Recurrence rotational

The principle of the acquisition system is to translate the probe into a tube (including hemispherical drilled holes) step by step, every 0.04 mm, after a forwards and backwards 360 rotation of the tube trigging every 0.2° angular step a 360° electronic scanning of tube with the 160 acoustic apertures. During the electronic scanning the tube is assumed to stay at the same place. The acquisition lasts about 30 minutes for a C-scan acquisition with a 14 kHz recurrence frequency. 

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. 

Figures 3a and 3an show fs DFWM spectra of acetic acid (CH3COOH) and per-deuterated acetic acid (CD3COOD) vapor from a gas cell experiment (300K). In contrast to formic acid, acetic acid shows only J-type recurrences from dimeric species in the fs DFWM spectra at room temperature. The difference of 45 ps between the position in time of the recurrences from (CH-)C00H)2 and (CD3COOD)2 is determined by the smaller rotational constants B, C of (CD3COOD)2. From the non-linear fitting (Fig 3b, 3bo) the rotational constants A=5,7 0.3GHz B+C=1657.2 1.2MHz of the acetic acid dimer (CH3COOH)2 and...

The former feature is demonstrated by a part of the fs DFWM spectrum of benzene as depicted in Fig. 3. The data displayed is an extension to the published spectra in Ref. [5]. The experimental trace in Fig. 3a shows regions around the J-type recurrences at a total time delay of ca. 1.5 ns. In Fig. 3b a simulated spectrum is given, computed on the basis of a symmetric oblate rotor with the rotational constant B" = 5689 MHz and the CDs Dj- 1.1 kHz and Djk = -1.4 kHz. For comparison in Fig. 3c the same recurrences are calculated with all CDs set to zero. It can be seen that the CDs cause a strong modulation, splitting and time shift in the recurrences. Even recurrences are differently affected than odd ones. One can conclude that high temperatures do not prevent the occurrence of rotational recurrences and thus, the application of RCS. On the contrary, they enable the determination of CDs by analysis of spectral features at long time delay and hence, reflect the non-rigidity of molecules. 

In extending the studies of vibrational coherence to rotational coherence in isolated molecules, we formulated the concept of rotational recurrences (echoes ), which led to rotational coherence spectroscopy. A polarized picosecond (and later femtosecond) pulse was used to orient a molecular ensemble (Fig. 7). The molecules then rotate freely with different speeds... 

Figure 7. (a) Concept of time-dependent alignment as a method for structural determination. Top Initial alignment at t = 0, dephasing, and recurrence of alignment at later times. Bottom Classical motion of a rigid prolate symmetric top. (b) Structures of stilbene and tryptamine-water complex from rotational coherence spectroscopy transients are shown, [see ref. 13]. 

The development of a new form of spectroscopy based on the exploitation of the time evolution of the coherence associated with the rotational motion of an excited molecule. Conventional spectroscopies depend on the measurement of differences between the energy levels of a molecule, which become more and more difficult to measure and to interpret as the size of the molecule increases. In contrast, the intervals between recurrences in the coherent rotational motions of large molecules are directly related to the moments of inertia of the molecules and can be used to determine their structures. 

The experiments described above used nanosecond laser pulses, which are much longer than the rotational period of the molecules. At the termination of the pulse, the pendular state that is formed relaxes adiabatically to a free-rotor eigenstate. If instead picosecond laser pulses are used, a rotational wave packet is formed by successive absorption and re-emission of photons during the laser pulse. Such wave packets are expected to display periodic recurrences of the alignment after the end of the pulse. 

The matrix Un represents a simple rotation with rotation angle /3 . Using (4.3.4) in (4.3.2), we obtain a recurrence relation for the rotation angles... 

Among of the systems under study, clusters with three, five, six and twelve atoms possess higher symmetries having more than a twofold rotational axis. As a result doubly degenerate librational states occur in these complexes. In Fig. 10 one of the possible representations of these degenerate states in the HCl(Ar)i2 cluster is displayed, however, any linear combination of these functions is also valid. Three pronounced recurrences shown in Fig. 10 demonstrate repeated collisions of hydrogen with the cluster. The situation becomes dramatically different for photolysis started from the first non-degenerate librationally excited state. In this... 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

The result given here may also be extended to rotation in space. As far as the cage motion is concerned, the complex susceptibility will still be governed by the Rocard equation because the equations of motion factorize. However, the solution for the dipole correlation function is much more complicated because of the difficulty of handling differential recurrence relations pertaining to rotation in space in the presence of a potential. 

In conclusion, we remark that a detailed review of matrix continued frachon methods for the solution of differential recurrence relations is available in Ref. 70, while a detailed account of the rotational Brownian motion of the sphere is available in Ref. 33. 

Equation (136), which is a three-term algebraic recurrence relation for the ci, (i), can be solved in terms of continued fractions [21], thereby yielding the generalization of the familiar Gross Sack result [18,19] for a fixed axis rotator to fractional relaxation, namely. 

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