consider, for example, the ESR spectrum of vanadium(iv) in acidic aqueous solution (Figure 3.1), where the species is [V0(H20)5]2+. 

The anisotropies that lead to line broadening in isotropic ESR spectra influence solid-state spectra more directly. Accordingly a more complex spin Hamiltonian is required to interpret such spectra ... 

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. 

A stress that is describable by a single scalar can be identified with a hydrostatic pressure, and this can perhaps be envisioned as the isotropic effect of the (frozen) medium on the globular-like contour of an entrapped protein. Of course, transduction of the strain at the protein surface via the complex network of chemical bonds of the protein 3-D structure will result in a local strain at the metal site that is not isotropic at all. In terms of the spin Hamiltonian the local strain is just another field (or operator) to be added to our small collection of main players, B, S, and I (section 5.1). We assign it the symbol T, and we note that in three-dimensional space, contrast to B, S, and I, which are each three-component vectors. T is a symmetrical tensor with six independent elements ... 

As demonstrated by Griffin, Levitt, and coworkers in the late 1980s [21, 93], it is also possible to recouple homonuclear dipolar couplings through interference between isotropic chemical shifts and the rotor revolution. This phenomenon, called rotational resonance, occurs when the spinning frequency is adjusted to a submultiple of the isotropic chemical shift difference, i.e., ncor = ct> so o) °. To understand this experiment, the dipolar coupling Hamiltonian in (10) is transformed... 

Let us calculate the frequencies of transitions between Zeeman eigenstates s) and r), assuming that the nuclei are only subjected to an isotropic chemical shift and the first- and second-order quadrupolar interaction. As seen in Sect. 2.1, the Hamiltonian that governs the spin system in the frame of the Zeeman interaction (the rotating frame) is... 

The Mu spin Hamiltonian, with the exception of the nuclear terms, was first determined by Patterson et al. (1978). They found that a small muon hyperfine interaction axially symmetric about a (111) crystalline axis (see Table I for parameters) could explain both the field and orientation dependence of the precessional frequencies. Later /xSR measurements confirmed that the electron g-tensor is almost isotropic and close to that of a free electron (Blazey et al., 1986 Patterson, 1988). One of the difficulties in interpreting the early /xSR spectra on Mu had been that even in high field there can be up to eight frequencies, corresponding to the two possible values of Ms for each of the four inequivalent (111) axes. It is only when the external field is applied along a high symmetry direction that some of the centers are equivalent, thus reducing the number of frequencies. 

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... 

Molecules in ordinary liquids average out all anisotropic spin interactions due to isotropic Brownian motions, and their NMR spectra are governed by the Hamiltonian in units of Hz due to the Zeeman interaction, the isotropic chemical shift (a) and the isotropic indirect spin-spin coupling (7)... 

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