Electron spin vectors

Sharp and Lohr proposed recently a somewhat different point of view on the relation between the electron spin relaxation and the PRE (126). They pointed out that the electron spin relaxation phenomena taking a nonequilibrium ensemble of electron spins (or a perturbed electron spin density operator) back to equilibrium, described in Eqs. (53) and (59) in terms of relaxation superoperators of the Redfield theory, are not really relevant for the PRE. In an NMR experiment, the electron spin density operator remains at, or very close to, thermal equilibrium. The pertinent electron spin relaxation involves instead the thermal decay of time correlation functions such as those given in Eq. (56). The authors show that the decay of the Gr(T) (r denotes the electron spin vector components) is composed of a sum of contributions... 

The term S(S + )(A/h)2 is proportional to the square of the coupling energy between an electron spin vector S and the nuclear spin vector /. The same equations hold for lanthanides. 

For irradiation times of J short with respect to the relaxation time of / the NOE extent is independent of the relaxation time of the nucleus and provides a direct measurement of time required to saturate signal J is not negligible compared with t, the response of the system is not linear [18]. The truncated NOE is independent of paramagnetism as it does not depend on p/, which contains the electron spin vector S in the R[m term, and only depends on gkj), which does not contain S. If then the steady state NOE is reached, the value of p/ can also be obtained. This is the correct way to measure p/ of a nucleus, provided saturation of J can be considered instantaneous. In general, measurements at short t values minimize spin diffusion effects. In fact, in the presence of short saturation times, the transfer of saturation affects mainly the nuclei directly coupled to the one whose signal is saturated. Secondary NOEs have no time to build substantially. As already said, this is more true in paramagnetic systems, the larger the R[m contribution to p/. 

We will present the effective Hamiltonian terms which describe the interactions considered, sometimes using cartesian methods but mainly using spherical tensor methods for describing the components. These subjects are discussed extensively in chapters 5 and 7, and at this stage we merely quote important results without justification. We will use the symbol T to denote a spherical tensor, with the particular operator involved shown in brackets. The rank of the tensor is indicated as a post-superscript, and the component as a post-subscript. For example, the electron spin vector A is a first-rank tensor, T1 (A), and its three spherical components are related to cartesian components in the following way ... 

The initial singlet or triplet RP is a nonstationary state that evolves under the inhuence of the interactions described above, primarily the hyperhne interachon. This process is readily simulated exactly, even for systems with many nondegenerate nuclei. To perform calculations, the electron spin vector operator is written in terms of Hilbert space matrices, typically... 

In EPR spectroscopy, samples are brought into a homogeneous external magnetic field. The electron spin vector of the paramagnetic species in the sample can only take on distinct orientations with respect to the direction of this external field. 

In quantum mechanical terms, the allowable electron spin states are quantized, and the component of the electron spin vector in any given direction can only take one set of discrete values which are +s or —s,... 

For an electron in an orbital centered on the nucleus in question (Fig. 6), the anisotropic hyperfine coupling follows from Eq. 14, but having in addition a term <3cos a — 1> which represents the average direction of the electron spin vector within the orbital. The anisotropic hyperfine splitting can now be defined as the separation between adjacent energy levels, viz. Mg, Mj and Mg, and equals... 

Figure 2. Precession of electron spin vectors S/ and S of a radical pair viewed in...

FIGURE 10.1 Possible orientations of the electron spin vector with reflect to the z axis. In each case, S lies on the surface of a cone whose axis is the z axis. 

When one adds together two electron spin vectors - - S2> then the maximum z component of the spin angular momentum will be (in h units) Ms = m i + mg2 = + 5 = h This corresponds to the vectors Si,S2, called parallel to each other, while the minimum Ms = msi -h ms2 = — = 0 is an antiparallel configuration ofsi ands2 (Fig. 1.12). 

The energies of the two spin levels in Fig. 19.1b can be calculated using the Schrddinger equation, = E, in which H is the so-called spin Hamiltonian (Eq. 19.3) with Sz being the component of the electron spin vector along the direction... 

Initially, the small value of the blue Cu sites had been attributed to the Cu 4pz mixing into the Cu 3d 2 orbital due to the distorted tetrahedral structure of the blue copper site. The spin dipolar interaction of the Cu nuclear spin with the electron in the Cu 4p orbital would oppose that of an electron spin vector in the d 2 orbital [47]. The experimental probe of the 4p mixing into the d 2 2 or-... 

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