Transit time frequency function

Fig. 4-3 The age frequency function rp(r) and the transit time frequency function (r) and the corresponding average values Ta and for the three cases described in the text (a) Ta < ry, (b) Ta = r (c) Adapted from Bolin and...

Here, I(co) is the Fourier transform of the above C(t) and AEq f is the adiabatic electronic energy difference (i.e., the energy difference between the v = 0 level in the final electronic state and the v = 0 level in the initial electronic state) for the electronic transition of interest. The above C(t) clearly contains Franck-Condon factors as well as time dependence exp(icOfvjvt + iAEi ft/h) that produces 5-function spikes at each electronic-vibrational transition frequency and rotational time dependence contained in the time correlation function quantity <5ir Eg ii,f(Re) Eg ii,f(Re,t)... 

All of these time correlation functions contain time dependences that arise from rotational motion of a dipole-related vector (i.e., the vibrationally averaged dipole P-avejv (t), the vibrational transition dipole itrans (t) or the electronic transition dipole ii f(Re,t)) and the latter two also contain oscillatory time dependences (i.e., exp(icofv,ivt) or exp(icOfvjvt + iAEi ft/h)) that arise from vibrational or electronic-vibrational energy level differences. In the treatments of the following sections, consideration is given to the rotational contributions under circumstances that characterize, for example, dilute gaseous samples where the collision frequency is low and liquid-phase samples where rotational motion is better described in terms of diffusional motion. 

In order to study the resonant behavior of spectral density, let us plot the SNR as function of driving frequency go. From Fig. 22 one can see, that SNR as function of co has strongly pronounced maximum. The location of this maximum at co = mlnax approximately corresponds to the timescale matching condition mlnax 7i/Tmm, where Tmm is the minimal transition time from one state to another one. 

Time resolution of the enthalpy changes is often possible and depends on a number of experimental parameters, such as the characteristics of the transducer (oscillation frequency and relaxation time) and the acoustic transit time of the system, za, which can be defined by ra = r0/ua where r0 is the radius of the irradiated sample, and va is the speed of sound in the liquid. The observed voltage response of the transducer, V (t) is given by the convolution of the time-dependent heat source, H (t) and the instrument response function,... 

For the dynamical distribution it will in general be necessary to consider both the auto and cross time correlation functions of the 0-1 and the 1-2 frequencies (117). For example, if the fluctuations, <5A(t), in the anhar-monicity are statistically independent of the fluctuations in the fundamental frequency, the oscillating term (1 — elAt3) in Equation (18) would be damped. In a Bloch model the fluctuations in anharmonicity translate into different dephasing rates for the 0-1 and 1-2 transitions that were discussed previously for two pulse echoes of harmonic oscillators. Thus we see that even if A vanishes, the third-order response can be finite (94). 

Another factor that may be beneficial is physical activity, since it affects the immune function and antioxidant defenses, transit time of digestion, hormones, and body fat, and it improves energy balance. Therefore, it may have a protective effect on prostate cancer and it may even slow progression and metastasis (G14, H8, K7, 02, 03). In a 9-year follow-up study performed by Hartman et al., the relative risk for physical exercise in prostate cancer was compared with sedentary workers and found to be 0.6 (Cl = 0.4-1.0), 0.8 (Cl = 0.5-1.3), and 1.2 (Cl = 0.7-2.0) for occupational workers, walker/lifters, and heavy laborers, respectively. Except for heavy laborers, an inverse association was observed (RR = 0.7, Cl = 0.5-0.9) compared to men who were sedentary at work and leisure (H8). However, other studies indicate a positive association between vigorous exercise and prostate cancer Cl), and therefore further study is necessary to provide an activity optimum. Frequency, duration, intensity, type of exercise, and the period during a man s lifetime when exercise may be beneficial, must be investigated (02, 03). 

Variation of Xl between the high- and low-frequency limits is characterized by two exponentially decaying functions with different transition times [34]. In the case of the alcohols the difference between these limits is significant and can lead to complex relaxation behavior in these solvents. For methanol, the high-frequency value of Xl is 1.0 ps and the low-frequency value, 4.0 ps. For 1-pro-panol, the difference is greater, the corresponding values being 5.3 and 36.5 ps, respectively. 

In a previous work (J ) the frequency distribution of transit times was obtained by following the concentration of Pu-239,240 outflowing from a layer of loamy soil. Although the resulting function was not a perfectly monotonically decreasing one an approximation of the transit time density distribution by the form exp(-kt), with k = 0.17 a-1 as turnover rate and t = time in years, seemed to be reasonable enough. 

The Cross-Section for the i —> j transition is expressed as one of three frequently confused quantitites, alj(i/ — v0), <7y(0), and a°p where (Jij v — v0) is the lineshape function (units of area), cross section at line center (units of area), and cr°j is the integrated cross section (units of area times frequency). One commonly encounters Lorentzian... 

Fig. 5. Pulsed-nozzle FT microwave measurements. A molecule-radiation interaction occurs when the gas pulse is between mirrors forming a Fabry-Perot cavity. If the transient molecule has a rotational transition of frequency vm falling within the narrow band of frequencies carried into the cavity by a short pulse (ca. 1 (is) of monochromatic radiation of frequency v, rotational excitation leads to a macroscopic electric polarization of the gas. This electric polarization decays only slowly (half-life T2 = 100 (is) compared with the relatively intense exciting pulse (half-life in the cavity t 0.1 (is). If detection is delayed until ca. 2 (is after the polarization, the exciting pulse has diminished in intensity by a factor of ca. 106 but the spontaneous coherent emission from the polarized gas is just beginning. This weak emission can then be detected in the absence of background radiation with high sensitivity. For technical reasons, the molecular emission at vm is mixed with some of the exciting radiation v and detected as a signal proportional to the amplitude of the oscillating electric vector at the beat frequency v - r , as a function of time, as in NMR spectroscopy Fourier transformation leads to the frequency spectrum [reproduced with permission from (31), p. 5631.

When an+ 1 ions traverse the sheath in a short time compared to the field oscillations. Under this condition, an ion traversing the sheath experiences the sheath voltage prevailing at the time the ion entered the sheath. In the absence of collisions, the lED function will reflect precisely the variation of the sheath voltage with time. This quasi steady-state condition of cut+ 1 is satisfied for low RF frequencies or short ion transit times, i.e., thin sheaths (low sheath voltage or small Debye length). 

In principle, the ultrasonic techniques described for solid-liquid flow measurement can be applied to measure air flow rate and particle velocity. Direct measurement of air flow rate by measuring upstream and downstream transit times has been demonstrated. But, the Doppler and cross-correlation techniques have never been applied to solid/gas flow because the attenuation of ultrasound in the air is high. Recent developments have shown that high-frequency (0.5-MHz) air-coupled transducers can be built and 0.5-MI Iz ultrasound can be transmitted through air for a distance of at least 1 in. Thus, the cross-correlation technique should be applicable to monitoring of solid/gas flow. Here, we present a new cross-correlation technique that does not require transmission of ultrasonic waves through the solid/gas flow. The new technique detects chiefly the noise that interacts with the acoustic field established within the pipe wall. Because noise may be related to particle concentration, as we discussed earlier, the noise-modulated sound field in the pipe wall may contain flow information that is related to the variation in particle concentration. Therefore, crosscorrelation of the noise modulation may yield a velocity-dependent correlation function. 

It is instructive to compute the time correlation function in the simple case that the ground and excited state potentials are harmonic but differ in their equilibrium position and frequency. This is particularly simple if the initial vibrational state is the ground state (or, in general, a coherent state (52)) so that its wave function is a Gaussian. We shall also use the Condon approximation where the transition dipole is taken to be a constant, independent of the nuclear separation, but explicit analytical results are possible even without this approximation. A quick derivation which uses the properties of coherent states (52) is as follows. The initial state on the upper approximation is, in the Condon approximation, a coherent state, i /,(0)) = a). The value of the parameter a is determined by the initial conditions which, if we start from a stationary state, are that there is no mean momentum and that the mean displacement (x) is the difference in the equilibrium position of the two potentials. In general, using m and o> to denote the mass and the vibrational frequency... 

Figure 18 Time cross-correlation functions for three Raman transitions in iodobenzene (from the ground state to the B excited electronic state with v = 1, 2, 3 quanta in the vu vibrational mode. (Left) Computed for a harmonic B state potential and convoluted with a 125-fs-wide window function. The spectrum is computed from this cross-correlation function. (Right) The time correlation function determined from the Raman frequency spectrum (the excitation profile) via the maximum entropy formalism, as discussed in the text, using nine Lagrange multipliers kr. (From Ref. (102).)...

If the experimental data are sufficiently accurate, then from the susceptibility function 3 /(m- t) it is possible to evaluate the transit time probability density, (Puu id-, t), by means of numerical inverse Fourier transformation. If the experimental data are not very accurate, then it is still possible to evaluate the first two or three moments and cumulants of the transit time. Since the mean transit time is nonnegative, it follows that (puu d < 0 r) = 0, and thus the characteristic function of the probability density (Puu id-, t) is identical with the susceptibility function in the frequency domain, Sr/MdsT t), experimentally accessible from eqs. (12.116)-(12.117). It follows that the moments < 0" (r) > , m = 1, 2,..., and the cumulants m = 1,2,...,... 

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