Accidental perturbation

In the cobweb cycle, as drawn in Fig. X.2, the equilibrium is unstable. Any small perturbation will set up an ever-widening cycle. By the same token, if the farmers begin out of equilibrium, they will never come near it. If we draw the diagram differently, with the supply curve steeper than the demand curve, the opposite is true. After a while, the farmers converge to the equilibrium and return to it after any accidental perturbation. A preliminary conclusion might be that the realization of an equilibrium depends on details of the interaction. Some deviations from equilibrium correct themselves, while others get out of hand. 

In this discussion, two mutually canceling simplifications have been made. For the transition value of the radius ratio the phenomenon of double repulsion causes the inter-atomic distances in fluorite type crystals to be increased somewhat, so that R is equal to /3Rx-5, where i has a value of about 1.05 (found experimentally in strontium chloride). Double repulsion is not operative in rutile type crystals, for which R = i M + Rx- From these equations the transition ratio is found to be (4.80/5.04)- /3i — 1 = 0.73, for t = 1.05 that is, it is increased 12%. But Ru and Rx in these equations are not the crystal radii, which we have used above, but are the univalent crystal radii multiplied by the constant of Equation 13 with z placed equal to /2, for M++X2. Hence the univalent crystal radius ratio should be used instead of the crystal radius ratio, which is about 17% smaller (for strontium chloride). Because of its simpler nature the treatment in the text has been presented it is to be emphasized that the complete agreement with the theoretical transition ratio found in Table XVII is possibly to some extent accidental, for perturbing influences might cause the transition to occur for values a few per cent, higher or lower. 

Although most of the water in the meniscus evaporates once the tip has been retracted, residual structures can be observed in a radius of several tens of micrometers (depending on humidity and contact time) around the original contact point. For the tip radius and loads used in these experiments, the contact radius is approximately 10 A. The residual structures are in the form of flat islands and sometimes droplets. In our first experiments the perturbation created by a brief tip contact was not fuUy appreciated. Accidental tip contacts during approach of the tip to the surface do often occur. In such cases the tip is subsequently moved to an adjacent area, several micrometers away, to study the unperturbed surface. However, as stated already, the perturbed areas can extend over tens of micrometers away from the contact point. Droplets can be observed when the relative humidity is... 

The ex model has been elaborated in a number of ways. An electrostatic perturbation was added (33) to account for band splittings in the d-d spectra of tetragonal copper(II) ammine complexes where the simple AOM predicted accidental degeneracy the merits of this refinement will be discussed in 2.5.1. Another development has been the introduction of d—s and d—p mixing, which is apparently necessary to account for the d-d spectra of chlorocuprates(II) (34). This requires the additional parameters e, edpa and edpv. 

F ERMI RESONANCE. In polyatomic molecules. Hvo vibrational levels belonging to different vibrations lor combinations of vibrations) may happen lo have nearly die same energy, and therefore be accidentally degenerate. As was recognized hy Fermi in the case of CO such a "resonance" leads to a perturbation of the energy levels that is very similar to the vibrational perturbations of diatomic molecules. 

In this theory, the dynamics of the intrinsic-surface-confined excitons account surprisingly well—in a natural way, without introducing ad hoc parameters—for the surface emissive properties, and they allow, a contrario, a very sensitive probing of various types of surface disorders, whether residual, accidental, or induced. The disorder may be thermal, substitutional, chaotic owing to surface chemistry, or mechanical owing to interface compression. It may be analyzed as a specific perturbation of the surface exciton s coherence and of its enhanced emissive properties. 

The evaluation methodology must be capable of providing statistically reliable results at the levels in which we are interested. The measurement processes that are used to obtain responses need to be robust in the sense that they are resistant to both determinate accidental and random perturbations. This requirement for robustness applies to the processes for producing, representatively sampling, and evaluating products, be they materials, processes or otherwise. 

Recently Simmons and Tllford (126) have presented spectroscopic evidence for an accidental predissociation of CO at 94,872 cm. This energy is below the 99,650 cm threshold energy for production of 0( D) and just above that for process 3. They observe that the R(30) doublet in the 0,0 band of the E-X system is enhanced in absorption and missing in emission and attribute the predissociation to a perturbing state which correlates with ground state atoms. 

The rate of conventional (single-center) two-photon absorption depends on the square of the focussed laser intensity, and as long ago as 1968 Gontier and Trahin showed that in the absence of accidental resonances an intensity factor of (/// ) is introduced for each additional photon involved in a multiphoton atomic excitation process. The constant / is a characteristic irradiance whose value depends on the sample, and corresponds to the situation where perturbation theory breaks down and all multiphoton processes become equally feasible. A similar trea ment of molecules leads to an intensity factor per photon of y = where If is an irradiance that... 

In a crystal, perturbations can be classified as internal and external. The internal perturbations are disturbances from an equilibrium condition, taken as an ideal uniform distribution of impurities or defects which do not modify the crystal lattice and the average electronic density. Mechanical perturbations can be microscopic, like those introduced by impurities or defects producing large local volume changes, which reflect on crystal lattice spacings when their concentration is large, or macroscopic due to residual or accidental stresses. Permanent perturbations can also be produced by unrelaxed stresses... 

One of the most dramatic manifestations of an interference effect is the vanishing of a line or of an entire band that, on the basis of known Franck-Condon factors and inappropriately simple intensity borrowing ideas, should be quite intense (see Fig. 6.6). This effect can easily be mistaken as an accidental predissociation (Section 7.13). Yoshino, et al, (1979) have studied the valence Rydberg N2 b, E+ cVE+ perturbations. Abrupt decreases in emission intensity for c 4 — X1E+ (v = 1 and 4) and b — X (v = 4) bands had been attributed to weak predissociation rather than perturbation effects (Gaydon, 1944 Lofthus, 1957 Tilford and Wilkinson, 1964 Wilkinson and Houk, 1956). The b (v = 4) C4 (v = 1) and b (v = 13) C4 (v = 4) deperturbation models of Yoshino et al., (1979) provide a predissociation-ffee unified account of both level shift and intensity effects. Weak predissociation effects cannot be ruled out, but are not needed to account for the present experimental observations. 

When a predissociation is weak, its interpretation is often difficult small first-order effects can be masked by second-order effects. If only a few lines are missing or weakened, it is necessary to consider the possibility of an accidental predissociation, or, in other words, a three-state interaction involving a local perturbation by a weakly predissociated level (See Section 7.13). Predissociation of normally long-lived (metastable) states detected in emission may originate from very small interactions such as spin-spin or hyperfine interaction, as is the case for the I2 B3II0+ state (Broyer, et al., 1976). 

Anomalous isotope effects occur at accidental or indirect predissodations, which are discussed in Section 7.13. The accidentally predissociated v, J-level is perturbed by a v, 7-level that is directly predissociated by a third (unbound) state. The accidentally predissociated level, having acquired an admixture of the perturber s wavefunction, borrows part of the characteristics of the perturber,... 

In Section 7.8 the possibility of predissociation of isolated lines was mentioned. This is usually called accidental predissociation and can be interpreted as perturbation of a nominally bound rotational level by a predissociated level that lies nearby in energy for this value of J. This type of predissociation should more generally be called indirect predissociation, since the predissociation takes place through an intermediate state (or doorway state, see Section 9.2). 

This formula shows how interference between terms in the sum over predissociated perturbers can increase or decrease the accidental width (Lefebvre-Brion and Colin, 1977). 

The earliest pulsed laser quantum beat experiments were performed with nanosecond pulses (Haroche, et al., 1973 Wallenstein, et al., 1974 see review by Hack and Huber, 1991). Since the coherence width of a temporally smooth Gaussian 5 ns pulse is only 0.003 cm-1, (121/s <-> 121 cm"1 for a Gaussian pulse) nanosecond quantum beat experiments could only be used to measure very small level splittings [e.g. Stark (Vaccaro, et al., 1989) and Zeeman effects (Dupre, et al., 1991), hyperfine, and extremely weak perturbations between accidentally near degenerate levels (Abramson, et al., 1982 Wallenstein, et al., 1974)]. The advent of sub-picosecond lasers has expanded profoundly the scope of quantum beat spectroscopy. In fact, most pump/probe wavepacket dynamics experiments are actually quantum beat experiments cloaked in a different, more pictorial, interpretive framework,... 

Accidental degeneration is a rare and remarkable exception in astronomy the odds against (1) being exactly fulfilled are infinite. A close approach to it is found in the case of perturbations of some minor planets (Achilles, Patroclus, Hector, Nestor) which have very nearly the same period of revolution as Jupiter. In atomic theory, on the other hand, where the Jfc° s can only have discrete values, accidental degeneration is very common. 

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