Extreme-narrowing limit

As stated earlier, since tt]/ = yff2yr and since the gyromagnetic ratio of proton is about fourfold greater than that of carbon, then if C is observed and H is irradiated (expressed as C H ), at the extreme narrowing limit Ti, = 198.8% i.e., the C signal appears with a threefold enhancement of intensity due to the nOe effect. This is a very useful feature. For instance, in noise-decoupled C spectra in which C-H couplings are removed, the C signals appear with enhanced intensities due to nOe effects. 

While the assumption of an isotropic rotational motion is reasonable for low molecular weight chelates, macromolecules have anisotropic rotation involving internal motions. In the Lipari-Szabo approach, two kinds of motion are assumed to affect relaxation a rapid, local motion, which lies in the extreme narrowing limit and a slower, global motion (86,87). Provided they are statistically independent and the global motion is isotropic, the reduced spectral density function can be written as ... 

The cross-relaxation in the rotating frame (dashed lines in fig. 1) mono-tonically increases with correlation time. For looTc < 0.382 two crossrelaxation rates, cr" and (T have similar dependence on correlation time, and at the limit when loqTc —> 0, the extreme narrowing limit, their ratio... 

Since 13C relaxation of polymers has already been reviewed (Schaefer, 1974) we can restrict ourselves here to a few general remarks and to more recent papers in this field. Although i3C relaxation in polymers is generally dominated by dipolar relaxation, the NOE factors do not reach the maximum of tj = 2 commonly observed for small molecules. This is due to the fact that polymers usually do not satisfy the extreme narrowing limit only for which this NOE maximum is valid. In this case it is helpful to compare H relaxation data (Cutnell and Glasel, 1977). 

For small or medium-sized molecules undergoing rotational motion in the extreme narrowing limit, the spectral density is field-independent and the n.0.e. 

For a rigid spherical or nearly spherical molecule undergoing diffusive rotational motion in the extreme narrowing limit, a single correlation time, given by Eq. 16, is adequate to describe the overall motion. Equation 16 can be modified to take into account intramolecular interactions from other protons attached to other carbons in the molecule. Assuming that tc is the correlation time for each such... 

By substituting the expressions for spectral densities in Eq. 8.11, we obtain an equation that is algebraically cumbersone in general but that can be simplified in either of two regimes (1) homonuclear spins (I = S) or (2) rapid tumbling (extreme narrowing limit). 

As with dipolar relaxation, the interaction energy for CSA (in this case from Eqs. 6.3 and 7.17) enters quadratically, leading to the following expression for an axially symmetric shielding tensor in the extreme narrowing limit ... 

Take [Pg.225]

The spin-lattice relaxation times Ti due to the quadrupole interaction mechanism can be ctmverted to die rotational correlation times T2R ui a simple maimer. In the extreme narrowing limit attained by nqnd molecular rotational motions, the spin-lattice relaxation rate 1/Tj forthe m nucleus with the spin 1=1 is expressed by... 

We have applied the discrete-exchange model to these data. An exchange between Na ion under a particular slow-motion condition and in the extreme narrowing limit is assumed. Transverse relaxation time and diffusion coefficient are written as follows ... 

In the extreme narrowing limit (See Appendix A.2), the expressions for the relaxation times are very simple. The coefficients in front of the spectral densities vary depending on whether the interactions are between equivalent or nonequivalent spins [6]. 

The quadrupole tensor is proportional to the 2nd rank spin tensor of a single spin, and an expression for the relaxation time in the extreme narrowing limit is easily derived (See Appendix A.2). 

For small molecules (the extreme narrowing limit), the maximum increment in intensity Hmax Tirr/ Tobs SO that an initial intensity of unity (Iq = 1.0) increases up to (1 + irimax)- [In our example, A was irradiated ( irr ) and X observed ( obs ).] The maximum enhanced intensity, obtained by rearrangement of eq. 5-4, is given by... 

Second, in small molecules, the NOE builds up slowly and attains a theoretical maximum of only 50%, as noted earlier in the ID context. (See Section 5-4 and Appendix 5.) Because a single proton may be relaxed by several neighboring protons, the actual maximum normally is much less than 50%. (Of course, the same problem exists in the ID NOE experiment.) Moreover, as the molecular size increases and behavior departs from the extreme narrowing limit, the maximum NOE decreases to zero and becomes negative. Thus, particularly for medium-sized molecules, the NOESY experiment may fail. For larger molecules, whose relaxation is dominated by the Wq term, not only is the maximum NOE —100% rather than +50%, but also the NOE buildup occurs more rapidly. The NOESY experiment thus has been of particular utility in the analysis of the structure and conformation of large molecules such as proteins and polynucleotides. 

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