Modulated perturbation

At higher pressures only Raman spectroscopy data are available. Because the rotational structure is smoothed, either quantum theory or classical theory may be used. At a mixture pressure above 10 atm the spectra of CO and N2 obtained in [230] were well described classically (Fig. 5.11). For the lowest densities (10-15 amagat) the band contours have a characteristic asymmetric shape. The asymmetry disappears at higher pressures when the contour is sufficiently narrowed. The decrease of width with 1/tj measured in [230] by NMR is closer to the strong collision model in the case of CO and to the weak collision model in the case of N2. This conclusion was confirmed in [215] by presenting the results in universal coordinates of Fig. 5.12. It is also seen that both systems are still far away from the fast modulation (perturbation theory) limit where the upper and lower borders established by alternative models merge into a universal curve independent of collision strength. 

Modulated structures, corresponding to substances in which there is an average structure with a three-dimensional periodicity showing modulated perturbations. 

Here 6 is the Heaviside step function and the effective mass is M = m/Ak2L. In (6) we have divided U(q) between Ih,. whose eigenstates are strictly bound if their energy fix, < Um, and a potential V < 0 in the time-modulated perturbation e(t)V. This perturbation allows the particle to tunnel periodically from the bound state to the the unbound (continuum-energy) eigenstates of H whenever e(t) = 1 in Eq. (4). This form conforms to Eq. (1) and answers query (a) (Fig. 1-insets). 

Montfort and Hadziioannou also studied the effect of interfacial interactions, ie., surface forces, on the rheological response of films as thin as 200 nm with sinusoidal modulated perturbation. They considered the following forces ... 

In Chapter 19 we saw that kinetic processes cannot be studied with a conventional rapid-scanning interferometer when the reaction rate is so fast that the reaction is essentially complete by the time just one or two interferograms have been measured. Instead, very fast reactions must be repeated at each retardation step of a step-scan interferometer. A different but related approach to the measurement of reversible dynamic systems can also be made with step-scan interferometers. In this case, however, very small changes in the state of the sample are introduced by subjecting it to a modulated perturbation of some type. Dynamic information can be obtained when the phase of the induced signal lags behind the phase of the perturbation by several degrees. An example of a reversible modulated process is when a polymer film is subjected to a modulated uniaxial strain, and this measurement is discussed in some detail in the first four sections of this chapter. 

If the index of refraction of a thin material were modulated in Heu of its absorption, the resultant transmittance function for a gra ting prepared as in the absorption case is given by equation 9 where n is the average index of the thin film. An is the amphtude of the index perturbation, and T is the thickness of the film. 

It should be noted that there is a considerable difference between rotational structure narrowing caused by pressure and that caused by motional averaging of an adiabatically broadened spectrum [158, 159]. In the limiting case of fast motion, both of them are described by perturbation theory, thus, both widths in Eq. (3.16) and Eq (3.17) are expressed as a product of the frequency dispersion and the correlation time. However, the dispersion of the rotational structure (3.7) defined by intramolecular interaction is independent of the medium density, while the dispersion of the vibrational frequency shift (5 12) in (3.21) is linear in gas density. In principle, correlation times of the frequency modulation are also different. In the first case, it is the free rotation time te that is reduced as the medium density increases, and in the second case, it is the time of collision tc p/ v) that remains unchanged. As the density increases, the rotational contribution to the width decreases due to the reduction of t , while the vibrational contribution increases due to the dispersion growth. In nitrogen, they are of comparable magnitude after the initial (static) spectrum has become ten times narrower. At 77 K the rotational relaxation contribution is no less than 20% of the observed Q-branch width. If the rest of the contribution is entirely determined by... 

Fano theory, FIR spectra of NH3 128 far infrared (FIR) spectra 62 Fano theory 128 MD calculations 47 Poley absorption 74, 82 fast modulation limit, perturbation theory 182... 

When a pure sinusoidal AC current passes across the electrode/solution interface, the cell voltage (a two electrode arrangement is used) shows a sinusoidal perturbation. It contains multiples of the fundamental frequency of the modulation, the first harmonie dominates. The magnitude of the effect is comparable to Faradaie rectification, but experiments may be easier to perform. Measurement and evaluation have been described in detail [60Old, 72Hil2]. (Data obtained with this method are labelled FD.)... 

Under low-frequency excitation, the flame front is wrinkled by velocity modulations (Fig. 5.2.5). The number of undulations is directly linked to frequency. This is true as far as the frequency remains low (in this experiment, between 30 and 400 Hz). The flame deformation is created by hydrodynamic perturbations initiated at the base of the flame and convected along the front. When the velocity modulation amplitude is low, the undulations are sinusoidal and weakly damped as they proceed to the top of the flame. When the modulation amplitude is augmented, a toroidal vortex is generated at the burner outlet and the flame front rolls over the vortex near the burner base. Consumption is fast enough to suppress further winding by the structure as it is convected away from the outlet. This yields a cusp formed toward burnt gases. This process requires some duration and it is obtained when the flame extends over a sufficient axial distance. If the acoustic modulation level remain low (typically v /v < 20%),... 

The "M"-flame case shows a different kind of flame interaction illustrated in Figure 5.2.4 (MF). The "M"-shape comprises two reactive sheets separated by fresh reactants. This gives rise to flame-flame interactions between neighboring branches of the "M"-shape [41]. The case presented corresponds to an equivalence ratio O = 1.13, a mixture flow velocity v/v = 1.13 m/s, a modulation level fixed to v = 0.50m/s, and a modulation frequency/= 150 Hz. The description of the flame motion over a cycle of excitation starts as in the flame-plate interaction. A velocity perturbation is generated at... 

Dynamics of a methane-air conical flame submitted to a convective wave of equivalence ratio perturbations,/= 175 Hz, and = 0.1. The mass fraction perturbation is shown on the color scale. The flame is represented by temperature isocontours. Two cycles of modulation are displayed. 

The variational calculations were performed using the Alchemy II package [67] while the further perturbation calculations used a code derived from the original CIPSI module. Proper interfaces between the two programs were developed. 

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