The continuous spectrum

Thus far we have discussed the direct mechanism of dissipation, when the reaction coordinate is coupled directly to the continuous spectrum of the bath degrees of freedom. For chemical reactions this situation is rather rare, since low-frequency acoustic phonon modes have much larger wavelengths than the size of the reaction complex, and so they cannot cause a considerable relative displacement of the reactants. The direct mechanism may play an essential role in long-distance electron transfer in dielectric media, when the reorganization energy is created by displacement of equilibrium positions of low-frequency polarization phonons. Another cause of friction may be anharmonicity of solids which leads to multiphonon processes. In particular, the Raman processes may provide small energy losses. 

While being very similar in the general description, the RLT and electron-transfer processes differ in the vibration types they involve. In the first case, those are the high-frequency intramolecular modes, while in the second case the major role is played by the continuous spectrum of polarization phonons in condensed 3D media [Dogonadze and Kuznetsov 1975]. The localization effects mentioned in the previous section, connected with the low-frequency part of the phonon spectrum, still do not show up in electron-transfer reactions because of the asymmetry of the potential. 

Consider a potential V x) having a single minimum separated from the continuous spectrum by a sufficiently large barrier satisfying (1.1), e.g., a cubic parabola (fig. 19)... 

Again we use the ImF method in which the tunneling rate is determined by the nontrivial instanton paths which extremize the Eucledian action in the barrier. Let for deflniteness the potential V Q) have a single minimum at = 0, F(0) = 0, separated from the continuous spectrum... 

Fig. 15-2. Contrast between the continuous spectrum of a hot tungsten ribbon and the line spectrum of a hydrogen discharge tube.

The continuous spectrum is thus characterized by a short-wavelength limit and an intensity distribution. Experiments on other target materials have shown that these characteristics are independent of the target material although the integrated intensity increases with atomic number. (See Equation 1-3.) The continuous spectrum, therefore, results generally from the interaction of electrons with matter. Attempts (none completely successful) have been made to treat this interaction theoretically by both classical and quantum mechanics. 

The short-wavelength limit of the continuous spectrum is clearly a quantum phenomenon. X-ray generation by electron bombardment in principle resembles cathodoluminescence, and both processes are inverse photoelectric effects. The short-wavelength limit, Xq, discovered by Duane and Hunt6 obeys the relationship... 

In order to study as effectively as possible the mechanism of generation of the continuous spectrum, researches have been conducted with the use of extremely... 

Fig. 1-15. The molybdenum spectrum excited by 35-kv electrons and by the polychromatic beam from a 35-kv x-ray tube. With x-ray excitation, most of the energy appears in the characteristic lines. With electron excitation, most of it is wasted in the continuous spectrum.

Figure 1-15 illustrates an outstanding difference between electron and x-ray excitation of characteristic spectra. The former method more conveniently yields spectra of high intensity, but (as might have been expected from Figure 1-3) it unfortunately leads to a much higher background, the continuous spectrum, than is obtained with x-ray excitation. 

The introductory treatment of x-ray spectra, characteristic (1.8) and continuous (1.5), needs expansion to make it a more useful basis for discussing x-ray optics. This expansion brings with it other benefits such as the deeper insight into the excitation of x-rays by electrons (1.4) that results from a better understanding of the intensity-wavelength relationship in the continuous spectrum. 

The discussion so far applies alsr> to the excitation of characteristic x-rays by electron bombardment of a sample. In such excitation, the continuous spectrum will always appear as an appreciable and often objectionable background. The discussion also bears on the excitation of x-rays by x-rays to the extent that it describes the kind of x-ray beam often used for excitation. 

The spectrum of the operator q consists of the points in euclidean three space. The eigenfunctions x > are not normalizable in the usual way as they correspond to eigenvalues in the continuous spectrum, but are normalized to a 8-function... 

Figure 2.1. The X-ray spectrum of molybdenum, showing Ka and Kp lines superimposed upon the continuous spectrum. The quantum energy is shown in the upper scale the intensity is in arbitrary units.

Dispersing the radiation from the star into its component wavelengths reveals that the spectrum of a star is not the continuous spectrum of a black body but there are... 

Hence, it is pertinent to mention here that the continuous spectrum cannot be employed effectively for spectrochemical analysis and these spectra may be eliminated completely by volatalizing the material (sample) before excitation. 

Balmer jump spect The sudden decrease in the intensity of the continuous spectrum of hydrogen at the Balmer limit. Also known as Balmer discontinuity, bob mor ijamp )... 

We notice that it is the analytic continuation which has the effect of breaking the time-reversal symmetry. If we contented ourselves with the continuous spectrum of eigenvalues with Re = 0, we would obtain the unitary group of time evolution valid for positive and negative times. The unitary spectral decomposition is as valid as the spectral decompositions of the forward or... 

For our purposes, it is interesting to note that the perovskite ErAlo.9Cro.1O3 emits the continuous spectrum of Cr(III) without the red and green Er(III) bands in spite of the fact that the aqueous solution has e of erbium(III) at 5230 A equal to 3.2 to be compared with a-tenth of e, 1.3 of chromium(III) at 5750 A. The absorption spectrum of the nitrate solution is even more striking in a spectroscope, because the transitions of Er(III) are so much narrower. Quite generally, Cr(III) seems to be very effective to prevent narrow band luminescence of the lanthanides. 

VF is to convert the sum over discrete states to an integral over the continuous spectrum of states ... 

Grishin, N. I., 1956. Research on the continuous spectrum of noctilucent clouds, All Union Ast. Geod. Soc. Bull. USSR, 19, 3-16. 

It is evident that this potential leads to a logarithm squared contribution of order a (Za) after substitution in (3.71). One may obtain one more logarithm from the continuous spectrum contribution in (3.71). Due to locality of the potential, matrix elements reduce to the products of the values of the respective wave functions at the origin and the potentials in (3.72). The value of the continuous spectrum Coulomb wave function at the origin is well known (see, e.g., [94]), and... 

---