Spatial oscillations

Figure Bl.19.6. Constant current 50 mn x 50 mn image of a Cu(l 11) surface held at 4 K. Tliree monatomic steps and numerous point defects are visible. Spatial oscillations (electronic standing waves) with a...

Figure 12.3b shows the widths of the Bragg layers as a function of the layer number of the CBNL depicted in Fig. 12.3a. There are two notable properties of the Bragg layers (1) the width of the high-index layers is smaller than the width of the low-index layers, and (2) the width of the layers decreases exponentially as a function of the radius, converging asymptotically to a constant value. The first property exists in conventional DBRs as well and stems from the dependence of the spatial oscillation period, or the wavelength, on the index of refraction. The second property is unique to the cylindrical geometry and arises from the nonperiodic nature of the solutions of the wave equation (Bessel or Hankel functions) in this geometry.

A chemical reaction can be designated as oscillatory, if repeated maxima and minima in the concentration of the intermediates can occur with respect to time (temporal oscillation) or space (spatial oscillation). A chemical system at constant temperature and pressure will approach equilibrium monotonically without overshooting and coming back. In such a chemical system the concentrations of intermediate must either pass through a single maximum or minimum rapidly to reach some steady state value during the course of reaction and oscillations about a final equilibrium state will not be observed. However, if mechanism is sufficiently complex and system is far from equilibrium, repeated maxima and minima in concentrations of intermediate can occur and chemical oscillations may become possible. 

An acidic bromate solution can oxidize various organic compounds and the reaction is catalyzed by species like cerous and manganous ions that can generate 1-equivalent oxidants with quite positive reduction potential. Belousov (1959) first observed oscillations in Celv]/[Cem] during Ce (III) catalysed oxidation of citric acid by bromate ion. Zhabotinskii made extensive studies of both temporal and spatial oscillations and also demonstrated that instead of Ce (III), weak 1- equivalent reductants like Mn(II) and Fe (II) can also be used. The reaction is called Belousov-Zhabotinskii reaction. This reaction, most studied and best understood, can be represented as... 

Fig. 9.7. Non-stationary behaviour in the diffusive autocatalysis model showing sustained temporal and spatial oscillations with D = 5.2 x 10 3, / = 0.08, and k2 = 0.05 (a) z = 0 or 465 (the oscillatory period) (b) z = 115 (c) z = 140 (d) z = 160 (e) z = 235. The limit cycle obtained by plotting the concentrations at the centre of the reaction zone, a,s(0) and /J (0), versus each other is shown in (f). The broken curve in (a) is the unstable stationary-state profile about...

This statement could be proved in the manner similar to that used in Section 8.2. It is important to note that the correlation dynamics of the Lotka and Lotka-Volterra model do not differ qualitatively. A stationary solution exists for d = 3 only. Depending on the parameter k, different regimes are observed. For k kq the correlation functions are changing monotonously (a stable solution) but as k < o> the spatial oscillations of the correlation functions (unstable solution) are observed. In the latter case a solution of non-steady-state equations of the correlation dynamics has a form of the non-linear standing waves. In one- and two-dimensional cases there are no stationary solutions of the Lotka model. 

Figure 24 Constant-current STM image of a Cu(lll) surface measured at 4K (Vt = 0.1 V and It = 1.0 nA). Spatial oscillations with a periodicity of 15 A are clearly emanating from monatomic step edges and point defects. (From Ref. 53.)...

Lenz P, Sogaard-Andersen L (2011) Temporal and spatial oscillations in bacteria. Nat Rev Microbiol 9 565-577... 

IIIC) Yoshikawa, K. Distinct Activation Energies for Temporal and Spatial Oscillations in... 

Example Spatial Oscillator.—A massive particle is restrained by any set of forces in a position of stable equilibrium (t.g. a light atom in a molecule otherwise consisting of heavy, and therefore relatively immovable atoms). The potential eneigy is then, for small displacement, a positive definite quadratic function of the displacement components. The axes of the co-ordinate system (x, y, z) can always be chosen to lie along the principal axes of the ellipsoid corresponding to this quadratic form. The Hamiltonian function is then... 

In our treatment of the spatial oscillator, for example, each co-ordinate depended on one w only. 

In the case of the spatial oscillator the path tills, in tho general case, a parallelepiped. In tho absence, then, of identical coinmcnsurabilities, the rectangular co-ordinatos, or functions of them, are tho only separation variables, and tho integrals Jx, Jy, and have an absolute significance. 

In tho case of tho spatial oscillator with vx vy, we could rotate the coordinate system arbitrarily about the z-axis without destroying tho property of separation in x, y, z co-ordinates. We obtained, in the various co-ordinate systems, different Jz s and Jv s. Further, rectangular co-ordinates are not the only ones for which the oscillator may be treated by the separation method. 

In order to show this and at the same time to give an example of the solution of the Hainilton-Jacobi equation by separation, in a case whore it does not resolve additively (i.e. is not of the form (1)), we shall use cylindrical coordinates in treating the spatial oscillator for which vx= vv—v. The canonical transformation (12), 7 ... 

The Belousov-Zhabotinsky reaction provides an interesting possibility to observe spatial oscillations and chemical wave propagation. If a little less acid and a little more bromide are used in the preparation of the reaction mixture, it is then a stable solution with a red color. After introducing a small fluctuation in the system, blue rings propagate, or even more complex behavior is observed. 

UP2 regimes turn out to be p2 = 1-75 and ps = 2.4 instead of p 2 = 1.45 and P3 = 1.75 when the backflow is neglected [39, 40]. In [9], it was also shown that the precession frequency /o for UPl states actually increases when the backflow is included (as expected because 71 effectively decreases). An unanticipated spatial oscillations of the backflow in the UP2 regime were also found which results from spatial oscillations of the director twist dz. They are a consequence of oscillations in the torque resulting from interference phenomena between ordinary and extraordinary light. The backflow behaves very differently for the three types of the director motion and thus can act as a sensitive diagnostic to distinguish them. 

Quantum transients are temporary features that appear in the time evolution of matter waves before they reach a stationary regime. They usually arise as a result of a sudden switch interaction that modifies the confinement of particles in a spatial region or after the preparation of a decaying sfafe [48]. The archetypical quantum transient phenomena is diffraction in time which consists of the sudden opening of a shutter to release a semi-infinite beam producing temporal and spatial oscillations of the time evolving wave [49]. A common feature in the mathematical description of quantum transient phenomena is the Moshinsky function, which as we have seen is closely related to the Faddeyeva function. Since in a recent review [48] there appears a discussion on transient phenomena for the dynamics of tunneling based on the present resonant state formalism [54, 66-76], here we restrict the discussion to the time evolution of quantum decay. 

Figure 7.13 Plot of Breathing mode of the probability density I (x,t)p, along the internal region of the quadruple barrier system when the initial state is placed on the central well. One sees that the internal dynamics consists of a quasi-periodical processes involving spatial oscillations leading to the reconstruction of the initial state. At each oscillation there is a leakage through the ends of the open system.

Based on these qualitative arguments, we may predict that for 8 < 1, in which the diffusion coefficient of the autocatalyst is greater than that of the reactant, the overall tendency will be a stabilization of the planar front. For 8 > 1, however, we anticipate that the planar front will lose stability as the destabilizing influence of the reactant diffusion becomes dominant. To test this prediction, Eqs. [91] were numerically integrated with 8 = 1 and 8 = 5. The initial conditions correspond to a discontinuity in the concentrations of A and B a = 0, p = 1 for X s Xq and a = 1, p = 0 for x > Xg at some convenient Xq for all y. To perturb the planar front, the middle third in the y direction was displaced sli tly forward. For 8 = 1, the perturbation eventually completely decays away to yield a planar front. For 8 = 5, the perturbation evolves to produce a distinctly nonplanar front in which four spatial oscillations in the y direction are displayed. The concentration profiles of a and P for the front are... 

Again the time scale of these spatial oscillations is slow so that there is no great difficulty in controlling the fluctuations, whose temperature peaks might be damaging, so long as instrumentation to measure the fluctuations and control rods to compensate for them are sufficiently widely distributed in the reactor. 

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