Two-site jump

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. 

The temperature dependent T data are shown in Fig. 9. 7j values decrease from 28 ms at 21°C with increasing temperature, and show a minimum of 6.4 ms at 80° C. These results indicate the presence of the motion with a Larmor frequency of 30 MHz at this temperature. This minimum was found to be attributed to the flipping motion of a phenyl ring from the result of our other experiments discussed in later section.13 The jump rates of the flipping motion were estimated with a two-site jump model that a C-2H bond jumps between two equivalent sites separated by 180°, and that the angle made by the C-2H bond and the rotational axis is 60°. The quadrupole coupling constant of 180 kHz and the asymmetry parameter approximated to zero were used in the calculation. The calculated values for fitting with the... 

The line shapes were calculated for the flipping motion with the two-site jump model described above, and the calculated spectra are shown in Fig. 11 for the higher temperature region. The experimental line shapes at 20 and 30° C are well reproduced showing the motional mode and rates obtained by T analysis are reasonable at least around these temperatures. Above 40°C the calculated line shapes are invariable and remain in the powder pattern undergoing a rapid flipping motion, while the experimental ones... 

Valuable information on the geometry of the proton motion is offered by the Q-dependence of the elastic incoherent intensity (EISF) (see Fig. 4.34). For two site jumps this intensity is described by ... 

Selective inversion recovery experiments i.e. only select frequencies within the powder pattern are excited, have also been performed on 2H for the purposes of studying molecular motion. Initial experiments were performed on deuterated dimethylsulfone (DMS) to demonstrate the utility of the experiment.46 Selective inversion recovery curves were fitted to a suitable motional model, a two-site jump model in the case of DMS, to yield the motional rates as a function of temperature. A significant feature of this work is that the activation energy for the motion so obtained differs markedly from that obtained from earlier 13C chemical shift anisotropy lineshape studies. 

In this case a two-axis jump process is studied. The two-site jump around the first axis is characterized by a rate constant k, whereas the rate constant for the jump around the second axis is AT During free precession only the + IQ coherence is of interest and therefore the matrix of interest is... 

Likewise the L-matrix for a two-axis jump process performing a three-site jump around one axis and followed by a two-site jump around the other axis having rate constants k2 and k3, respectively, is then defined as... 

The example illustrates, for example, two rotating methyl groups undergoing a two-site jump—as in dimethyl sulfone. Grouping the six deuterons as 1,2 and 3 on the first carbon and 4,5, and 6 on the second carbon, the Aj matrices are calculated for the combined rotation in the three-site jump and the two-site jump. 

Computation times for 14N FIDs employing parameter set P3 and A —10n Hz varied between 2.41 min (two-site jump, QCPMG) and 12.6 min (six-site jump, QCPMG) for the calculation of a static QCPMG FIDs. The computation time for an MAS FID with ideal excitation was 14 min for a two-site jump and 83 min for a six-site jump. In these calculations, the density operator was set to — Iy when acquisition begins. Calculations of the rotor-synchronized 14N SQ- and DQ-FIDs corresponding to parameter set P7 required 20 and 32 min, respectively. 

Figure 1 Simulated MAS (rows Ml and M2) and QCPMG (rows Q1 and Q2) spectra corresponding to the static limit (fc=10-9 Hz) of a two-site jump process corresponding to parameter sets P1-P5 in Table 1. In rows Q1 and Ml, the Hamiltonian includes the only HQ(1) whereas both HQ(1) and Hq(2) are included for the spectra in rows Q2 and M2. Gaussian line broadenings of 30 (A), 50 (B) or 75 (C-E) Hz were applied prior to Fourier transformation.

Figure 2 Simulated 14N (43.34 MHz) QCPMG spectra corresponding to a two-site jump process (column A) or a six-site jump process (column C) and MAS spectra corresponding to a two-site jump process (column B) or a six-site jump process (column D). All simulations employed parameter set P3 in Table 1. The logarithm of the rate constant k is indicated at each row of spectra.

Figure 3 Maximum intensity as a function of log(fc) for simulated 14N (43.34 MHz) spectra using parameter set P3. The intensity profile for a two-site jump is shown by a solid line for the MAS experiments and by squares for the QCPMG experiments. For the six-site jump process, the triangles correspond to the intensity profile for the MAS experiment and the filled circles for the QCPMG experiment.

Figure 9 Maximum intensity as a function of log(fc) for simulated MAS spectra of a two-site jump process for 133Cs (parameter set P9, solid line), 23Na (parameter set P10, solid triangles) and 39K (parameter set Pll, solid squares).

This is in good agreement with the intensity profiles in Figure 12B that clearly shows that the intensity in the fast limit increases when the order of the jump process is increased. In the regime k=103-105 Hz, it is noted that the intensity of the two-site jump is not as heavily reduced as in the higher order jump processes. 

Figure 11 (A) Simulated 39K (23.325 MHz) QCPMG spectra employing parameter set Plla corresponding to a two-site jump (column A), three-site...

Figure 12 Maximum intensity as a function of log(fc) for simulated 39K spectra shown in Figures 10 and 11. (A) The intensity profiles for a two-site jump using the QE (solid triangles), the QCPMG (solid line), the QCPMG-MAS (solid circles) and the single-pulse MAS (solid squares) experiments. (B) The intensity profiles of the QCPMG experiment for a two-site (solid line), three-site (solid triangles), four-site (solid squares) and a six-site (solid circles) jump process.

The two-site jump model appears to work well for this polymer. However, it should be noted that the density of transitions is large here due to the larger spin quantum number of deuterium (/= 1), the fact that there are three of them in the isotopically substituted polymeric radical, and that the coupling constant for each deuterium is smaller by a factor of 6.4 compared to the protonated radical. Coupling these facts to the visual fitting process, these fits may not be unique. In fact, when the same model is applied to the temperature dependence of the protonated PMMA spectra (Fig. 14.2), reasonable visual fits could not be obtained with this model. Deuteration of the... 

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