Oscillation vanishing

When carbon (S) spins are locked following a CP 90° pulse, the HB term becomes nonsecular and, after these oscillations vanish, the S spin-lock magnetization is fractionally reduced. Thus, the observed decay is generally a sum of two Weibull functions, usually exponentials (Equation 20 p = 1) the initial slope reflects a Tle dominated by a spin-lattice process and the final slope yields a Tt dominated... 

Studies of CO oxidation over Pt/Al203 [72-74] showed that oscillations are observed only in the presence of admixed hydrocarbons or water. If the reaction mixture is thoroughly purified, self-oscillations vanish. These data are evidently an argument in favour of the fact that reversible formation steps of non-reactive species play a special role in the appearance of selfoscillations. In accordance with the assertions of Marshneva et al. [163,164],... 

In this section we discuss the model predictions for the ketone ethyl acetoacetate (1). With the ketone absent ([Ket]x = 0 mM), the extended model reproduces all previous results with oscillations of all system variables above [Glc]xo > 18.5 mM [53]. Figure 3.6 shows the system s response to a fixed glucose concentration [Glc]xo at 30 mM and an increase of [Ket]x to 1 mM. The oscillations vanish at [Ket]x = 0.23 mM in a supercritical Hopf bifurcation and the steady state is stable for [Ket]x > 0.23 mM. Figure 3.6a shows the minimum and maximum concentrations of NADH as two thick curves, while in all other panels the time averages of the plotted variables are shown, not the minimum and maximum values. Since the addition of ketone provides an alternative mode of oxidation of NADH, the concentration of NADH is decreasing in Fig. 3.6a whereas the fluxes of carbinol production are increasing in Fig. 3.6b. 

As before for confined fluids with short-range interaction potentials, the disjoining pressure / (sj) has damped oscillatory character with the oscillations vanishing when the confined systems become bulk-like at large wall... 

It is important to stress that the above three descriptions of the spiral wave practically coincide far away from the rotation center. Moreover, already at a relatively small distance ta from the rotation center, the Archimedean spiral becomes very close to the curvature affected spiral obtained from Eq. (9.3), as can bee seen in Fig. 9.1(d). In this example ta can be estimated as ss 9.0 A. Recent computations performed with the Oregonator model [40] and experiments with the BZ reaction [43] also confirm that an Archimedean spiral provides a suitable approximation of the wave front except in a relatively small region of radius A near the rotation center. Even the shape of a slightly meandering spiral waves exhibits only small oscillations around an Archimedean shape, and the amplitude of these oscillations vanishes very quickly with r [44]. Therefore, the Archimedean spiral approximation will be used below to specify the shape of the wave front. 

Fig. 3.5e Analytical integration method secondary distribution due to a small overpotential, the oscillation vanishes. The distribution agrees with that of figure 3.5c.

Further on, the situation is developing in the following way. After 2-3 days of such operation, delays occur in the intermediate region II while the system passes to the conditions of common anode effect I. Upon a time, these delays become more and more durable, and then osciUatimis appear in the region II, as shown in Fig. 5.17. Oscillations vanish after another 3-5 h of the electrolysis, and then, a few hours later, the system jumps into the conditions of intensive regime represented by line III in Fig. 5.16. Finally, after 4—5 days operation, the anode... 

On increasing to the limit value (k ), the frequency of the oscillations increases, while the amplitude remains nearly constant. At kj > (k ), the oscillations vanish jump-like and the system enters the stable steady state. Thus, the four-step mechanism with a buffer step, Eq. (7.122), described by Eqs. (7.123)-(7.125) can be termed the simplest eatalytic oscillator based on mass-action-law assumptions. 

The term cos(ngrml/2mo) is called the spin factor. When g = 2 (free-electron value) and 7W =0.5ffJo, this term becomes zero for the fundamental oscillation (r= 1) and the dHvA oscillation vanishes for all values of magnetic field. This is called the zero spinsplitting situation in which the up and down spin contributions to the oscillation cancel... 

The decrease in electronic correlation energy as a result of retardation depends on the phase shift involved, i.e. the frequency of the oscillator considered multiplied by the retardation 2ry/c of the reaction field. With increasing separation, the correlation of high frequency oscillators vanishes prior to that of low frequency oscillators finally at large separations the nearly electrostatic interactions are predominant. The findings on the dispersion energy between two particles in the case of retardation generally exhibit an additional factor l/rtj compared with the non-retarded case. 

If the second term on the right-hand side of the equation is omitted, the latter is transformed into Eq. (2.76). As the omission is possible only for t < tj, Fourier transformation of the reduced equation holds for co-tj 1 only. Consequently, the equality (2.75) is of asymptotic character, and may not be utilized to find full g(co) or its Fourier-transform Kj(t) at any times. When it was nevertheless used in [117], the rotational correlation function turned out to be alternating in sign. The oscillatory behaviour of Kj(t) occured not only in a compressed gas, but also at normal pressure, when Kj(t) should vanish monotonically, if not exponentially. The origin of these non-physical oscillations is easily... 

The interface between a liquid metal and an electrolyte solution can be vibrated by applying an oscillating external pressure variation. An electric signal picked up at the oscillating interface as a function of the applied electrode potential can be registered. It shows a particular dependency, at the signal vanishes [83Mly]. (Data obtained with this method are labelled VE). 

There are several possible ways of deriving the equations for TDDFT. The most natural way departs from density-functional perturbation theory as outlined above. Initially it is assumed that an external perturbation is applied, which oscillates at a frequency co. The linear response of the system is then computed, which will be oscillating with the same imposed frequency co. In contrast with the standard static formulation of DFPT, there will be special frequencies cov for which the solutions of the perturbation theory equations will persist even when the external field vanishes. These particular solutions for orbitals and frequencies describe excited electronic states and energies with very good accuracy. 

The first derivative vanishes since U(R) has a minimum at Re. Within this approximation the nuclear Schrodinger equation reduces to that of a harmonic oscillator, whose frequency to is given by ... 

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