Slow rotational oscillations

In Chap. 5 measurements of gas adsorption by slow rotational oscillations of the sorbent material are discussed. This method uses the inertia of mass to detect changes caused by gas adsorption. Combined with gravimetric or volumetric measurements it allows the measurement of gas solubilities in non-rigid, i. e. swelling sorbent materials as for example polymers. 

For these reasons we will be here restricted to consider only the slow rotational oscillations of sorbent masses in a sorptive gas atmosphere such that the masses moved geometrically are transformed into themselves and only initiate flows in the surrounding gas which are rotationally synunetric. 

In this section some of the advantages and disadvantages of sorption measurements hy using slow rotational oscillations of a pendulum coming from our experience are described in brief. 

Rotational pendulum for measurements of gas adsorption equilibria by observing slow damped oscillations. Height of instrument 1.5 m. More parameters of the instrument are given in Sect. 2.4. 

The periodic recurrence of cell division suggests that globally the cell cycle functions like an autonomous oscillator. An extended model incorporating the sequential activation of the various cyclin-dependent kinases, followed by their inactivation, shows that even in the absence of control by cell mass, this sequence of biochemical events can operate as a limit cycle oscillator [145]. This supports the union of the two views of the cell cycle as dominoes and clock [146]. Because of the existence of checkpoints, however, the cell cycle stops at the end of certain phases before engaging in the next one. Thus the cell cycle looks more like an oscillator that slows down and makes occasional stops. A metaphor for such behavior is provided by the movement of the round plate on the table in a Chinese restaurant, which would rotate continuously under the movement imparted by the participants, were it not for frequent stops. 

Let us consider the case of a = 30 corresponding to a weakly developed chaotic attractor in the individual nephron. The coupling strength y = 0.06 and the delay time T2 in the second nephron is considered as a parameter. Three different chaotic states can be identified in Fig. 12.16. For the asynchronous behavior both of the rotation numbers ns and n f differ from 1 and change continuously with T2. In the synchronization region, the rotation numbers are precisely equal to 1. Here, two cases can be distinguished. To the left, the rotation numbers ns and n/ are both equal to unity and both the slow and the fast oscillations are synchronized. To the right (T2 > 14.2 s), while the slow mode of the chaotic oscillations remain locked, the fast mode drifts randomly. In this case the synchronization condition is fulfilled only for one of oscillatory modes, and we speak of partial synchronization. A detailed analysis of the experimental data series reveals precisely the same phenomena [31]. 

Another study used H T, T2 and 13C T, T p measurements to assess the molecular dynamics in dry and wet solid proteins bacterial RNAase, lysozyme and bovine serum albumin.115 All relaxation time data were analysed assuming three components for the molecular motion methyl group rotation and slow and fast oscillations of all atoms. An inhomogeneous distribution of correlation times was found for all samples, not surprisingly given the inhomogeneous nature of the samples. Interestingly, it was found that dehydration affected only the slow internal motions of the proteins and that the fast ones remained unaltered. 

However, the NMR properties of solid-phase methane are very complex, due to subtle effects associated with the permutation symmetry of the nuclear spin set and molecular rotational tunnelling.55 Nuclear spin states ltotai = 0 (irred. repr. E), 1 (T) and 2 (A) are observed. The situation is made more complicated since, as the solids are cooled and the individual molecules go from rotation to oscillation, several crystal phases become available, and slow transitions between them take place. Much work has been done in the last century on this problem, including use of deuterated versions of methane for example see Refs. 56-59. Much detail has emerged from NMR lineshape analysis and relaxation time measurements, and kinetic studies. For example, the second moment of the 13C resonance is found to be caused by intermolecular proton-carbon spin-spin interaction.60 Thus proton inequivalence within the methane molecules is created. 

For rotational diffusion of the protein molecules two extreme situations can be described analytically in the case of slow reorientation of the EFG, the amplitude of the perturbation function is damped exponentially, but oscillations are stiU present, and in the other extreme of fast reorientation of the EFG, the oscillations in the perturbation... 

Successive high-resolution AMRO experiments shown in Fig. 4.39 verified the proposed FS in an impressive way [376]. As mentioned in Sect. 3.3, 0-(ET)2l3 was one of the first compounds where AMRO, i. e., resistance oscillations periodic in tan (9, were observed [258]. These results, which were reproduced later [377], are understood by the warped FS model explained above. The period of the oscillations is related to the Fermi wave vector via (3.18). In the experimental data shown in Fig. 4.39 not only the previously reported fast AMROs but also slow ones (indicated by small dashes) were observed [376]. The insets of Fig. 4.39 show the peak numbers of the (a) fast and (b) slow oscillation frequency vs tan O. From the slopes for different field rotation planes fcr(0) could be constructed. The resulting two ellipsoidal FSs are in good agreement with the proposed topology of Fig. 4.37c with respect to both form and area. 

Fig. 4.39. The main panel shows the angular dependence of the resistance of 0-(ET)2l3 in B = 13T for field rotation in the cb plane (orthorhombic notation). Two superimposed AMROs are visible. The insets show the peaks of (a) the fast and (b) the slow oscillation vs tan0. Prom [376]...

The rotating wave approximation (RWA) is a useful simplification [465] which contains most of the features of coherent excitation it consists in assuming that only exp(iwt) terms are present (counter rotating terms are neglected), in which case the amplitudes Oi(t) and o/(t) experience only slow oscillations at the frequency Aw, according to the differential equations ... 

Fig. 4.a Theoretical evolution of Mu in a transverse magnetic field. The fast oscillation is the hyperfine oscillation of Mu and the slow osdllation, showing only half of the poiod, is the rotation of Mu. 

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