Sideband, rotational

In spin relaxation theory (see, e.g., Zweers and Brom[1977]) this quantity is equal to the correlation time of two-level Zeeman system (r,). The states A and E have total spins of protons f and 2, respectively. The diagram of Zeeman splitting of the lowest tunneling AE octet n = 0 is shown in fig. 51. Since the spin wavefunction belongs to the same symmetry group as that of the hindered rotation, the spin and rotational states are fully correlated, and the transitions observed in the NMR spectra Am = + 1 and Am = 2 include, aside from the Zeeman frequencies, sidebands shifted by A. The special technique of dipole-dipole driven low-field NMR in the time and frequency domain [Weitenkamp et al. 1983 Clough et al. 1985] has allowed one to detect these sidebands directly. 

With sleeve or Babbitt bearings, looseness is displayed as an increase in sub-harmonic frequencies (i.e., less than the actual shaft speed, such as 0.5x). Rolling-element bearings display elevated frequencies at one or more of their rotational frequencies. Excessive gear clearance increases the amplitude at the gear-mesh frequency and its sidebands. 

At low rotation rates, less than the chemical shifts anisotropy, however, the powder spectra contained disturbing side bands dispersed among the isotropic chemical shifts. In order to discriminate between sidebands and isotropic resonances two spectra obtained at different spinning speeds were multiplied together or the differentiation was made by visual inspection. 

Similar to the PIP, the Hamiltonian [Eq. (52a)] of a periodic pulse shows an infinite number of effective RF fields with both x and y components of the scaling factors X a and the phases 0na. The periodic pulse, however, acquires a different symmetry as that of the PIP. From Eq. (52c) and = ana, it follows that the scaling factor Xm, is symmetric in respect to the sideband number n, while the phase 6na is anti-symmetric according to Eq. (51c). These symmetries seem to be a coincidence arising from the mathematical derivations. As a matter of fact, they are the intrinsic natures of the periodic pulse. Considering the term f x i)Ix for instance, any Iy component created by the rotating field denoted by a> must be compensated at any time t by its counter-component oj n in order to reserve the amplitude modulated RF field. 

In polarization modulated ENDOR spectroscopy (PM-ENDOR)45, discussed in Sect. 4.7, the linearly polarized rf field B2 rotates in the laboratory xy-plane at a frequency fr fm, where fm denotes the modulation frequency of the rf carrier. In a PM-ENDOR experiment the same type of cavity, with two rf fields perpendicular to each other, and the same rf level and phase control units used in CP-ENDOR can be utilized. To obtain a rotating, linearly polarized rf field with a constant magnitude B2 and a constant angular velocity Q = 2 fr (fr typically 30-100 Hz), double sideband modulation with a suppressed carrier is applied to both rf signals. With this kind of modulation the phase of the carrier in each channel is switched by 180° for sinQt = 0. In addition, the phases of the two low-frequency envelopes have to be shifted by 90° with respect to each other. The coding of the two rf signals is shown in Fig. 8. 

In these spectra, the protein has been regenerated with retinal specifically 13 C labeled at positions 11,12 and 13, and in each case the retinal resonance exhibits a sharp centerband at the isotropic chemical shift and is flanked by rotational sidebands. Other lines in the spectrum are the natural-abundance 13C resonances of the protein carbonyls ca 175 ppm) and aliphatic carbons (0-100 ppm). Contributions from the Ammonyx-LO detergent in these spectra are seen in the different intensities in the 0-100 ppm region. Ammonyx-LO does not exhibit NMR resonances above 100 ppm. Spectra of the 9-cis pigment isorhodopsin are similar. Table 38 summarizes the isotropic chemical shifts from the solid-state NMR spectra of rhodopsin regenerated with retinal13 C labeled at each position along... 

FIGURE 39. MAS 13C NMR spectra of (a) 13C-11, (b) 13C-12 and (c) 13C-13 rhodopsin. Center-bands and rotational sidebands of the retinal resonances are marked with asterisks. Reprinted with permission from Reference 55. Copyright (1990) American Chemical Society... 

We have prepared a number of acylium ions on metal halide powders and measured the principal components of their chemical shift tensors (43-45). Most of these cations have isotropic l3C shifts of 154 1 ppm. Often insensitivity to substituents results from opposite and offsetting variations in the principal components. The acetylium ion has an axially symmetric chemical shift tensor because of its C3 rotation axis. When the symmetry is reduced from C3v to C2v or lower, a nonzero 27 value may be observed. The sensitivity of chemical shift tensors to symmetry is a powerful means of probing molecular structure and temperature-dependent molecular dynamics. Multiple orders of spinning sidebands may offend those who seek solution-like NMR spectra of solids, but discarding most of the information inherent in the chemical shift is a considerable concession to aesthetics. 

Table 5 Calculated dipolar rotational sideband intensities for an aromatic CH pair undergoing various molecular motions, under MAS at 1.894 kHz (from [40])...

When the sample is placed in a gas-driven rotor and spun at the rate of a few kilohertz under 6 equal 54.7° (Figure IB and C), the anisotropic part is zero. In such an experiment, the so-called MAS, narrowing of the resonance lines is observed. However, it is only true when the spin rotation is larger than anisotropy A8 given in hertz. In cases when the rotor rotation (cor) < AS, the isotropic signal is flanked on both sides by rotational sidebands. In a strong applied field B0, the only measurable quantity of the shift tensor is the component oriented along z-axis in the laboratory frame ... 

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