Interference oscillations

The corresponding differential cross sections f will therefore exliibit interference oscillations. The integral cross sections are... 

On the other hand, one must recognize that quantal interference oscillations may result in transition probabilities that oscillate around the simple classical-like scaling laws (Levine and Wulfman, 1979). Figure 2.4 is an illustration of such a phenomenon for the overtone transitions in HF. 

Second, the calculated (as well as the measured) distributions are remarkably smooth although often more than fifty or so rotational states are populated. If so many quantum states take part in a collision, one intuitively expects pronounced interference oscillations. The reason for the absence of interferences is the uniqueness between 70 and j one and only one trajectory contributes to the cross section for a specific final rotational state. If two trajectories that lead to the same j had comparable weights, the constructive and destructive interference, within a semiclassical picture, would lead to pronounced oscillations (Miller 1974, 1975 Korsch and Schinke 1980 Schinke and Bowman 1983). These so-called supernumerary rotational rainbows are well established in full collisions (Gottwald, Bergmann, and Schinke 1987). If the weighting function W (70) is sufficiently wide that both trajectories contribute to the dissociation cross section, similar oscillations may also exist in photodissociation (see, for example, Philippoz, Monot, and van den Bergh 1990 and Miller, Kable, Houston, and Burak 1992). 

The behavior of the functions r(co) and t(w) is complicated by a series of interference oscillations due to the complex cotangent. Fora transparent layer [n(co) real], the oscillations have zero minima for r to) at n(co)a)e/nc = integers. In the case of weak absorption by the phonon continuum (n = v + in, k 0), the back-face reflection component, for a sufficiently thick sample, is absorbed, and we obtain... 

There Is also a small reduction In the Interference oscillations In the forward direction. It Is significant to note, however, that the background angular distribution Is sideways peaked, as a consequence of the generally Increasing behavior of the opacity functions of Figure 7 at low 1. 

Recall that even for that simple problem the amplitude of the interference oscillations does not vanish in the h - 0 limit. It is only the frequency of oscillations that becomes more rapid in that limit.61... 

The amplitude of a single contribution is just the classical cross section, so that the classical result is obtained for the cross section. If more than one point of stationary phase exists the cross section displays characteristic interference oscillations which arise from the different classical paths ... 

Buck (b) Semiclassical quantities 5 Interference oscillations Semiclassical Minimalization Rainbows, large B... 

The quality of the approximation is illustrated in Figure 9.6 for fhe case = 0.5. In the crossover region the interference oscillations are nicely reproduced, showing that their phases are correct. The next-to-leading order asymptotic correction makes a clear improvement in the description (upper curves). When = 0.1, there is almost no exponential region, while for large... 

Fig. 3.4.2. Raw recorder traces of interference oscillations due to the change in the sample birefringence with deformation for hexyloxyazoxybenzene at various temperatures. Polarizer and crossed analyser are inclined at 45° to the principal axes of the specimen. The sudden onset of oscillations occurs at the threshold field. The increase in the threshold for the successive traces illustrates the rapid temperature variation of the elastic constant. Sample thickness 45 m. T, = 128.5 °C. (After Gruler, Scheffer and Meier.

An extension of the semiclassical theory into the non-classical regime is possible by analytical continuation and complex-valued trajectories [9-11]. If 2 is close to classical probabilities become singular and the primitive semiclassical theory breaks down however, this failure can be corrected on a more sophisticated (uniform) level of the semiclassical theory. Thus, the probability has three different regimes the classically allowed region, in which the probability shows quantum interference oscillations the classically forbidden region, in which the probability decays exponentially to zero between these two regimes the probability shows a pronounced rainbowlike maximum. 

Figure 3 shows that at energies below the triplet barrier, the TSH-D probabilities have broad humps that are similar to the quantum results, except that the Stuckelberg interference oscillations are not correedy described. Tl problem with interference oscillations is to be expected, as the TSH model does not describe the coherence that would be associated with trajectories that react via the two pathways that are available. However the TSH-D results do predict approximately the correct peak probabilities, and the change in die probabilities that occurs in going from Model 1 to 7. 

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