Number of oscillations

Finally, semi-classical approaches to non-adiabatic dynamics have also been fomuilated and siiccessfLilly applied [167. 181]. In an especially transparent version of these approaches [167], one employs a mathematical trick which converts the non-adiabatic surfaces to a set of coupled oscillators the number of oscillators is the same as the number of electronic states. This mediod is also quite accurate, except drat the number of required trajectories grows with time, as in any semi-classical approach. 

Moreover, in this linear-response (weak-coupling) limit any reservoir may be thought of as an infinite number of oscillators qj with an appropriately chosen spectral density, each coupled linearly in qj to the particle coordinates. The coordinates qj may not have a direct physical sense they may be just unobservable variables whose role is to provide the correct response properties of the reservoir. In a chemical reaction the role of a particle is played by the reaction complex, which itself includes many degrees of freedom. Therefore the separation of reservoir and particle does not suffice to make the problem manageable, and a subsequent reduction of the internal degrees of freedom in the reaction complex is required. The possible ways to arrive at such a reduction are summarized in table 1. 

With a new software program it is possible to measure the Texture Constant" of pectins. This Texture Constant K is calculated by the ratio of the maximum force during the time interval of the measurement and the measured area below the force-time curve. The resulting constants K correlate well with the dynamic Weissenberg number of oscillating measurements carried through with the same pectin gels. 

One should note that the phase shift becomes time-independent and maximal for a = 1, i.e., at the resonance condition v = vG. The frequency spectrum 4>(a) bears a sine shape with a bandwidth inversely proportional to the number of oscillations of the gradient field (Fig. 4). Such a behaviour was also predicted in Ref. 15. Recording in a systematic way the phase shift as a function of vG without space encoding would be a very fast and efficient method to scan in a whole object the possible frequencies of spin motions. 

Unfortunately, as with all oversimplified theories, there are limitations for the application of the latter equation to ions close to the dissociation threshold, hi these cases, the number of degrees of freedom has to be replaced by an effective number of oscillators which is obtained by use of an arbitrary correction factor. [7] However, as long as we are dealing with ions having internal energies considerably above the dissociation threshold, i.e., where E - Eo)/E = 1, the relationship is valid and can even be simplified to give the quasi-exponential expression... 

Ion trajectory simulations allow for the visualization of the ion motions while travelling through a quadrupole mass analyzer (Fig. 4.36). Furthermore, the optimum number of oscillations to achieve a certain level of performance can be determined. It turns out that best performance is obtained when ions of about 10 eV kinetic energy undergo a hundred oscillations. [110]... 

The last method has been pushed to an impressive sensitivity by putting the probe inside the cavity of a cw dye laser oscillating on several modes close above threshold. The sensitivity of such a broad-band dye laser to selective intracavity absorption on a single mode is proportional to the number of oscillating modes due to... 

Planck showed that the mean energy of a great number of oscillators, each with a characteristic angular frequency w, = 2ttv,, is given by... 

Roughly speaking, the Q factor is the number of oscillations the oscillator can sustain after an initial push. The stronger the damping, the smaller the quality factor. 

Given the response of a single oscillator to a time-harmonic electric field, the optical constants appropriate to a collection of such oscillators readily follow. The induced dipole moment p of an oscillator is ex. If 91 is the number of oscillators per unit volume, the polarization P (dipole moment per unit... 

For any given value of 0, x varies with the angle of observation 6. The number of oscillations in this curve is greater for larger values of 0 and n. 

Though reduced to the barest of essentials, the scheme shows many features observed in real examples of oscillatory reactions a pre-oscillatory period, a period of oscillatory behaviour, and then a final monotonic decay of reactant and intermediate concentrations to their equilibrium values. We can identify from the model such features as the dependence of the length of the pre-oscillatory period on the initial reactant concentration and the rate constants, an estimate for the number of oscillations, and the length of the oscillatory phase. By tuning the parameters we can obtain as many oscillations as we wish. 

The number of oscillations will be independent of the initial reactant concentration and is determined most strongly by the rate constant ratio k0/k2 the faster the decay of the reactant the fewer oscillatory excursions one would expect to see. For ku = 0.01 and ex 10 3 we predict aproximately 140 oscillations which is of the correct order of magnitude compared with the results in Fig. 3.10(a) (82 excursions). 

If the value of the reactant decay rate e is not very small, higher-order correction terms will become significant more quickly. Exact (i.e. precisely computed) concentration histories will not be well appproximated by the pseudo-stationary forms (3.72) and (3.73) even when the state is locally stable. During any possible period of oscillatory behaviour, the number of oscillations will naturally decrease as e increases, as expressed by eqn (3.79). In addition to this, however, the time for the first excursion to develop, which... 

A= wavelength in nanometres (1 nm= 10-B metres) v = frequency or number of oscillations per second in Hertz (Hz)... 

The QRRK rate constant in Fig. 10.7 certainly fits the experimental data well. However, this is to be expected given the origin of the parameters in the model. Specifically, the high-pressure Arrhenius parameters were obtained from fits to the experimental data. The number of oscillators was taken as an adjustable parameter, as was the collision cross section used in ks. Thus the QRRK curve in Fig. 10.7 should match the experiment in the high-pressure limit, and two parameters were varied to enable a fit to the pressure fall-off behavior. 

Compute the fall-off curve using QRRK theory. For this calculation, assume a collision diameter of 4.86 A. Assume that the average energy transfer per N2-C-C5H5 collision is -0.69 kcal/mol (needed to calculate the parameter /5 used in the model). Take the number of oscillators to be, v = actual, with the frequency calculated above. Assume the reaction barrier to be E0, given above. 

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