Time spectra

When a time window twice the duration of the delay time is used, perfect coincidence is at the centre of the time window and it is possible to make an accurate assessment of the background by considering the region to either side of the perfect coincidence region. An example of a time spectrum is shown m figure Bl.10.8. 

Figure Bl.10.8. Time spectrum ftom a double coincidence experiment. Tln-ough the use of a delay in the lines of one of the detectors, signals that occur at the same instant in botii detectors are shifted to tlie middle of the time spectrum. Note the unifonn background upon which the true comcidence signal is superimposed. In order to decrease the statistical uncertainty in the detemiination of the true coincidence rate, the background is sampled over a time Aig that is much larger than the width of the true coincidence signal. Ax.

Referring to figure BLIP. 7 consider electrons from the event under study as well as from other events all arriving at the two detectors. The electrons from the event under study are correlated in time and result in a peak in the time spectrum centred approximately at the delay time. There is also a background level due to events that bear no fixed time relation to each other. If the average rate of tlie background events in each detector is R and i 2> then the rate that two such events will be recorded within time Ax is given by i g, where... 

There are a number of important concepts which emerge in our discussion of viscosity. Most of these will come up again in subsequent chapters as we discuss other mechanical states of polymers. The important concepts include free volume, relaxation time, spectrum of relaxation times, entanglement, the friction factor, and reptation. Special attention should be paid to these terms as they are introduced. 

HHLM = high heat-load monochromator, (b) NLE time spectrum of the radiation, which was scattered by a Fe foil rotating at 10 kHz and which is obtained after background correction of the image plate data. The spectrum was obtained during a 1 min exposure. (Taken from [75])... 

The linear viscoelastic behavior of liquid and solid materials in general is often defined by the relaxation time spectrum 11(1) [10], which will be abbreviated as spectrum in the following. The transient part of the relaxation modulus as used above is the Laplace transform of the relaxation time spectrum H(l)... 

The scaling of the relaxation modulus G(t) with time (Eq. 1-1) at the LST was first detected experimentally [5-7]. Subsequently, dynamic scaling based on percolation theory used the relation between diffusion coefficient and longest relaxation time of a single cluster to calculate a relaxation time spectrum for the sum of all clusters [39], This resulted in the same scaling relation for G(t) with an exponent n following Eq. 1-14. 

The relaxation modulus is often expressed with the relaxation time spectrum, Eq. 1-4 ... 

Fig. 11. Schematic of relaxation time spectrum of the critical gel of PBD 44 (Mw = 44 000). The entanglement and glass transition is governed by the precursor s BSW-spectrum, while the CW spectrum describes the longer modes due to the crosslinking [60]. denotes the longest relaxation time of PBD44 before crosslinking...

This most simple model for the relaxation time spectrum of materials near the liquid-solid transition is good for relating critical exponents (see Eq. 1-9), but it cannot be considered quantitatively correct. A detailed study of the evolution of the relaxation time spectrum from liquid to solid state is in progress [70], Preliminary results on vulcanizing polybutadienes indicate that the relaxation spectrum near the gel point is more complex than the simple spectrum presented in Eq. 3-6. In particular, the relation exponent n is not independent of the extent of reaction but decreases with increasing p. 

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 ... 

We assume that the above solution is valid in about the same time range as the self-similar relaxation time spectrum, Eq. 1-5. The retardation time spectrum is also self-similar. It is characterized by its positive exponent n which takes on the same value as in the relaxation time spectrum. 

As an example of the measurement of cross-correlated relaxation between CSA and dipolar couplings, we choose the J-resolved constant time experiment [30] (Fig. 7.26 a) that measures the cross-correlated relaxation of 1H,13C-dipolar coupling and 31P-chemical shift anisotropy to determine the phosphodiester backbone angles a and in RNA. Since 31P is not bound to NMR-active nuclei, NOE information for the backbone of RNA is sparse, and vicinal scalar coupling constants cannot be exploited. The cross-correlated relaxation rates can be obtained from the relative scaling (shown schematically in Fig. 7.19d) of the two submultiplet intensities derived from an H-coupled constant time spectrum of 13C,31P double- and zero-quantum coherence [DQC (double-quantum coherence) and ZQC (zero-quantum coherence), respectively]. These traces are shown in Fig. 7.26c. The desired cross-correlated relaxation rate can be extracted from the intensities of the cross peaks according to ... 

The shear relaxation modulus can in general be written as an integral over the relaxation time spectrum H. At the same time Equation 3 can also be used. Thus, we have... 

Andrews,R.D., Tobolsky.A. V. Elastoviscous properties of pdyisobutylene. IV. Relaxation time spectrum and calculation of bulk viscosity. J. Polymer Sci 7, 221-242 (1951). 

Fig. 1 (right) shows the transient spectra obtained after excitation of 6HQ(C) in a 3x1 O 2 M HCIO4 aqueous solution (pH 1.5). The shortest time spectrum presents a strong absorption at 391 nm and a shoulder around 500 nm. It decays with a characteristic time of 2.2 ps to yield a new spectrum with 2 absorption bands at 403 and 486 nm and a negative one... 

As a liquid is cooled at a finite rate, the relaxation time spectrum will shift to longer times and a temperature region will eventually be reached where the sample is no longer in volume equilibrium. If the sample continues to be cooled at this rate it will become a glass. A glass is a nonequilibrium, mechanically unstable amorphous solid. If the sample is held at a fixed temperature near Tg the volume will relax towards its equilibrium value. In this section we will restrict our attention to equilibrium liquids at temperatures near... 

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