Isotropic motion

In order to generate a set of calculated structure factors Fc(Q) from a set of coordinates, it is necessary to introduce a model for the time variation of the electron density. The usual assumptions in macromolecular crystallography include harmonic isotropic motion of the atoms and in addition, the molecular scattering factor is expressed as a superposition of atomic scattering factors. With these assumptions the calculated structure factor (equation III.2) is given by.27... 

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

Given the specific, internuclear dipole-dipole contribution terms, p,y, or the cross-relaxation terms, determined by the methods just described, internuclear distances, r , can be calculated according to Eq. 30, assuming isotropic motion in the extreme narrowing region. The values for T<.(y) can be readily estimated from carbon-13 or deuterium spin-lattice relaxation-times. For most organic molecules in solution, carbon-13 / , values conveniently provide the motional information necessary, and, hence, the type of relaxation model to be used, for a pertinent description of molecular reorientations. A prerequisite to this treatment is the assumption that interproton vectors and C- H vectors are characterized by the same rotational correlation-time. For rotational isotropic motion, internuclear distances can be compared according to... 

The 13C NMR sensitivity can sometimes be a problem, but for the kind of samples studied here the effective concentration of monomer units is several molar which does not place excessive demands on present Fourier transform NMR spectrometers. In addition to the sensitivity of the chemical shift to structure (9), the relaxation of protonated carbons is dominated by dipole-dipole interaction with the attached proton (9). The dependence of the relaxation parameters T, or spin-lattice, and Tor spin-spin, on isotropic motional correlation time for a C-H unit is shown schematically in Figure 1. The T1 can be determined by standard pulse techniques (9), while the linewidth at half-height is often related to the T2. Another parameter which is related to the correlation time is the nuclear Overhauser enhancement factor, q. The value of this factor for 13C coupled to protons, varies from about 2 at short correlation times to 0.1 at long correlation... 

Completely isotropic motion of the probe in the interlayer would produce AB= A and 9 = 54.7°, the "magic angle". Thus, hectorite has the greatest anisotropy of motion, with a tendency of the z-axis of the nitroxide to tilt toward the z axis. This orientation may facilitate direct surface-methyl group interaction as well as close approach of the -NH3+ group to surface charge sites. 

For isotropic motions in an isotropic medium, the values of the instantaneous and steady-state emission anisotropies are linked to the rotational diffusion coefficient Dr by the following relations (see Chapter 5) ... 

In the standard mathematical expressions for the contribution of quadrupolar relaxation to the relaxation rates of the quadrupolar nucleus (in nuclear magnetic resonance), rapid isotropic motion is assumed to occur. This behavior, in most cases, will not be true in the solid or liquid crystalline state ". ... 

The functions /"(tc) and f Tc) depend on the type of molecular motion, and for a rigid body with isotropic motion, they become [9, 10] ... 

Fig. 1. Contour plot of the dependence of the cross-relaxation rate in the laboratory frame, a" (solid line), and rotating frame, cr (dashed line), on the interproton distance, r, and correlation time, Tc. Rigid body isotropic motion is assumed (eqs. (1) and (2)). Top panel shows the dependence of the ratio of the two cross-relaxation rates on correlation time, a is always positive, whereas cr" is positive for tuoTc < v/5/2 and negative for woTi < /5/2.

Cross-relaxation rates in the two frames, o" and o, can provide estimates of r and /(tc) for the observed spin pair [9-11]. Because the ratio of crossrelaxation rates from the two frames, 5(tc) (eq. (3)), does not depend on the distance, it can be used for determination of the correlation time. For the rigid body isotropic motion, the ratio is ... 

Figure 14. Relation between correlation time and relaxation times of protons of water in isotropic motion at 200 MHz spectrometer.

The appearance of a sharp central peak in the deuteron NMR spectrum for unaligned lyotropic liquid crystalline samples has been observed by several authors (30, 33, 34). This has been interpreted in terms of phase inhomogenities (35) or isotropic motion (36). However, recently Wennerstrom et al. showed for a lamellar amphiphile-water mesophase that this could be referred to double quantum transitions (37). It is expected that double quantum transition peaks appear in NMR spectra... 

The shift in the two tensors is expected to be effective for carbohydrate molecules bearing a number of polar groups and hydrogen-bonding centers. Hence, serious difficulty for quantitative analysis may arise if the molecule does not contain three or more nonequivalent C—H vectors that relax predominantly via the overall motion. If this fact is ignored, qualitative treatment may lead to an erroneous motional description. Thus, one should be very cautious in interpreting the relaxation data for overall motion, especially when discrepancies well outside the experimental error are observed for the T, values. When the relaxation times are nearly similar and within the experimental error, isotropic motion may be considered as a first approximation to the problem. 

The effective correlation times for an approximately isotropic motion, tr, ranged from 40.3 ps in methanol to 100.7 ps in acetic acid for 5a, and from 61.6 ps to 180.1 ps for 5b in the same solvents. Neither solvent viscosity nor dielectric constant bore any direct relationship to the correlation times found from the overall motion, and attempts to correlate relaxation data with parameters (other than dielectric constant) that reflect solvent polarity, such as Kosover Z-values, Win-stein y-values, and the like, were unsuccessful.90 Based on the maximum allowed error of 13% in the tr values derived from the propagation of the experimental error in the measured T, values, the rate of the overall motion for either 5a or 5b in these solvents followed the order methanol N,N-dimethylformamide d2o < pyridine < dimethyl sulfoxide. This sequence appears to reflect both the solvent viscosity and the molecular weight of the solvated species. On this basis, and assuming that each hydroxyl group is hydrogen-bonded to two molecules of the solvent,137 the molecular weights of the solvated species are as follows in methanol 256, N,N-dimethylformamide 364, water 144, pyridine 496, and dimethyl sulfoxide 312. 

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