Observed relaxation times

Measurements of relaxation times fall broadly into two classes, those which monitor the populations of some chosen states, and those which measure in some way the impedance of the system to the propagation of a thermal disturbance many laser experiments fall into the first class, whereas ultrasonic dispersion or shock-tube measurements fall into the second. Although artefacts can occur if unsuitable population v. time profiles are used [76.P3], there is, in general, no real difficulty in using equation (2.14) to obtain the vibrational relaxation rate we need not discuss this point further at the moment. Problems may well arise, though, in the determination of rotational relaxation rates in this way, as I will show. 

4 Notice also that in any practical set of normal modes there is a non-physical one (Mj in this case) which arises from the truncation of the levels and the orthogonality requirement on the vectors there is never, in practice, any significant flux associated with such a mode. 

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The disturbance thus decays always according to first-order kinetics, the observed relaxation time t being related to the rate constants Iclh and /chl... 

Predictions using the observed relaxation time spectrum at the gel point are consistent with further experimental observations. Such predictions require a constitutive equation, which now is available. Insertion of the CW spectrum, Eq. 1-5, into the equation for the stress, Eq. 3-1, results in the linear viscoelastic constitutive equation of critical gels, called the critical gel equation ... 

By comparison of the observed relaxation times with those observed for water solutions doped with the corresponding paramagnetic ion, these authors defined a catalyst accessibility which was taken to be a measure... 

A relaxation (and relaxation time) that contributes to the observed relaxation times (Ti and T2) in NMR experiments. Scalar coupling relaxation is the dominant relaxation process in P( O) NMR. 

Experimental results show reasonable agreement with the above equations even for deformations considerably larger than those for which the theory might be expected to apply. In practice, breakup occurs for E < 0.26 0.05 (R12), whereas observed relaxation times are longer than predicted from Eq. (12-50) (TIO). Experimentally no breakup occurs when /c > 3 instead a drop becomes aligned and elongated in the flow direction with its aspect... 

It is thus more difficult to investigate experimentally by acoustic methods, which will require very high fjp values at which classical dispersion and absorption must also be taken into account87. Theoretical treatment is also much less satisfactory molecules are usually distributed in a variety of rotational energy levels, so that a number of different transition probabilities have to be considered, and observed relaxation times usually represent averages over sets of transitions involving a range of /-states. 

There are several reports of T1 measurement techniques in specialized applications such as single line spectra, where the FID is directly integrated (Cantor and Jonas, 1976 Matson, 1977), a rather fast single scan FT method to observe relaxation times of unstable compounds (Kaptein et al., 1976), and a very special technique to measure extremely long 7j values of very weak signals such as that from 109Ag and other metal nuclei (Kronenbitter and Schwenk, 1977). 

In a single-domain particle of a-Fe203 the magnetization vector is held in the c-plane perpendicular to the c-axis by the magnetocrystalline field. Mossbauer studies use the 57Fe nucleus as the "observer to record when the relaxation time t becomes shorter than the period for precession of the nuclear spin about the direction of the effective field. Substitution into the equation for the Larmor frequency, or observer relaxation time, with an expression for the frequency factor proportional to the specific volume and anisotropy constant of the oxide gave (26, 27) the relationship ... 

The common occurrence is that t12 tx t2, in which the lowest-energy mode provides the channel through which energy is taken up from, and returned to, translational degtees of freedom. In this case, the single observed relaxation time t is given by... 

It has been demonstrated that the model works well with a wide range of heterogeneous environments including pores, channels, surfaces, and gels and a collection of hosting materials such as silica, alumina, titania, and clays [63-66]. As the exact molar fraction and relaxation time for the surface-adsorbed molecules are difficult to determine, the experimental results are often used as a relative comparison among samples with similar compositions. For example, relative surface area or particle size can be estimated for two slurry samples with a similar Tij value based on their observed relaxation times [67]. 

Although the rate coefficients for the formation of the first three base pairs are large, the dimers formed are unstable and break up rapidly also, so that there is only a low concentration of the dimer containing two base pairs, and the observed relaxation time for the overall equilibrium... 

The observed relaxation time is time-averaged value of s e and Sj. The fraction of Na ion under the slow-motion condition is expected small, but the transverse relaxation rate constants (s ) are expected larger than Sfr j by 10 to 100 times. Therefore, we applied this equation to the observed relaxation rate constants to estimate the 2 compartments. One problem is accurate knowledge of the value of s for 23Na+ in the agar gel. 

A stig — dielectric increment per gm. protein per liter /r = dipole moment in debye units t H O is the relaxation time in water at 25° (correcting for the relative viscosity of water and the solvent actually employed) To = relaxation time of a sphere, of volume equal to that of the protein, in water at 25° ajb = ratio of major to minor axis, calculated from r and observed relaxation times, by the equations of Perrin (92) [Cohn and Edsall (Jd)], neglecting hydration. 

The curve for insulin is unusual in that the observed relaxation time is shorter than would be calculated for an anhydrous sphere of molecular weight 36.OOO. At the time when these observations were made in 1939, no explanation of this phenomenon was available. However, it is now known — see the discussion on page 134 — that insulin under certain conditions dissociates into smaller units of molecular weight 12.000 and there is evidence that in the solution of 80% propylene glycol in which the dielectric measurements were made, the insulin molecule actually dissociates into units of the smaller size. 

In general, the quadrupole mechanism is a highly effective means of relaxation. The sign of Q relates to the distribution of charge around the nucleus while its magnitude depends upon several factors. Hius, observed relaxation times can vary over a wide range of values. The effects of the quadrupole relaxation mechanism on NMR lineshapes and linewidths have already been treated in appreciative detail. A very recent theoretical... 

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... 

Table 16. Contributions to observed relaxation times and rates of protons in poly(methyl methacrylate) (PMMA).

In general, the longitudinal relaxation rate of a hydride resonance is dominated by dipolar and chemical shift anisotropy contributions, the former being by far the most important in determining the observed relaxation time.[14] Thus a first look to the T value may provide a useful tool for structural assignments, since it immediately reports on the distances with respect to other magnetically active... 

Relaxation times are commonly measured for porous media that have been saturated with a fluid such as water or an aqueous brine solution. The observed relaxation times are strongly dependent on the pore size, the distribution of pore sizes, the type of material (e.g. content of paramagnetic ions) and the water content. While relaxation times in porous media have been modelled using random walk methods and finite-element methods, simplified models are usually needed to obtain information on pore space. Section 3.2 reviews the standard model used to analyse relaxation behaviour of fluid in macroporous samples such as rocks. Mesoporous materials such as porous silica will be discussed in Section 3.3. 

Rocks tend to have a very broad distribution of pore sizes, and there is therefore a distribution of observed relaxation times. In general, the magnetization signal, M(t), from a spin-echo experiment (used to determine the Ti distribution) is expected to obey (based on Eq. (9))... 

Fig. 3. Details of the model analysis of Fig. 2. The upper diagram shows, in addition to the overall Tx fit for 5CB, the three T zm contributions M = OF, Rot and SD) according to equations (la) and (lb) and indicates that, with a maximum Tiz. Tip ratio of 3, the contribution by equation (4a) cannot describe the angular dependent results in the kHz regime. The lower diagram illustrates the differences between the Nordio small-step and the Void third-rate rotational model and the preference of the second concept, which becomes visible mainly by the A = 90° data in the MHz regime. In the first case (Nordic model), the experimental error limits of 7% are smaller than the standard deviation of 17% between the calculated and the observed relaxation times. In the second case (Void model), both limits are of comparable magnitude.

As stated in section 5.7.2, measured mechanical and dielectric relaxation times are not expected to be equal to the fundamental relaxation times of the corresponding relaxing units in the polymer because of the effects of the surrounding medium. These effects are different for the two types of measurement, so it is unlikely that the measured relaxation times will be exactly the same. There are arguments that suggest that, in order to take this into account, the observed relaxation times for dielectric and mechanical relaxation should be multiplied by Soo/ s and respectively, before... 

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