Coincidences correlation time

We shall take the Heisenberg and Schrodinger pictures to coincide at time 7 = 0. We next correlate with the complete hamiltonian 27(0) (i.e., the total hamiltonian in the Heisenberg picture at time 1 = 0, which by the above convention is also the total hamiltonian in the Schrodinger picture) an unperturbed hamiltonian 27o(0). We shall write... 

Since relaxation is initially non-exponential, the true correlation time (1.69) does not coincide with r = l/ny, but is equal to... 

Figure 4.47 Typical electronic circuit for the measurement of electron-electron coincidences with two spectrometers (SP1, SP2) placed at the positions 0 , and 5, , respectively. The pre- and main amplifiers are together represented by a triangle. The delay retards the signal from SP1, thus providing a STOP of the time-to-digital converter (TDC) if this time measuring device has been initiated by a START signal from a time-correlated event registered in SP2. The output of the TDC, i.e., the number of time-correlated events as function of the correlation time is stored in a histogramming memory (HIS. MEM.) which then is read out by a computer (COMP.).

This time difference, called the correlation time, is stored in a histogramming memory, and after the necessary processing time (dead time rdead) the TDC is reset and waits for the next START signal. The histogramming memory collects all individual coincidences by sorting them according to their correlation times. In this way, a time spectrum is obtained which can be transferred to a computer. 

Upon substitution of these functions into the expressions (3.5.4) and (3.5.6) for Tj and T2, respectively, the dependence of the relaxation times on the correlation time depicted in Fig. 3.5.2 is obtained for a given Larmor frequency >l. Regimes of slow and fast motion are discriminated by the Ti minimum, where the correlation time is of the order of the Larmor period. In the regime where t/

molecular motion is fast on the NMR timescale, and the homogeneous linewidth is highly reduced. This situation is typical for small molecules at room temperature. It is called the extreme narrowing limit, where both Ti and T2 coincide. 

HBA. The T p values display minima, coincident with the temperatures of the y and p relaxations previously observed by dielectric and dynamic mechanical analysis. The Tjp - temperature data have been fitted to the Cole-Davidson distribution function, which indicates a broad distribution of correlation times. The activation energies obtained from the fitting are higher than from dielectric data, but, in a qualitative sense support the contention that the y relaxation is associated with the motion of HBA units, and the p relaxation with the motion of HNA units. 

Figure 1.2 shows the experimental and theoretical EPR spectra of spin probes in dry and water-saturated of both isotropic and textured morphology. A comparison shows that the theoretieal spectra of both Tempo and Tempol spin probes qualitatively deseribe the main features observed in the corresponding experimental speetra. This coincidence indicates that the spectra actually represent a superposition of the EPR signals from radi-eal molecules with different rotation correlation times. 

In figure 9 we have plotted [ ( -+1)] lnC (t) as a function of time for CS2 at 192K. At short times the curves will be quadratic and coincident for all . If the Debye model for reorientation is obeyed the curves will be linear and coincident at long times. The curves are almost linear for t > 1.5ps but are not coincident for all . The curves contain a point of inflexion at the time when C (t), (the angular velocity a.c.f.) passes through zero. The fact that Cj (t) lies above its assymptotic exponential value means that the reorientational correlation times obtained by integrating under C Ct) are shorter than those obtained from the limiting slopes of the functions. 

In the discussion of the ionic contribution the rod-like model for polyelectrolytes will be kept in mind. In addition to the laboratory fixed coordinate system S a second frame S will be introduced. The origins of the systems coincide but the z axis of S is parallel to the nearest polyion and the x axis passes through the rod radially. This s stem should be particularly useful if deviations of the charge distribution from c lindrical symmetry (around the rod) are small and/or short lived with respect to the correlation time of At the origin of S the field gradient will have C2V sym-... 

Coincidence experiments explicitly require knowledge of the time correlation between two events. Consider the example of electron impact ionization of an atom, figure Bl.10.7. A single incident electron strikes a target atom or molecule and ejects an electron from it. The incident electron is deflected by the collision and is identified as the scattered electron. Since the scattered and ejected electrons arise from the same event, there is a time correlation... 



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