Relaxation times calculations

There is a second relaxation process, called spin-spin (or transverse) relaxation, at a rate controlled by the spin-spin relaxation time T2. It governs the evolution of the xy magnetisation toward its equilibrium value, which is zero. In the fluid state with fast motion and extreme narrowing 7) and T2 are equal in the solid state with slow motion and full line broadening T2 becomes much shorter than 7). The so-called 180° pulse which inverts the spin population present immediately prior to the pulse is important for the accurate determination of T and the true T2 value. The spin-spin relaxation time calculated from the experimental line widths is called T2 the ideal NMR line shape is Lorentzian and its FWHH is controlled by T2. Unlike chemical shifts and spin-spin coupling constants, relaxation times are not directly related to molecular structure, but depend on molecular mobility. 

Table I shows the values of the relaxation time calculated using Eqs. (249) and (250). Both the inertial time and the long time decrease with increasing density. This is in agreement with the trend of the curves in Fig. 6. Indeed, the actual estimates of the relaxation times in Table I are in semiquantitative agreement with the respective boundaries of the plateaux in Fig. 6. The estimate of Xiong, the upper limit on x that may be used in the present theory, is perhaps a little conservative.

Fig. 10.25. 13C CPMAS and 19F MAS SSNMR spectra of the drug formulation (top) and the physical mixture of API with excipients (bottom). Shown above each peak are the relaxation times calculated from the 13C detected...

Cases with the same K0, K, combination tend to the same equilibrium state. (2) Points mark the relaxation time calculated from Eq. (13). 

Note Points mark the relaxation time calculated according to Eq. (13). 

The origin of the n2 measured using the 10 ns pulses could be electronic or molecular rotation. These can be distinguished by measuring the ratio of the critical power for self-focusing for linear and circular polarised light. The observed ratio of 2.1 is consistent with a molecular rotation (11-13.161 and relates to the anisotropic polarisability of the molecule. The rotational relaxation time, calculated from the Debye formula (H), is about 0.5-2 ns, consistent with these results. 

I) The H-bond mean lifetime, thb> from mf-18- (2) The probability that an hydrogen bond is randomly intact, pg, from Eq. (S.2). (3) The relaxation time calculated from Eq. (6.1) with conditions (6.2). (4) Measured relaxation time r, from refs. 52 and S3 (+) tg, values foreseen by our model using the parameters in columns 5-7, which list the rotational diffusion parameters utilized for a more accurate calculation of dielectric behavior. (8) The principal relaxation band amplitude Ag. (9) and (10) The characteristic time l/Xj and the amplitude A of the only other significant band, as calculated from Eqs. (6.1) and (6.4), respectively. 

H-bond lifetimes tiq from ref. 46. (2) Probability pg calculated ftdlowing the procedure in Section V.A. (3) Relaxation times calculated from Eq. (6.1) with the same conditions (6.2) for H2O and D2O. (4) Experimental values of dielectric relaxation times from ref. 75 ( ) at 10°C (t) extrapolated. (5) Self-diffiision coefficients calculated from Eq. (7.5) assuming for D2O the same Dj-(ij/) used for H2O. (6) Self-diffusion experimental data from ref. 63. 

The disaccharides such as trehalose, maltose, and leucrose are useful in biopreservation and life science, and the polysaccharides are important in other areas. On elevating pressure, fructose, D-ribose, 2-deoxy-D-ribose , and leucrose have a secondary relaxation shifting to lower frequencies with applied pressures, mimicking the behavior of the a-relaxation. The one in leucrose is sensitive to the thermodynamic history of measurements. There is also good agreement of the observed relaxation time of the secondary relaxation with the primitive relaxation time calculated from the Coupling Model for D-ribose and 2-deoxy-D-ribose. These results indicate that this secondary relaxation in the mono- and di-saccharides is connected to the a-relaxation in the same way as in ordinary glassformers, and hence it is the JG p-relaxation of... 

FIGURE 5.22 The segmental relaxation times for PEO neat (bold solid line) and in blends with PMMA (dashed lines) containing 3% to 30% PEO (from top to bottom) from deuteron NMR measurements. Fits to the data by the VET law given by Lutz et al. (2003) are shown but not the data themselves. Also shown is the independent relaxation time for PEO (dotted Une, using n = 0.5), which lies close to the characteristic time, tc = 2 ps. The most probable relaxation times calculated for a number of temperatures by the CM equation for concentrations of PEO at 3% (diamonds, n = 0.76), 10% (triangles, n = 0.75), and 30% (squares, n = 0.715), respectively. 

From a practical point of view, all the above expressions for the orientation autocorrelation function lead to very similar numerical results for NMR spin-lattice relaxation time () calculations. 

A straightforward way to assess the relaxadon time from easily obtainable DSC data is outlined in the literature (Mao et al. 2006a). Despite the improvement of the AG equation over the KWW equation, relaxation times calculated from the AG equation also did not show the capability of predicting the physical stability for a set of drugs. 

Fig. 5. Temperature dependence of effective spin-lattice Tj and spin-spin Ti relaxation times calculated from Eqs. 8 as well as of intrachain and interchain diffusion rates calculated from Eq. 10 for neutral solitons diffusing in trans-PA along and between randomly oriented chains and chains oriented with their c-axis with respect to an external magnetic field by / = 90°, 60°, 30°, and 0°. [From Refs. 12(b) and 61 with permission.]...

This observation is well known and in general there are two explanations. The first one is based on stating that a viscoelastic measurement is a macroscopic displacement while the dielectric measurement is a microscopic displacement. For this reason, several thousand microscopic displacements (dielectric) are required to make one macroscopic displacement (viscoelastic). Another method of comparison has been suggested and that is to compare the dielectric relaxation time to the viscoelastic relaxation time calculated form the modulus related to the compliance through G (w) = It is the view of the present authors that such... 

C.3.1. Cubic Symmetry When K- Is Not Negligible C.3.2. Mixing of Cubic and Uniaxial Symmetries C.4. Conclusions Relaxation Time Calculation... 

D.2. Relaxation Time Calculation Neel s Model D.3. Relaxation Time Calculation Brown s Model D.3.1. Hypothesis Discussion... 

FIG. 12-23. forage and loss shear moduli of a polyfdimethyl siloxane) with number-average molecular weight Mn = 16,000, plotted logarithmically against frequency v with reduced scales, r i is the terminal relaxation time calculated as it]oM /ir pRT. Solid curves, experimental dashed curves, from modified Rouse theory for most probable distribution of molecular weights. (Barlow, Harrison, and Lamb.3 )... 

Doi and Edwards analysed the described disentangling process of the primitive chain in more detail. As in the case of the Rouse-motion, the dynamics of the disentangling process can also be represented as a superposition of independent modes. Again, only one time constant, the disentangling time Td, is included, and it sets the time scale for the complete process. In the Doi-Edwards treatment, ra is identified with the longest relaxation time. Calculations result in an expression for the time dependent shear modulus in the terminal flow region. It has the form... 

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