Zeeman anisotropic term

The electron-electron dipolar term, Ho, equals S1.D.S2. The tensor D is completely anisotropic and only mixes T-states with one another. It is therefore dropped. The nuclear Zeeman term, tlzi =... 

The restricted summation in equation (7.221) means that, for this contribution, the Zeeman energy depends on the x and y components of S only (i.e. it has cylindrical symmetry). It is therefore referred to as the anisotropic correction to the electron spin Zeeman term, which has the isotropic form gsH sB S. 

The isotropic g and a values are now replaced by two 3x3 matrices representing the g and A tensors and which arise from the anisotropic electron Zeeman and hyperfine interaction. Other energy terms may also be included in the spin Hamiltonian, including the anisotropic fine term D, for electron-electron interactions, and the anisotropic nuclear quadrupolar interaction Q, depending on the nucleus. Usually the quadrupolar interachons are very small, compared to A and D, are generally less than the inherent linewidth of the EPR signal and are therefore invisible by EPR. They are readily detected in hyperfine techniques such as ENDOR and HYSCORE. All these terms (g. A, D) are anisotropic in the solid state, and must therefore be defined in terms of a tensor, which will be explained in this section. 

Here, each Vj. term is due to each of the dipole-dipole interaction between two components radicals, the anisotropic Zeeman of the two radicals, and their anisotropic HFC terms, T, is each rotational correlation time, and co is given by gfigB h. 

Since the ground state of the compound is the singlet (vide infra), it is not necessary to introduce local anisotropy or anisotropic interaction terms in Eq. (83). The Zeeman perturbation is ... 

The Nuclear Zeeman Interaction The nuclear Zeeman term represents the direct interaction between the external magnetic field and the nuclear magnetic moment. This is usually neglected since it cancels for transitions between states with identical values of m/. When forbidden transitions are being considered, however, it is sometimes necessary to take account of this effect. It is applicable to both anisotropic and isotropic spectra. 

The diagonal term gel arises from the spin-Zeeman term S Bext, while the anisotropic part has two contributions. The direct term arises from the P operator (eq. (10.79)) and the relativistic correction from the spin-Zeeman term (eq. (10.83)), with additional contributions coming from the combination of V2LG and the spin-orbit operators P and Pe . 

This procedure (neglect of nuclear Zeeman term) is sometimes also adopted to obtain hyperfine coupling tensors from ESR measurements of free radicals. The method is, however, not suitable for the analysis of hyperfine structure due to a-H in jr-electron radicals of the type )Cc-H at X-band, and other cases where the anisotropic hyperfine coupling and the nuclear Zeeman energy are of comparable magnitudes, as discussed for case 3 below. 

Fig. 3.39 5 = 1/2 ESR line pattern for an / = Vi nucleus with an anisotropic hyperfine coupling and a nuclear Zeeman term of comparable magnitude. The intensities 1 and Ip depend on the values of Va and vp and the nuclear Zeeman energy vn in frequency units according to Eqs. (3.38) and (3.39)...

Fig. 4.25 Effective fields B+ ms = +V2) and B. ms = -V2) acting on a nucleus with an anisotropic hfc and a nuclear Zeeman term of comparable magnitude. Ba is due to the hypetfine couphng, Bn to the applied magnetic field...

Anisotropic A and g, and non-negligible b The general case with appreciable g- and hyperfine anisotropy and a nuclear Zeeman term that is comparable with the hyperfine coupling is rarely considered in practical apphcations. 

Hq, Hd, Hcs and Hj describe the quadrupolar, dipolar, chemical shift and indirect electron coupled interactions, respectively, and are listed according to their typical magnitude. The Zeeman term was given by equ. (1) and is not influenced by the local environment. Quadrupolar interactions occur for nuclei with I> A only and are described in the next chapter for the case of 2H with 1=1, thus the dominant interaction for IH and 13C is given by Ho and Hcs. The dipole-dipole interaction between spins is a strongly anisotropic interaction. Similarly, the chemical shift depends on the orientation of the molecule with respect to Bq, this part is the so-called chemical shift anisotropy (CSA). In a static sample without any molecular motions... 

The isotropic parameters are given by giso = iTr and Aiso = j Tr /f. With an increase in the magnetic field, the anisotropic part of the Hamiltonian proportionally increases due to the Zeeman term and this leads to an overall enhanced sensitivity of the nitroxide spectra to molecular motion at high magnetic fields. Specifically, at high magnetic fields the fast motion condition ... 

Let us discuss first the case in which only the first term is present. In the Solomon and Bloembeigen equations for / , (i = 1, 2) there is the cos parameter at the denominator of a Lorentzian function. Up to now cos has been taken equal to that of the free electron. However, in the presence of orbital contributions, the Zeeman splitting of the Ms levels changes its value and cos equals xs / o or (g/h)pBBo- When g is anisotropic (see Fig. 1.16), the value of cos is different from that of the free electron and is orientation dependent. The principal consequence is that another parameter (at least) is needed, i.e. the 0 angle between the metal-nucleus vector and the z direction of the g tensor (see Section 1.4). A second consequence is that the cos fluctuations in solution must be taken into account when integrating over all the orientations. Appropriate equations for nuclear relaxation have been derived for both the cases in which rotation is faster [40,41] or slower [42,43] than the electronic relaxation time. In practical cases, the deviations from the Solomon profile are within 10-20% (see for example Fig. 3.14). 

The second additional term is an anisotropic correction to the electron spin Zeeman interaction, which in a molecule-fixed axis system is given by... 

The steps to be followed may be summarized. Secular determinants must be constructed for each of the doubly degenerate levels in both directions. First-order Zeeman coefficients must be evaluated for each direction. Matrix elements connecting the three secular determinants must be evaluated to yield second-order Zeeman coefficients. The first-and second-order Zeeman coefficients must be substituted into the Van Vleck equation to yield the anisotropic magnetic susceptibilities x and x - Generally, anisotropic magnetic properties are discussed in terms of /x and n since the variation of these anisotropic components are much more easily visuaUzed. 

The intensity ratio between the main and satellite lines has been used to estimate the distance between a paramagnetic centre and protons at surrounding molecules. The method is based on the assumptions that the point dipole approximation applies for the anisotropic hfc, and that this coupling is much smaller than the nuclear Zeeman energy of the proton. It was found advisable to measure this ratio at as high microwave frequency as possible to achieve the latter condition, i.e. Ba < Bn in terms of the direct field model. The procedure was adequate at Q-band but not at X-band to obtain the distance between a proton and the P03 radical in a single crystal of Na2HP03-51120, see [37] for details. 

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