Extreme narrowing condition

Small molecules in low viscosity solutions have, typically, rotational correlation times of a few tens of picoseconds, which means that the extreme narrowing conditions usually prevail. As a consequence, the interpretation of certain relaxation parameters, such as carbon-13 and NOE for proton-bearing carbons, is very simple. Basically, tlie DCC for a directly bonded CH pair can be assumed to be known and the experiments yield a value of the correlation time, t. One interesting application of the measurement of is to follow its variation with the site in the molecule (motional anisotropy), with temperature (the correlation... 

Furthermore, as will be seen later, under the extreme, narrowing condition, and for isotropic motion, the following simple relationships hold. 

If we assume that the extreme narrowing condition exists, then the following simpler expression applies ... 

When the condition co x 1 is fulfilled (and this is the case for fast motions since the NMR measurement frequency is lower than 10 Hz), the spectral density becomes frequency independent. In such cases, one says that extreme narrowing conditions prevail. 

Fig. 9. Same as in figure 8, but for the evolution with the static magnetic field Bq (or, equivalently, with the measurement frequency Vq). For = 1 ps, extreme narrowing conditions are seen to be fulfilled.

As immediate consequences, we can state that (i) Ri and R2 vary according to the square of the static magnetic field. This contribution can therefore be sorted out by experiments performed at different magnetic fields and (ii) under extreme narrowing conditions ... 

Bo is the measurement frequency. Rapid exchange between the different fractions is assumed the bulk, water at the protein surface (s) and interior water molecules, buried in the protein and responsible for dispersion (i). In fact, protons from the protein surface exchanging with water lead to dispersion as well and should fall into this category Bulk and s are relevant to extreme narrowing conditions and cannot be separated unless additional data or estimations are available (for instance, an estimation of fg from some knowledge of the protein surface). As far as quadrupolar nuclei are concerned (i.e., and O), dispersion of Rj is relevant of Eqs. (62) and (63) (this evolves according to a Lorentzian function as in Fig. 9) and yield information about the number of water molecules inside the protein and about the protein dynamics (sensed by the buried water molecules). Two important points must be noted about Eqs. (62) and (63). First, the effective correlation time Tc is composed of the protein rotational correlation time and of the residence time iw at the hydration site so that... 

The assumption of a single electron spin and a single T2 holds usually for S = 1/2 and for S > 1 in certain limits. Let us assume that the instantaneous distortions of the solvation sphere of the ion result in a transient ZFS and that the time-dependence of the transient ZFS can be described by the pseudorotation model, with the magnitude of the transient ZFS equal to At and the correlation time t . The simple picture of electron relaxation for S = 1 is valid if the Redfield condition (Att <5c 1) applies. Under the extreme narrowing conditions ((Os v 1), the longitudinal and transverse electron spin relaxation rates are equal to each other and to the low-field limit rate Tgo, occurring in Eqs. (14) and (15). The low field-limit rate is then given by (27,86) ... 

Thus, if Pi is known, then the measurement of 77 provides the maximum value (minimum for 71 < 0) and can be used to determine the dipole-dipole contribution to the total relaxation rate ... 

In the case of free water, the extreme narrowing condition is fulfilled and thus Rij = Rij = Rj. Thus, from Equation 4, we can write... 

Interaction Relaxation rate (extreme narrowing condition) Remark... 

At the other limit of correlation times, Eqs. 8.6 and 8.9 show that for small tc, the denominators approach unity, and T2 = T,.The region of Tc< 1 /(o0 is often called the extreme narrowing condition. Note that we are considering here only dipolar interactions. Other relaxation mechanisms discussed subsequently may cause T2 to be smaller than 7), even under the extreme narrowing condition. 

Traditional ways of probing dynamics in liquids include the measurement of the spin-lattice (Ti) and spin-spin relaxation time (T2). Under extreme narrowing conditions 1), i.e., at high temperatures in the fluid liquid, both 7) and T2... 

The occurrence of a negative NOE when both nuclei have magnetogyric ratios of the same sign can be attributed to a breakdown of the extreme narrowing condition so that equation (16) is no longer valid. (222-227)... 

Thus when (cua + (Ux)Tr is not much less than 1, where is the molecular rotational correlation time, the potential maximum enhancement is not attained and it is not possible directly to use the size of the observed NOE to deduce the relative contribution of the dipole-dipole mechanism to the overall longitudinal relaxation of the observed nucleus. (228) The extreme narrowing condition is more stringent at higher measuring fields, but even for protons at say 300 MHz it is easily fulfilled (229) by small molecules for which Tr 10 -10 s. However, with large molecules such as globular... 

Fairly recently it has been pointed out that, even when the extreme narrowing condition is satisfied, equation (16) may not be correct if cross-correlation of the H and relaxation processes is at all important. Consequently equation (19) should be used instead (cf. refs. 257 and 258) ... 

NMR relaxation method is a very powerful tool for the study of ionic interactions. Many nuclei studied are with spin quantum number I > 1/2 and their relaxations are governed by quadrupole interaction between the nuclear electric quadrupole moment and the electric field gradient (efg) at the nuclear site. Under extreme narrowing condition, the longitudinal... 

Where, 1/T2i and D are the observed values of 1/T2f0r IT2s 2nd diffusion coefficient, respectively, sj and Db are the transverse relaxation rate constants (S or S2) and the diffusion coefficient of Na ion under the slow-motion condition, respectively. S2 and Sj correspond to the I1/2X-1/2I coherence and the l-l/2x-3/2l, 13/2x1/21 coherences, respectively. Pb is the fraction of the Na ion under the slow-motion condition (1> Pb ), which is proportional to the fraction of agar. Sj g and 2re the relaxation rate constant and the diffusion coefficient of Na+ under the extreme narrowing condition, respectively. 

Chemical shift anisotropy is caused by the interaction of the nuclear spin with the field arising from the perturbation of the shells of molecular electrons by an external field. It is modulated by the rotational tumbling of molecules in liquids and solutions and the anisotropy in chemical shifts [4]. At the extreme narrowing conditions (See Appendix A.2), we have ... 

In this expression, y is the gyromagnetic ratio and is the root-mean-square average of the X component of the fluctuating local field. Note that wo replaces o) in the expression for the relaxation time, which, under the extreme-narrowing condition, becomes 27 [b2l] Tc-Thus, the relaxation time T decreases as Tc increases (i.e., as mobility decreases with, for example, larger molecules or lower temperatures). This regime is depicted for Tj at the left of Figure A5-2 note that Ti is independent of the resonance fi equency coq here. 

Under extreme narrowing conditions the molecular correlation time, rc is related to the nuclear resonance frequency by equation (23). These conditions are usually found in low viscosity solutions and within their... 

Magnetic multipoles of rank higher than one become active in spin systems with I > 5 and their contribution to relaxation depends on dynamics. The appearance of multipole terms complicates the relaxation description and supports the multiexponential behavior of relaxation. Nosel et al. presented the effects of high rank multipoles on lineshape and longitudinal relaxation of 7=3 systems. Results obtained from both numerical simulation and experimental data show that longitudinal and transverse relaxation are strongly influenced by these multipole terms, especially at lower temperatures where, due to molecular mobility, the extreme narrowing condition is not fulfilled. 

Small molecules in low viscosity solutions have, typically, rotational correlation times of a few tens of picoseconds, which means that the extreme narrowing conditions usually prevail. As a consequence, the interpretation of certain relaxation parameters, such as carbon-13 and NOE for proton-bearing carbons, is... 

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