Motional narrowing, line broadening

Therefore, the absorjDtion line is massively inlromogeneously broadened at low temperature. An inliomogeneous lineshape can be used to detennine the static or quasistatic frequency spread of oscillators due to a distribution of environments, but it provides no dynamical infonnation whatsoever [94, 95]. As T is increased to 300 K, the absorjDtion linewidth decreases and increases. At 300 K, the lineshape is nearly homogeneously broadened and dominated by vibrational dephasing, because fast dephasing wipes out effects of inliomogeneous environments, a well known phenomenon tenned motional narrowing [951. 

In experimental measurements, sueh sharp 5-funetion peaks are, of eourse, not observed. Even when very narrow band width laser light sourees are used (i.e., for whieh g(co) is an extremely narrowly peaked funetion), speetral lines are found to possess finite widths. Let us now diseuss several sourees of line broadening, some of whieh will relate to deviations from the "unhindered" rotational motion model introdueed above. 

There is a second relaxation process, called spin-spin (or transverse) relaxation, at a rate controlled by the spin-spin relaxation time T2. It governs the evolution of the xy magnetisation toward its equilibrium value, which is zero. In the fluid state with fast motion and extreme narrowing 7) and T2 are equal in the solid state with slow motion and full line broadening T2 becomes much shorter than 7). The so-called 180° pulse which inverts the spin population present immediately prior to the pulse is important for the accurate determination of T and the true T2 value. The spin-spin relaxation time calculated from the experimental line widths is called T2 the ideal NMR line shape is Lorentzian and its FWHH is controlled by T2. Unlike chemical shifts and spin-spin coupling constants, relaxation times are not directly related to molecular structure, but depend on molecular mobility. 

In passing it is interesting to note that Fig. 5 qualitatively explains the reason for the difference in the effect of motion on spectral lines in radiofrequency and optical spectroscopy. In radiofrequency spectroscopy one refers to motional narrowing, while collisional broadening is used to... 

The line broadening caused by partial motional narrowing can be distinguished from that due to isotropic reorientation at a reduced rate by appropriate magic angle spinning experiments. 

In NMR experiments, molecular mobility leads to narrowing of the resonance lines. Conversely, restricted molecular motion, as occurs in crystalline phases, causes line-broadening. Until the advent of magic angle spinning and related techniques, tliis was a hindrance to NMR studies of solid polymers. We have used it to advantage in following the orientation effects in solid PTFE. 

For large spin-orbital interactions a marked anisotropic -factor is expected and provides an important mechanism for relaxation of the electron spin from the upper to the lower state. Once again, if this relaxation is too efficient, there will be an uncertainty-principle broadening of the lines, which may be so great that no absorption can be detected. The relaxation is due to coupling between the orbital component and vibrational and other motions of the lattice , which includes any inter- or intramolecular motions, and hence it may be necessary to cool to very low temperatures in order to obtain narrow lines. Fortunately this situation rarely arises for organic radicals since iS.g is almost invariably very small. 

Broad Lines. - The width of an n.m.r. line, At>1/2, is defined as the width in Hz at half signal height. Narrow lines, i.e., Ap1/2 < 10 Hz, are desirable in order to make use of chemical shift information and to follow chemical change. N.m.r. line widths in the liquid and the physisorbed state tend to be very narrow, with Ar>1/2 of the order of 10-1Hz. This fortuitous state arises because the molecular motion is sufficiently rapid and random in a liquid to average out the line broadening features present in solids, namely dipolar interactions, chemical shift anisotropy, quadrupolar interactions, and paramagnetic interactions which render the spectrum unusable under conventional or liquid-state experimental conditions. The mechanisms of each of these features will be described. The treatment will perforce be cursory, but an indication will be given to where a full theoretical treatment can be found. 

Solid-state 13C NMR spectra of carbon black filled, uncured and sulfur-vulcanised HR were recorded at 22.6 MHz. The line broadening of the filled polymer relative to the unfilled polymer is attributed to incomplete motional narrowing of the NMR lines [53, 54] Incorporation of filler also results in a decrease in the signal-to-noise ratios in the spectra, but fundamentally it does not obscure the qualitative and quantitative nature of the spectra for the moderately cured elastomer systems. 

The broad lines obtained for solid-state NMR spectra without applying any line-narrowing improvements are due to the different behaviour of nuclear spin interactions in solids compared to liquids. These interactions are averaged to zero or reduced to the isotropic values in liquids by the fast molecular motions, whereas the fixed (and different) orientations (with respect to the external magnetic field Bq) of the local environments of NMR active isotops in the rigid lattice of a solid cause line broadenings. The recorded broad NMR line patterns are superpositions of resonances from randomly oriented individual nuclei due to a random distribution of different orientations, since zeolitic materials usually are microcrystalline powders. Table 1 summarizes the nuclear spin interactions and their behaviour in liquids versus solids (17). 

Equation (3) has several other important implications which can be directly confirmed by finite-frequency probes. One example is the motion-narrowing effect in NMR experiments which is expected to disappear when l/r is below the chemical-shift-anisotropy (CSA) width. Indeed the NMR results of Tycko et al. [16] indicate that for a CSA width of 18.2 kHz the line broadens below 190 K and develops a powder pattern at lower temperature. This is in fair agreement with the 200 K calculated from Eq. (3). They also concluded that the thermal activation energy is around 260 meV below TV, again close to the values we calculated. The glassy dynamics can be probed by other experiments such as sound attenuation, microwave absorption, and thermal conductivity. In particular the characteristic temperature will depend on probe frequency. Such studies are essential to fully understand the low-temperature orientational dynamics. 

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