Dipolar local field

Figure 14.11 Basic 2D pulse sequence used for the measurement of residual dipolar local fields. After the excitation pulse the spins are allowed to evolve for some time tj (indirect dimension) under influence of the relevant spin interactions before crosspolarisation takes place. The direct detection during time t2 then takes place on the 13C side typically under proton dipolar decoupling (DD). The basic scheme can be extended by various spin manipulation techniques (not shown) during time tj. For instance, the heteronuclear dipolar contribution can be removed by a decoupling pulse...

One of the cross-polarization pulse sequences used to measure is shown in Figure 6.9. During the evolution time, At, of this pulse sequence, the magnetization is spin-locked along Any reorientation of the internuclear vectors induces fluctuations of the dipolar local fields, and the COcomponent of these fluctuating fields participates in the relaxation of the spins. This is a spin-lattice relaxation mechanism which is related... 

In conclusion, FMR experiments represent a useful tool for studying the dynamical behavior of fine magnetic particles and for obtaining information on the anisotropy energy and on the orientational distribution of easy axes. The proposed theoretical models are able to fit qualitatively the experimental spectra. However some important aspects, strongly affecting the thermal fluctuations of particle moments, have not yet been accounted for, for example, the effect of dipolar interactions in dynamical conditions (the dipolar local fields fluctuate in time in a system of nonidentical particles) and the effect of surface layers. Moreover, some models are valid only if H and cannot be applied when H is small. 

Apart from the small Boltzmann factor, up and down states are equally probable, thus we may imagine that resonance absorption for spin i will occur at frequency Y[Bq + b ]. Since bj has a range of values, so then will Wi This is illustrated in Figure 2(b). The width of this resonance absorption thus reflects the root-mean-squared dipolar local field. 

The general problem of calculating the lineshape for dipolar-coupled spins is not capable of analytic solution except for very simple cases, such as isolated pairs. This is because it is a many-body problem (1,2). Useful quantities, in such circumstances, are the moments of the lineshape. In particular the second moment, M2, which is the mean-squared linewidth, is often used. This, we can see from the above discussion, is a measure of the strength of the dipolar local field, and its utility lies in the fact that its calculation does not require the solution of the many-body problem (1,2). For a single crystal eq. 3 holds. 

Here the ijk coordinate system represents the laboratory reference frame the primed coordinate system i j k corresponds to coordinates in the molecular system. The quantities Tj, are the matrices describing the coordinate transfomiation between the molecular and laboratory systems. In this relationship, we have neglected local-field effects and expressed the in a fomi equivalent to simnning the molecular response over all the molecules in a unit surface area (with surface density N. (For simplicity, we have omitted any contribution to not attributable to the dipolar response of the molecules. In many cases, however, it is important to measure and account for the background nonlinear response not arising from the dipolar contributions from the molecules of interest.) In equation B 1.5.44, we allow for a distribution of molecular orientations and have denoted by () the corresponding ensemble average ... 

The dipolar contributions to the local field diverges if the terms are replaced by their absolute values. 

As Figure 2 shows, the two models differ only by a vertical displacement, which is a measure of the difference in the mean square interparticle dipolar fields (second moments) operative in the two models. The values of these mean square local fields can be calculated easily from the minimum value of Ti or from the rigid lattice values of T2. Because of the 1000-fold ratio between electronic and nuclear magnetic moments, even... 

The broad line spectra of nuclei with spin I = 1/2 in the solid state are mainly a consequence of the dominant contribution of the dipolar Hamiltonian HD (Eq. (4)), which gives rise to a local field B)oc. Its magnitude varies as a function of the angle 0.. between the intemuclear vector r.j and the applied magnetic field B0. Depending on the nature of spin system, two general types of interactions can be distinguished ... 

In the separated local field technique, dipolar I-S interactions are separated from chemical shifts of nucelus S. As dipolar interactions are highly sensitive to internuclear distances, the obvious use of the method is for the determination of molecular structure in the solid state. An example is provided by the work of Hester et al. (411) and Rybaczewski et al. (414) on... 

In the limit of the oriented gas model with a one-dimensional dipolar molecule and a two state model for the polarizability (30). the second order susceptibility X33(2) of a polymer film poled with field E is given by Equation 4 where N/V is the number density of dye molecules, the fs are the appropriate local field factors, i is the dipole moment, p is the molecular second order hyperpolarizability, and L3 is the third-order Langevin function describing the electric field induced polar order at poling temperature Tp - Tg. 

The interactions between the molecule and the environment can lead to distortions in the electrical properties due to the susceptibility of the molecules and the properties of the host matrix. The refractive index of the matrix acts as a screening factor, modifying the optical spectra and interaction between charges or dipoles embedded within it. Local field effects change the interaction with an electromagnetic field and should be considered along with orientation factors in the dipolar interaction. 

FIGURE 10.8 Pulse sequence for the separated local field experiment. After cross polarization, spin S precesses under the influence of dipolar interactions during , but is decoupled from I during (2. 

Consequently, polarisation due to dipolar orientation is directly proportional to the local field strength and inversely proportional to temperature. We can think of the quantity ifikT as an orientational polarisability and simply add... 

Thirdly, when two unpaired electrons are sufficiently close, as for example in the triplet state, the two magnetic dipoles interact magnetically the magnetic dipole-dipole interaction. The interaction can be viewed as an additional local magnetic field and therefore it changes the resonance frequency. The local field depends on the angle between the dipolar axes of the two unpaired electrons and the applied field, and therefore the change in resonance frequency is anisotropic. 

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