Longitudinal relaxation in the rotating frame

Cross-link density and parameters relating to the network structure can be measured by NMR by analysis of the transverse relaxation decay (cf. Section 1.3) and the longitudinal relaxation in the rotating frame [67]. Combined with spatial resolution, the model-based analysis of relaxation yields maps of cross-link density and related parameters [68]. Often the statistical distribution of relaxation parameters over all pixels provides a reduced data set with sufficient information for sample characterization and discrimination [68]. 

Similar to transverse relaxation longitudinal relaxation in the rotating frame is also sensitive to slow molecular motion (cf. Fig. 7.1.3) and is thus a good probe for elastomer... 

Slow motions can be stndied using so-called longitudinal relaxation in the rotating frame (Tip) (5). In these experiments, a radio-frequency field ( spin-lock pulse ) with field strength of aroimd 10-100 kHz plays the same role as the main magnetic field does for Ti. For instance, the changes in the phase structure imder mechanical deformation of thermoplastic elastomers have been investigated by Tip measurements of nuclei under MAS (49). 

In summary, then, all relaxation processes can ultimately be described as some linear combination of spectral density functions of the form shown in Eq. (11). We have here only considered explicitly the case of longitudinal or spin lattice relaxation in the laboratory frame (the so-called spin-lattice relaxation in the rotating frame being a different process), but a similar case can be made for transverse relaxation, relaxation processes in the rotating frame and crossrelaxation processes. The spectral densities involved in each case are J f( Mo> ) where co is the frequency of rotating frame transformation required to remove the stationary part of the total spin Hamiltonian in each case. This will be the Larmor frequency, co0, for any relaxation process taking place in the laboratory frame. For relaxation processes taking place... 

Moreover, precession under selective irradiation occurs in the longitudinal plane of the rotating frame, instead of rotation in the transverse plane, which occurs during the evolution of the FID. The magnitude of the vector undergoing precession about the axis of irradiation decreases due to relaxation and field inhomogeneity effects. 

In summary, then, all relaxation processes can ultimately be described as some linear combination of such spectral density functions. We have here only explicitly considered longitudinal relaxation processes in the laboratory frame, but a similar case can be made for transverse relaxation, relaxation processes in the rotating frame and cross-relaxation processes. The spectral densities... 

Physical parameters Molecular 0.1-1 nm Dipole-dipole interaction Second moment, fourth moment of lineshape Incoherent magnetization transfer characteristic times for cross-polarization and exchange Mesoscopic lnm-0.1 p,m Longitudinal relaxation time Ti Transverse relaxation time T2 Relaxation time Tip in the rotating frame Solid-echo decay time T2e Spin-diffusion constant Microscopic 0.1-lOp.m Molecular self-diffusion constant D Macroscopic 10 p,m and larger Spin density... 

The particular value of coi can be adjusted by the strength fii / y of the spin-lock field. The range of effective spin-lock field strengths can be increased if off-resonance techniques are applied. In this case, the effective field Beff forms an angle 0 with respect to the z-axis (cf. Fig. 2.2.7), and the effective relaxation rate (T p,efr) is a combination of the longitudinal relaxation rate (7i) and the relaxation rate in the rotating frame (Tip) [Kim2],... 

Heteronuclear incoherent magnetization transfer is the transfer of longitudinal magnetization. It can proceed in the laboratory frame and in the rotating frame. The nuclear Overhauser effect (NOE) [Nogl] is a manifestation of polarization tranter in the laboratory frame. In the extreme narrowing limit saturation of dipolar relaxation of the I doublet of a heteronuclear IS two-spin- system leads to an enhancement of the S-spin polarization by a factor... 

Tip filters for the longitudinal relaxation time in the rotating frame by applying spin-lock fields or multisolid echoes combinations of relaxation-time filters and multipulse excitation chemical-shift selective excitation ... 

NMR parameter images can be translated to material property images by calibration or relationships known from theory. For example, cross-link density can be linked to the transverse relaxation decay [101-103] and the longitudinal relaxation decay in the rotating frame [104, 105]. Relaxation of transverse magnetization in cross-linked elastomers is nonexponential (Fig. 

Figure 2.19. Longitudinal relaxation. The recovery of a magnetisation vector (shown on resonance in the rotating frame) diminishes the transverse (x-y) and re-establishes the longitudinal (z) components.

Bearing in mind that the nuclear spin system is an ensranble perturbed by the rf-field, we have to consider the possibility that energy from the rf-field is transferred to the nuclear spins and dissipated further to the crystal lattice. These effects can be described by relaxation times that characterize the rates with which the system returns to thermal equilibrium after the perturbation has been switched off. There are the longitudinal or spin-lattice relaxation time Tj and the transverse or spin-spin relaxation time Tj. Including the relaxation effects the equations of motion in the rotating frame (cf. Eq. (19)) are... 

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