Fast-motion limit

In impure metals, dislocation motion ocures in a stick-slip mode. Between impurities (or other point defects) slip occurs, that is, fast motion limited only by viscous drag. At impurities, which are usually bound internally and to the surrounding matrix by covalent bonds, dislocations get stuck. At low temperatures, they can only become freed by a quantum mechanical tunneling process driven by stress. Thus this part of the process is mechanically, not thermally, driven. The description of the tunneling rate has the form of Equation (4.3). Overall, the motion has two parts the viscous part and the tunneling part. 

At temperatures around 50-60°C the three-site jump model is not a good approximation to the multi-site jump model, because the motion is not sufficiently rapid to be in the fast motion limit. However, the calculated spectra are fairly fitted with the observed ones. This is because the calculated spectrum is a superposition of constituent spectra whose rates are spread over several orders, so that the resultant spectrum is governed by the constituent spectra in the fast and slow motion limits having greater intensity than that in the intermediate exchange regime. 

Finally, the pattern shown in Fig. 5c is in the fast motion limit, and is characteristic of acetone-d6 molecules undergoing rapid 2-fold re-orientations about the carbonyl bond ( 107 Hz). The measured values of Avxx, Av>7 and Avzz in this case were 7 000 Hz, 30 000 Hz and 37 000 Hz, respectively, which corresponds to r = 0.62. The fast motion limit was actually achieved between 40 and 50°C. 

R2M differs from R M principally for the first and last terms. It is well known in NMR that frequencies near zero contribute to R2 but not to R (first term). This term originates from a 7(0) term (Eq. (3.3)), as described in Section 3.2. The last term is present because R2M must also contain the probabilities of transitions connecting states with the same lz but different Sz. When cos and [Pg.92]

Some qualitative guidelines can be given to make an a priori estimate of the relative weight of dipolar, contact, and Curie relaxation contributions. Consider first the fast motion limit where Rim = Rim and none of the frequency-dependent terms is dispersed. The equations take the simple form already noted ... 

Outside the fast motion limit the relative weight of contact and dipolar interactions on R m and R2M may also be different. The following considerations are particularly relevant to proton relaxation. Curie contributions to / 2m can be sizable. By comparing Eqs. (3.20) and (3.27) on the one hand and Eq. (3.30) on the other, it can be noted that they contain terms of the type... 

Once the Curie contribution to R2M is estimated and subtracted, the contribution of contact and dipolar interactions can be estimated by examining the correlation time dependence of the paramagnetic relaxation depicted in Figs. 3.9 and 3.11. It appears that the maximum for R m occurs at dipolar term and at contact term. Taking for simplicity xf Ip = r °", this means that in the intermediate situation where ft>s T p > 1 > relative importance of the contact term is even smaller than that estimated in the fast motion limit. The equation for R2M has non-dispersive terms in both the dipolar and contact contributions (accounting for one-fifth and one-half of the total effect measured in the fast motion limit respectively), and therefore the conclusions drawn in the fast motion limit are still qualitatively correct. 

Fig. 7.10. Maximal intensity of the ROE as a function of (A) magnetic field for x, = 10 ns and (B) tr at a field of 800 MHz, for two hydrogen nuclei at 1.8 A distance. Curves (a), (b) and (c) refer to a field-independent R pM (Eq. (3.21)) of 0, 50 and 200 s-1, respectively. For R pM 0, the fast motion limit is the same as that of transient ROE the slow motion limit is positive and equal to 0.675.

The polymer then undergoes slower internal rotations, and the TREPR spectrum of the main-chain radical is broadened. The similarity in the spectra from PFOMA and PAMA suggests that the conformational mobility of the polymeric radical in solution plays a major role in the intensity and spectral shape of the TREPR signal from these polymeric radicals and that side chain size and structure can completely prevent access to the fast motion limit, at least at temperatures below 135°C. Higher temperatures are not currently available to us because our high temperature flow system in limited to a maximum reservoir temperature of 150°C, for safety reasons. 

Similar to fluorescence depolarization and NMR, two limiting cases exist in which the molecular motion becomes too slow or too fast to further effect the ESR lineshape (Fig. 8) (35). At the fast motion limit, one can observe a narrow triplet centered around the average g value igxx + gyy + giz with a distance between lines of aiso = Axx- -Ayy- -A2,z)l3, where gu and Ajj are principal values of the g-tensor and the hyperflne splitting tensor A, respectively. At the slow motion limit, which is also referred to as the rigid limit, the spectrum (shown in Fig. 8) is a simple superposition of spectra for all possible spatial orientations of the nitroxide with no evidence of any motional effects. Between these limits, the analysis of the ESR lineshape and spectral simulations, which are based on the Stochastic Liouville Equation, provide ample information on lipid/protein dynamics and ordering in the membrane (36). 

Thus, M2XC is the relaxation rate 1/ T2, which has been calculated in (3.5.6) by the BPP theory in the fast motion limit. In the slow motion limit of (3.5.6) only the spectral density (3.5.8a) at frequency zero needs to be considered, and... 

In the fast motion limit, the homogeneous decay is much slower than the inhomogeneous one, and the exponential relaxation factor can be neglected in (7.1.7). Assuming a Gaussian distribution of end-to-end vectors, the magnetization decay assumes the form [Sot2]... 

Fig. 3. Simulations of the lineshape expected for carbon 4 in Fig. 2 assuming phenyl ring flips in the fast-motion limit and a Gaussian distribution of flip angles centred at 180° with standard deviation a ...

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