Jump angle

When a sodium cation jumps between two sites with the same EFG tensor components and the same residence times on both positions, but different orientations, then the two-site exchange process can be described by tensor averaging of the two positions. Depending on the jump angle of the tensor, the motionally averaged principal components, Vxx, Vyy, and Vzz, may eventually have to be redefined in ordering to fulfill the requirement in parenthesis in (4). 

Figure 8 a shows the motionally averaged quadrupole coupling constant, (Cq)/Cq, and asymmetry parameter, ( ), for a two-site jump between axially symmetric equivalent sites. At jump angles of 70° and 109° the principal components (V, Vyy, Vzz) have to be rearranged in order, which leads to the discontinuities in the curve shapes of Fig. 8a. 

Due to the tetrahedral arrangement of the windows within the cage, tetrahedral jumps of the cations are feasible. This jump model is confirmed by the averaged parameters obtained by lineshape simulations of the broad lines at 295 and 813 K. Cq is reduced at 813 K to about half of its original value at 295 K, and the averaged asymmetry parameter, (77), is 1. Especially (77) is a very sensitive parameter on jump angle due to the steepness of the curve in Fig. 8a. 

Thus, the stimulated echo technique allows one to estimate the elementary angular jump angle a/. For selected aj, the crossover in the stimulated echo correlation function, observed between the tp 0 to tp > oo limits in a simulation, is displayed in Fig. 8. 

Figure 8. Double logarithmic plot of the ratio of the normalized correlation time i[tp)/xi as a function of the normalized evolution time tp, determined from random walk simulations for different jump angle aj. (From Ref. 12 cf. also Ref. 189.)...

While the advantages of the stimulated-echo technique lie in the resolution of small jump angles, measurement of two-dimensional (2D) NMR spectra is best suited to the study of large angular displacements, as typically found in crystalline rotator phases, for example. They lead to characteristic off-diagonal patterns [cf. 

Fig. 41 (top)], which are straightforwardly related to the jump angles [11,72]. In contrast, the dynamics involved in the a-process results in complete randomization of the molecular orientation so that the NMR frequencies at tm — 0 and tm xa> respectively, are uncorrelated and a box-like 2D spectrum with off-diagonal intensity spread all over the frequency plane is observed [cf. Fig. 41 (bottom)]. Information about the course of the reorientation process is available when 2D spectra are measured for different mixing times tm [72,94]. 

Fig. 5. Time evolutions of doubly normalized CODEX E(SNt,Jm)/E(J, ) curves, showing clear differences between simple jumps and diffusive motion (for a uniaxial interaction, tj=0). As indicated at the top, the line thickness decreases with increasing tjrc (=0.1,0.3, 1.0, 3.0, and 10). (a) Two- or three-site jumps, with a 109° (or equivalently 71°) jump angle. The shape of the normalized curves is the same at all mixing times, (b) Uniaxial rotational diffusion on a cone with an apex angle of 141° (or equivalently 2 x 109.5°). Characteristically for diffusive motions, the shape of the curves changes with tm.

Fig. 12. (a) Schematic representation of DQ exchange experiments for elucidation of slow molecular dynamics.45 (b) Calculated (black) and observed (gray) 13C-I3C DQ MAS NMR sideband patterns of crystalline poly(ethylene) yielding the dipolar coupling strength and DQ-DQ exchange sideband patterns for different jump angles. For details, see ref. 45. 

The evolution of the many-molecule dynamics, with more and more units participating in the motion with increasing time, is mirrored directly in colloidal suspensions of particles using confocal microscopy [213]. The correlation function of the dynamically heterogeneous a-relaxation is stretched over more decades of time than the linear exponential Debye relaxation function as a consequence of the intermolecularly cooperative dynamics. Other multidimensional NMR experiments [226] have shown that molecular reorientation in the heterogeneous a-relaxation occurs by relatively small jump angles, conceptually simlar to the primitive relaxation or as found experimentally for the JG relaxation [227]. 

Fig. 6. Theoretical integrated reduction factor of the powder spectrum (amplitude at the top of solid echo) for a two-site jump motion (Pa=Pb) as the function of jump frequency 1/Tc and for different jump angles 2/ . (Reprinted with permission from Ref. 112. Copyright 1990 American Chemical Society.)...

Fig. 9. Top H-2D exchange spectrum of dimethylsulfone. The inset shows the corresponding theoretical spectrum for a two-site jump with a jump angle of 106°. The angular information is encoded in the shape of the elliptical ridges perpendicular to the main-diagonal spectrum. Bottom Theoretical ridge pattern (contour plots) for different jump angles. (Adapted with permission from Refs. 138,140.)...

Fig. 11. Rotor-synchronized H-2D MAS exchange spectra of dimethylsulfone. (a), (b) and (c) are experimental and were recorded with z/r = 4.06 kHz at 30°C and nominal mixing times as indicated, (d), (e) and (f) are calculated using an axially symmetric quadrupole tensor with a jump angle of 108°. (Adapted with permission from Ref. 152.)...

This approach offers a detailed and model-free description of the distribution of jump-angles, and can cover an enormous range of timescales only limited by relaxation [1, 4, 61-64]. In the following section, both ID and 2D NMR lineshape analysis will be described. 

Four different models of molecular motion were in agreement with the jump angle determined by NMR. However, of these possible motions only one was in agreement with the dielectric relaxation results of Miyamoto et al. [69], This motion is defined by a dipole-moment transition and a conformational change (tg" tg <->g tg" t) yielding an effective dipole-moment reversal only along the chain axis and a reorientation angle of 113° for the C—bond directions. 

Another model of rotational reorientation is the jump-diffusion model first described by Ivanov (1964). In this model the molecule reorients by a series of discontinuous jumps (with an arbitrary distribution of jump angles). This should be contrasted with the Debye model, which involves infinitesimal jumps, and the Gordon model, which involves continuous free rotations between collisions. This model is probably applicable to the situation where the molecular orientation is frozen until a volume fluctuation occurs, at which time the molecular orientation jumps to a new frozen value. We present our own version of the jump model here. It is assumed that (a) the jump takes place instantaneously, (b) successive jumps are uncorrelated in time with an average time tv between jumps, and (c) the dihedral angle between the two planes defined by the orientation vector u in two successive jumps is randomized. 

Now if the distribution function of jump angles is very peaked for small angles, then we can expand Pi cos /) around / = 0. This gives... 

Figure 12.3. Top Jump model for the motion of the a-CD2 group in the crystalline phase bottom jump angle ( ) as a function of temperature for PD4S (squares) and PD5S (circles).

Figure 7 Two-dimensional exchange spectrum of deuterated dimethylsulfoxide. The strong diagonal ridge reflects molecules that have not reoriented during the mixing time, while the pattern of ellipses off the diagonal shows those that have. The ellipses arise due to the orientational dependence of the quadrupole interaction, the dominant anisotropy here. The distribution of jump angles is shown on the right, and sharply peaked at zero (static molecules) and 72°, the included angle of the C-D bonds as the entire molecule executes hops about its symmetry axis. With this type of experiment, slow to moderate dynamics of molecules and polymers can be followed in atomic level detail. (Reproduced with permission from Schmidt-Rohr K and Spiess HW (1994) Multidimensional Solid-Stale NMR and Polymers. London Academic Press.)...

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