Measurement of Relaxation Times

In an analogous manner, the decay toward zero of an NMR signal (i.e., thex,y component ofM) by spin-spin (transverse) relaxation follows the equation 

We have already hinted at two ways to measure T. In cases where the observed line width is due solely to effective (field-inhomogeneity-caused) spin-spin relaxation, the reciprocal of the halfwidth of the peak is T [Eq. (3.12)]. Also, in Section 3.4 we noted that the envelope of the FID signal follows exponential decay governed by T. In principle, we could measure the amount of time (t W2) required for the FID signal to decay to exactly half its initial magnitude (M, = i Mq), at which point Eq. (3.14b) gives 

In the usual case where T is less than 7), the latter value cannot be determined from normal FID data. However, there is a way to measure true (as opposed to effective) T2 values, and it involves a frequently encountered technique known as the spin echo. 

20b the result of a 90v pulse M has rotated (precessed) clockwise around the x axis (as viewed along the x axis toward the origin) and now lies along the +/ axis. 

At this point, with B, off, the individual nuclear magnetic moments that comprise M begin to dephase (relax transversely) because inhomogeneities in the magnetic field B0 

As we saw in Section 3.2, the intensity of an NMR signal is determined in part by the number of nuclei giving rise to the signal. We normally use solution samples to obtain reasonable relaxation times and halfwidths. It is typical to dissolve approximately 5-10 mg of a substance of average molecular weight (say, 200 daltons) in 0.4 mL of deuteriated solvent. Using special microtechniques, it is possible in some cases to obtain H spectra on as little as 10 pg of sample  

When picking a solvent there are several considerations. First, and most obvious, the solvent must be inert, and it must dissolve the sample Of course, if your sample is sparingly soluble or if there is only a fraction of a milligram of sample available, you can usually resort to signal-averaging techniques to obtain the desired 5/M Besides that, it is best if the solvent does not contain any nuclei of the isotope to be examined. 

Make sure you use thoroughly cleaned and dried NMR tubes, and never assume that a new tube, fresh out of the box, is clean to NMR standards. 

At this point it would be useful to review many of the important topics covered in the first three chapters by discussing a few of the actual techniques used to measure relaxation times. 

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Highly sophisticated pulse sequences have been developed for the extraction of the desired information from ID and multidimensional NMR spectra [172]. The same techniques can be used for high-resolution 1-NMR, s-NMR and NQR. Pulse experiments are commonly used for the measurement of relaxation times [173], for the study of diffusion processes [174] and for the investigation of chemical reactions [175]. Davies et al. [176] have described naming and proposed reporting of common NMR pulse sequences (IUPAC task group). An overview of pulse sequence experiments has been given [177],... 

The measurement of relaxation times 7j and T2 and the subsequent application of the theory formulated by Bloembergen et al. (236), and extended by Kubo and Tomita (272) and Torrey (288), leads to the determination of motional and thermodynamic parameters such as mean times between molecular jumps, diffusion coefficients, and activation enthalpies for translation. For example, Resing and Thompson (289, 290) used this... 

Despite the enormous problems encountered in interpreting relaxation times of molecules of low molecular weight as described in this article, the measurement of relaxation times of even more complex systems commonly studied in biochemistry is currently a very widely accepted method. In general, biochemical application of relaxation time measurements can be divided into three main areas ... 

These are obtained by introducing an explicit time dependence of the permittivity. This dependence, which is specific to each solvent is of a complex nature, cannot in general be represented through an analytic function. What we can do is to derive semiempirical formulae either by applying theoretical models based on measurements of relaxation times (such as that formulated by Debye) or by determining through experiments the behaviour of the permittivity with respect to the frequency of an external applied field. 

Until recently, measurements of relaxation times of aqueous solutions of non-electrolytes had not been extended to sufficiently low frequencies to allow the calculation of any but the principal relaxation time of the water. We have seen that this relaxation time is not identical with that of pure water, but relaxation times widely separated from that of pure water have not been reported. 

Studies of molecular dynamics have focused on the effect of temperature due largely to experimental convenience. Isobaric measurements of relaxation times and viscosities are carried out routinely as a function of temperature. From these experiments it is well established that the shape of the a-dispersion (i.e., the KWW stretch exponent fiKWW), when compared at Tg or some other reference value of xa, varies among different glass-formers [45,46]. Many experimental studies have shown also that for a given material, very often the distribution of relaxation times systematically broadens with decreasing temperature [47-50]. 

Because a large number of transients are needed to obtain even a nonquantitative 13C-spectrum of humic substances in solution, the time needed for measurement of relaxation times can be prohibitive. 

Ideally, measurements should be made at as many magnetic field strengths as possible, so that a unique fit of the experimental data to a particular model, may be obtained. As already mentioned, in addition to measuring the p values, measurement of relaxation times will help to substantiate the model chosen. 

The rapid development of lasers has led to the publication of increasing numbers of papers concerned this year with such subjects as superfluorescence and co-operative radiation processes,451 the thermodynamics of co-operative luminescence,452 saturation, collisional dephasing, and quenching of fluorescence of organic vapours in intense laser excitation studies,453 a theoretical model for fluorescence in gases subjected to continuous i.r. excitation,454 a quantum treatment of spontaneous emission from strongly driven two-level atoms,455 the development of site-selection spectroscopy,45 and measurements of relaxation times 457 using laser excitation. 

Nuclear Magnetic Resonance Spectroscopy.—As noted above, conformational analysis of bicyclo[3.3.1]nonanes is still a topic of considerable interest. A variable-temperature n.m.r. analysis now provides the first case in which the boat-chair-chair-boat equilibrium is directly observed in the amines (17) and (18). In a related case, re-examination of the acetal (19) suggests that the preferred conformation involves a chair carbocyclic ring and a boat heterocyclic ring. This conclusion was made by n.m.r. analysis, using lanthanide shift reagents, by a study of nuclear Overhauser effects, and by measurement of relaxation times of protons. Details have been reported for other 3-azabicyclo[3.3.1]nonanes, and the non-additivity of substituent effects on chemical shifts in 9-thiabicyclo[3.3.1]non-2-enes has been analysed. Both and n.m.r. data have been reported for a series of 9-borabicyclo[3.3.1]non-anes and their pyridine complexes. 

Jones JA (1997) Optimal sampling strategies for the measurement of relaxation times in proteins. J Magn Reson 126 283-286... 

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