Relaxation time integral times

Xl is known as the longitudinal relaxation time. Integration of equation (4.5.28) leads to the result that... 

T(f) corresponds to the actual temperature at the time t, At is the integration time step, and the relaxation time represents the strength of the coupling (smaller values mean stronger coupling to the bafli). If the coupling is too strong (r smaller... 

Finally, if a very large number of relaxation times is present, the summation in Eq. (3.61) can be replaced by an integral ... 

It should be observed that Eq. (3.102) may be viewed as a distribution function for relaxation times. In fact, if N,. is large enougli, integer increments in p may be approximated as continuous p values. This makes Tp continuous also. The significance of this is that Eq.(3.90) can be written as an integral in analogy with (3.62) if p is continuous ... 

The relative number of equivalent nuclei associated with each chemical shift is obtained from the integrated spectmm by normalizing the areas so that the area corresponding to the smallest peak in the spectmm is defined as 1. This relation may not be exactly correct in ft experiments where signals may be affected by significant differences in relaxation times for nuclei in different environments. 

In this expression. Ait is the size of the integration time step, Xj is a characteristic relaxation time, and T is the instantaneous temperature. In the simulation of water, they found a relaxation time of Xj = 0.4 ps to be appropriate. However, this method does not correspond exactly to the canonical ensemble. 

If the velocity relaxation time is short relative to the integration timestep, the following result is obtained ... 

Reactive trajectories, 43-44,45, 88,90-92,215 downhill trajectories, 90,91 velocity of, 90 Relaxation processes, 122 Relaxation times, 122 Reorganization energy, 92,227 Resonance integral, 10 Resonance structures, 58,143 for amide hydrolysis, 174,175 covalent bonding arrangement for, 84 for Cys-His proton transfer in papain, 141 for general acid catalysis, 160,161 for phosphodiester hydrolysis, 191-195,... 

This is the relaxation time of the polymer oxidation under electro-chemically stimulated conformational relaxation control. So features concerning both electrochemistry and polymer science are integrated in a single equation defining a temporal magnitude for electrochemical oxidation as a function of the energetic terms acting on this oxidation. A theoretical development similar to the one performed for the Butler-Volmer equation yields... 

It is once again the non-Markovian equation in a sense that the relaxation rates are time-dependent. They become constant for the times which are long enough to extend the integration over t to 00. This leads... 

It is taken into account that relaxation of different Cartesian components of a>Xl proceeds independently, and (coa,)=0. One can easily see that every cumulant in (Al.lb) when integrated yields the corresponding power of Ta i.e., of rotational relaxation time of the oc th component. Therefore,... 

Luminescence lifetime spectroscopy. In addition to the nanosecond lifetime measurements that are now rather routine, lifetime measurements on a femtosecond time scale are being attained with the intensity correlation method (124), which is an indirect technique for investigating the dynamics of excited states in the time frame of the laser pulse itself. The sample is excited with two laser pulse trains of equal amplitude and frequencies nl and n2 and the time-integrated luminescence at the difference frequency (nl - n2 ) is measured as a function of the relative pulse delay. Hochstrasser (125) has measured inertial motions of rotating molecules in condensed phases on time scales shorter than the collision time, allowing insight into relaxation processes following molecular collisions. 

This integral equation can be solved exactly with the stress relaxation time approximated as... 

In the spectrum of fully reductively [ C] methylated glycophorin A, the resonance at 42.8 p.p.m. must correspond to the N, N -di[ C]methylated, N-terminal amino acid residue. The ratio of the integrated intensities of the N, N -di[ C]methylLeu resonance to the N, N -di[ C]methyllysine resonances is 5 1, as expected. The integration values determined were valid, because the recycle times of spectra in Figs. 3B, 3C, and 3D were twice the spin-lattice relaxation-times (Tj values) of those of the di[ C]methyl carbon atoms, and also because the n.O.e. values of the N, N -di[ C]methyl and N, N -di[ C]methyl carbon atoms were equivalent. ... 

We carried out two sets of experiments in which we set the pulse angle first at 90°, then at 30°. Using these two values we then varied the relaxation delay. Since the greatest difference in the relaxation times is that between the OH proton and the aromatic protons, we show in Fig. 11 the comparison between the integration values of the aromatic protons (set equal to 2.0) and of the OH proton for 90° pulses and for 30° pulses. The values approach each other with a relaxation delay of 10 sec and are virtually equal for a delay of 25 sec, but the 90° pulses give values which are completely wrong if a conventional delay of 1-2 sec is used On the other hand, the error is quite low if the delay is set at 2 sec and the pulse length is 30°. 

For those new to the field of fluorine NMR, there are a number of convenient aspects about fluorine NMR that make the transition from proton NMR to fluorine NMR relatively easy. With a nuclear spin of j and having almost equal sensitivity to hydrogen along with sufficiently long relaxation times to provide reliable integration values, 19F nuclei... 

The diverging longest relaxation time, Eq. 1-6, sets the upper limit of the integral. The solid (gel) contribution is represented by Ge. The crossover to any specific short-time behavior for A < A0 is neglected here, since we are mostly concerned with the long-time behavior. 

The divergence of the longest relaxation time does not perturb the measurement. In comparison, steady state properties (the steady shear viscosity, for instance) would probe an integral over all relaxation modes and, hence, fail near the gel point. 

The first integral denotes the rest period, — oo < t < 0, where the strain rate is zero. The second integral contains a relaxation function which we chose very broad, including relaxation times much larger than the period In/iD. Integration and quantitative analysis clearly showed (without presenting the detailed figures here) that the effect of the start-up from rest is already very small after one cycle... 

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