Sideband states

As discussed in Chapter 10, we can think of the effect of the microwave field on the n( state as modulating its energy. Just as modulating the frequency of a radio wave produces sidebands, the microwave field can be thought of as modulating the energy of the nt state and breaking it into carrier and sideband states. 

Fig. 15.6 Energy levels of the Na 17p, 18s, and 18p states showing the first upper and lower sideband states of the p states. The numbers +2,..., -2 refer to the net number of photons emitted in a radiative collision at these tunings of the field. Note that there are several processes which lead to the net emission of, for example, zero photons (from ref. 3).

The m-th sideband state is displaced from the carrier, or the ( , 3) state in the absence of microwaves, by rtm. Note that the spatial wavefunction of Eq. (9) is unchanged. 

The carrier and sideband states cross the ( + 2)s state. If we ignore the AC Stark shift of the (n + 2)s state the intersection occurs at a static field Es = Ec — mw/k. When the core coupling is taken into account, the crossings become avoided crossings, and the magnitude of the avoided crossing between the n + 2)s state and the m-th sideband of the (n, 3) state is given by... 

The avoided crossing of the sideband state is the purely static field avoided crossing multiplied by the amplitude of the m-th sideband state. The size of the... 

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. 

We have referred to the various interactions which can cause line broadening in the solid state. One of these, which is normally not a problem in liquid state NMR, is due to the fact that the chemical shift itself is a tensor, i.e. in a coordinate system with orthogonal axes x, y and z its values along these axes can be very different. This anisotropy of the chemical shift is proportional to the magnetic field of the spectrometer (one reason why ultra-high field spectrometers are not so useful), and can lead in solid state spectra to the presence of a series of spinning sidebands, as shown in the spectra of solid polycrystalline powdered triphenylphosphine which follows (Fig. 49). In the absence of spinning, the linewidth of this sample would be around 75 ppm ... 

It has recently been demonstrated that the analysis of MAS sidebands patterns can be used to study molecular dynamics in the solid state [85-88]. Indeed, the line narrowing effect of MAS can be partly offset, or completely eliminated, if the 2H quadrupole tensor is reoriented due to motion on a time scale comparable to (first-order quadrupolar broadening, such motion-induced effects should be less evident in the DQMAS spectrum, as has indeed been observed by Wimperis and colleagues in several deuterated solids [87, 88]. For example, the simulation of the SQ spectrum of tetrathionate dihydrate-cfi yielded the same reorientational rate constant as the previously described quadrupolar echo approach (Fig. 6). 

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... 

The experimental data closely resemble the simulation based on the all-trans PSB values. The discrepancy between the two solid-state NMR studies on rhodopsin arises in part from a difference, in signal-to-noise ratio and in part from possible problems associated with a fatty acid resonance which overlaps with the centerband in the previous study. The simulations illustrate the sensitivity of the sideband intensities to changes in the chemical shift tensor, as well as the quality of data necessary to accurately determine the shift tensor values. 

An investigation of lithium diisopropyl amide (LDA) by solid state NMR led to the observation of dramatic differences between the spectra of the solid polymer and the complex crystallized from THF. Li as well as "C and "N MAS spectra showed large sideband patterns in the former case and only a few sidebands in the latter. For both materials X-ray data are available and establish a helix structure for the polymeric material, which is insoluble in hydrocarbon or ethereal solvents, and a dimer structure of the THF complex (25, 26, Scheme 4). The obvious difference between both structures, apart from the solvent coordination in the THF complex, is the magnitude of the structural N-Li-N angle, which is close to 180° in the first case and close to 90° in the second (176° and 107°, respectively). Thus, a large difference for the electric field gradient around the Li cation is expected for the different bonding situations. 

Alexandrite, the common name for Cr-doped chrysoberyl, is a laser material capable of continuously tunable laser output in the 700-800 nm region. It was established that alexandrite is an intermediate crystal field matrix, thus the non-phonon emitting state is coupled to the 72 relaxed state and behaves as a storage level for the latter. The laser-emitted light is strongly polarized due to its biaxial structure and is characterized by a decay time of 260 ps (Fabeni et al. 1991 Schepler 1984 Suchoki et al. 2002). Two pairs of sharp i -lines are detected connected with Cr " in two different structural positions the first near 680 nm with a decay time of approximately 330 ps is connected with mirror site fluorescence and the second at 690 nm with a much longer decay of approximately 44 ms is connected with inversion symmetry sites (Powell et al. 1985). The group of narrow lines between 640 and 660 nm was connected with an anti-Stokes vibronic sideband of the mirror site fluorescence. 

The chemical shift anisotropy of the 29Si nucleus is generally small, and thus unlike in l3C solid-state NMR at high fields, no sideband problems are encountered in MAS spectra of framework silicates. 

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