Line width calculation, relaxation

Fig. 3.15, The CARS spectrum rotational width versus methane density for various values of parameter y (1) y = 0, (2) y = 0.3, (3) y = 0.5, (4) y = 0.7, (5) y = 0.75, (6) y = 0.9, (7) y = 0.95, (8) y = 1. Curves (4) and (6) are obtained by subtraction of the dephasing contribution from the line width calculated taking account of vibrational broadening. The other dependences are found assuming purely rotational broadening (vibrational relaxation neglected).

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. 

An accurate determination of the magnetic moment of La and also of the minor isotope La has only very recently (100) been reported, along with the ratio of the quadrupole moments for which Q( La)/Q( La) = 2-15 is calculated from the respective line-widths, assuming quadrupolar relaxation for either isotope. The latter is confirmed on the basis of the relative line-widths in H2O and D2O. In a 0-6 M solution of LaClj in H2O, Av ( La) = 157 Hz or Tj = 2 03 ms, somewhat less than what Reuben (183) reported from pulsed studies (2-73 ms for LaClj, 018m in H2O, 23°C). 

Proton-driven spin diffusion (see also Appendix A) is the classical spin-diffusion experiment for low abundant spins. The line width of the one- and zero-quantum lines of the S-spins are mainly determined by the heteronuclear dipolar couplings while the homonuclear I-spin dipolar coupling makes the broadening of the levels homogeneous. Suter and Ernst [12] calculated an approximate value for the zero-quantum relaxation time... 

Most workers using relaxation measurements have shown the existence of ion-ion interactions but have not attempted to calculate equilibrium constants. Nevertheless, observation of line widths especially for quadrupolar nuclei, provides valuable corroboration for ion association in many cases. The stability constants of several paramagnetic ions and weak ligands in aqueous solution have been calculated from solvent proton relaxation measurements. ... 

In retrospection it is all too easy for a contemporary reviewer to point out a number of mistakes in Myers work. First of all, the use of Eq. (5.47) to describe the excess line width is only valid when C << 1 i,e, when the exchange rate is very much slower than the relaxation rate in I3. If a two-site treatment is applicable to the system and if n << 1 Eqs. (5.35) or (5.39) should have been equally possible. In the latter case, the calculated rate constant should in fact have been I/T2 the relaxation rate of iodide in the I3... 

In absence of experimental data on the relaxation rate in sites (c) and (d) in Cl, these were calculated from estimated electric quadrupole coupling constants and rotational correlation times. The excess line width of the chloride NMR signal was experimentally observed to be directly proportional to the total concentration... 

The experimental line width data were analyzed in very much the same way as for the zinc-nucleotide diphosphate complexes. The concentrations of the different liganded species ZnL, ZnL and Zn(HL), where HL symbolizes a protonated ligand, were calculated and employed to derive values of the molar relaxivities v. The results are summarized in Table 8.2. 

In Table 8.4 we have collected values of Cl quadrupole coupling constants and correlation times for a number of protein-chloride complexes. In a few cases, has been obtained experimentally from a comparison of T and T2 or from the frequency dependence of relaxation. In order to make possible a comparison between different proteins we have also calculated from experimental line width data using correlation times estimated by means of the Debye-Stokes-Einstein relation for spherical molecules. We then proceeded similarly as described in Ref. [4Z1],... 

In normal field-modulated spectra the modulation is kept smaller than the line width, and in careful work the effect of the modulation on the line shape is considered. However, the sidebands at 100 kHz or less are almost always within the envelope of the line shape, except for very narrow lines. The fast modulation that is required to obtain a relaxation time-dependent response in modulation spectroscopy inherently has more widely spaced sidebands. For example, at 200 MHz modulation the sidebands are approximately 71 G away fi om the center band. The modulation index and the d.c. offset can be varied to obtain a wide range of modulation conditions cf. Haworth Richards, 1966 Losee et al., 1997 Reference Data for Radio Engineers, 1968). It should be noted that the finite bandwidth caused by resonator Q can attenuate effects of sidebands and resultant signal amplitude as the modulation frequency is increased. This calculation is beyond the scope of this chapter but needs to be performed for a given experimental arrangement in order to be able to use the predictions of this chapter to actually measure relaxation times. 

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