The Continuous Relaxation Spectrum

If the number of elements in the generalized Maxwell model is increased toward infinity, one arrives at the continuous spectrum function, F(t), where F r) dr is the contribution to G(t) due to Maxwell elements having relaxation times between rand t+ dr. The relaxation modulus is related to the spectrum function as shown by Eq. 4.18. 

However, because of the concentration of relaxation information at very short times, it is generally preferable to work with a logarithmic time scale. This leads to a relaxation spectrum function, H( t), which is a time-weighted spectrum function defined as F T, so that the relaxation modulus is given by  

Relationships between the various material functions describing linear behavior and methods for converting among them are discussed by Ferry [ 1 ] and by Tschoegl [2j. 

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The continuous function II( n T) [often simply given the symbol H(r) as in this chapter) is the continuous relaxation spectrum. Although called, by long-standing custom, a spectrum of relaxation times, it can be seen that H is in reality a distribution of modulus contributions, or a modulus spectrum, over the real time scale from 0 to < or over the logarithmic time scale from - to +. ... 

A continuous retardation spectrum, L(t), can also be defined, which is analogous to the continuous relaxation spectrum, H(t). In terms of this function the creep compliance is given by Eq. 4.30. 

We performed a calculation of the relaxation rates using the phonon Green s functions of the perfect (CsCdBr3) and locally perturbed (impurity dimer centers in CsCdBr3 Pr ) crystal lattices obtained in Ref. [8]. The formation of a dimer leads to a strong perturbation of the crystal lattice (mass defects in the three adjacent Cd sites and large changes of force constants). As it has been shown in Ref. [8], the local spectral density of phonon states essentially redistributes and several localized modes appear near the boundary of the continuous phonon spectrum of the... 

Analysis of Eq. 180 shows that H(r) consists of continuous bands of relaxation times the number of bands increases with n, in other words with the length of the Rouse chains between the branching points [12]. To be noted is that in logarithmic scales in an intermediate regime the bands of H(t) show an almost linear behavior with slope 1/2 as a fimction of r the Rouse chains between branching points seem to be responsible for this behavior. Also, it is shown in [12] that the maximal relaxation time Tmax of the whole relaxation spectrum is approximately given by... 

To continue the investigation, carbon detected proton T relaxation data were also collected and were used to calculate proton T relaxation times. Similarly, 19F T measurements were also made. The calculated relaxation values are shown above each peak of interest in Fig. 10.25. A substantial difference is evident in the proton T relaxation times across the API peaks in both carbon spectra. Due to spin diffusion, the protons can exchange their signals with each other even when separated by as much as tens of nanometers. Since a potential API-excipient interaction would act on the molecular scale, spin diffusion occurs between the API and excipient molecules, and the protons therefore show a single, uniform relaxation time regardless of whether they are on the API or the excipients. On the other hand, in the case of a physical mixture, the molecules of API and excipients are well separated spatially, and so no bulk spin diffusion can occur. Two unique proton relaxation rates are then expected, one for the API and another for the excipients. This is evident in the carbon spectrum of the physical mixture shown on the bottom of Fig. 10.25. Comparing this reference to the relaxation data for the formulation, it is readily apparent that the formulation exhibits essentially one proton T1 relaxation time across the carbon spectrum. This therefore demonstrates that there is indeed an interaction between the drug substance and the excipients in the formulation. 

Figure 2.6. Energy level diagram (top) and spectra (bottom) illustrating the continuous model of relaxation. The energy of the emitted quanta decreases (hvF- kv F- hvF) and the position of the fluorescence spectrum (solid curves) moves smoothly as a result of relaxation.

In the classical limit h - 0, the spectrum of the Landau-von Neumann superoperator tends to the spectrum of the classical Liouvillian operator. If the classical system is mixing, the classical Liouvillian spectrum is always continuous so that we may envisage an analytic continuation to define a discrete spectrum of classical resonances. It has been shown that such classical resonances are given by the zeros of the classical zeta function (2.44) and are called the Pollicott-Ruelle resonances sn(E) [63], These classical Liouvillian resonances characterize exponential decay and relaxation processes in the statistical description of classical systems. The leading Pollicott-Ruelle resonance defines the so-called escape rate of the system,... 

Electron spin relaxation in aqueous solutions of Gd3+ chelates is too rapid to be observed at room temperature by the usual pulsed EPR methods, and must be studied by continuous wave (cw) techniques. Two EPR approaches have been used to study relaxation studies of the line shape of the cw EPR resonance of Gd3+ compounds in aqueous solution, and more direct measurement of Tle making use of Longitudinally Detected EPR (LODEPR) [70]. Currently, LODESR is available only at X-band, and the frequency dependence of relaxation is studied by following the frequency dependence of the cw EPR line shape, and especially of the peak-to-peak line width of the first derivative spectrum (ABpp). 

For the continuous-flow measurements, the pseudo-2D spectrum was recorded with a spectral width of 9616 Hz and 64 transients with 8K complex data points, thus resulting in an acquisition time of 0.42 s/transient along the 128 t increments. A relaxation delay of 1.2 s was used and the time resolution... 

The recent experimental core level spectrum by Cavell and Allison129 in Fig. 50 is very similar to that of N2 (Fig. 45), with a weak Is 3(1 jAr) satellite with 2% relative intensity at 7.2 eV above the main line, a strong 1 s 3(1 n 2 ri) satellite with 10% relative intensity at 12.2 eV, followed by a prominent discrete and continuous satellite spectrum at higher energies. The total relaxation shift is 16 eV, the MO HF ljs level lies well above the low-lying nn satellites and the limited range of the shake-up and shake-off spectrum shown in Fig. 50 only accounts for about one third of the relaxation shift (through Eq. (22)). 

The number N of retardation times needed depends on the required agreement between theory and experimental behaviour that is required. Instead of a description of viscoelastic behaviour with the aid of a discrete spectrum of relaxation and retardation times, also continuous relaxation or retardation time spectra can be used. In some cases these are easier to handle. 

Electronic Spectrum. Acetone is the simplest ketone and thus has been one of the most thoroughly studied molecules. The it n absorption spectrum extends from 350 nm and reaches a maximum near 270 nm (125,175). There is some structure observable below 295 nm, but no vibrational and rotational analysis has been possible. The fluorescence emission spectrum starts at about 380 nm and continues to longer wavelengths (149). The overlap between the absorption and the fluorescence spectra is very poor, and the 0-0 band has been estimated to be at - 330 nm (87 kcal/mol). The absorption spectra, emission spectra, and quantum yields of fluorescence of acetone and its symmetrically methylated derivatives in the gas phase havbe been summarized recently (101). The total fluorescence quantum yield from vibrationally relaxed acetone has been measured to be 2.1 x 10 j (105,106), and the measurements for other ketones and aldehydes are based on this fluorescence standard. The phosphorescence quantum yield is -0.019 at 313 nm (105). 

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