Spectrum, continuous

Hg. 1.14 The connection between the Fourier transform and the Fourier series can be established by gradually increasing the period of the function. When the period is infinite a continuous spectrum is obtained. (Figure adapted from Ramirez R W, 1985, The FFT Fundamentals and Concepts. Englewood Cliffs, NJ, Prenhce Hall.)... 

In 1817, Josef Fraunhofer (1787-1826) studied the spectrum of solar radiation, observing a continuous spectrum with numerous dark lines. Fraunhofer labeled the most prominent of the dark lines with letters. In 1859, Gustav Kirchhoff (1824-1887) showed that the D line in the solar spectrum was due to the absorption of solar radiation by sodium atoms. The wavelength of the sodium D line is 589 nm. What are the frequency and the wavenumber for this line ... 

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)... 

Equation (3.73) may also be obtained from (3.42). The contribution of Xi to the ratio of determinants in (3.70) is equal to. Apart from the bound states, there is a continuous spectrum of eigenvalues e whose contribution to (3.70) may be shown [Vainshtein et al. 1982] to equal Finally (3.70) gives... 

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... 

In a synchrotron, electrons are accelerated to near relativistic velocities and constrained magnetically into circular paths. When a charged particle is accelerated, it emits radiation, and when the near-relativistic electrons are forced into curved paths they emit photons over a continuous spectrum. The general shape of the spectrum is shown in Fig. 2.4. For a synchrotron with an energy of several gigaelectronvolts and a radius of some tens of meters, the energy of the emitted photons near the maximum is of the order of 1 keV (i.e., ideal for XPS). As can be seen from the universal curve, plenty of usable intensity exists down into the UV region. With suitable mono-... 

Because IPES maps the densities of unoccupied states, it is related to other techniques that do the same (e.g. STS and SXAPS). When used in conjunction with a technique that maps the densities of occupied surface states, e.g. UPS or ELS, a continuous spectrum of state density from occupied to unoccupied can be obtained. Just as in UPS, in which angular resolution enables elucidation of the three-dimensional occupied band structure, so in IPES angular resolution enables mapping of the three-dimensional unoccupied band structure. This version is called KRIPES (i. e. K-re-solved IPES). 

In crude distillation, there are thousands of different compounds present having a virtually continuous spectrum of boiling points. It would be impractical to consider each of these compounds in describing the crude or designing the equipment to process it. Instead the crude is treated as if it were composed of a manageable number (< 50) of pseudo components. These are defined by dividing the crude distillation curve into a series of adjacent boiling cuts. 

Strecken-spektrum, n. continuous spectrum, -teilchen, n. (Math.) linear element. 

Spectral Gamma Ray Log. This log makes use of a very efficient tool that records the individual response to the different radioactive minerals. These minerals include potassium-40 and the elements in the uranium family as well as those in the thorium family. The GR spectrum emitted by each element is made up of easily identifiable lines. As the result of the Compton effect, the counter records a continuous spectrum. The presence of potassium, uranium and thorium can be quantitatively evaluated only with the help of a computer that calculates in real time the amounts present. The counter consists of a crystal optically coupled to a photomultiplier. The radiation level is measured in several energy windows. 

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. 

Fig. 4-2. Continuous spectrum from a massive target obtained by superposition of spectra from thin targets. (After Compton and Allison, X-rays in Theory and Experiment D. Van Nostrand Co.)...

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. 

Usually, high intensity is desirable in a continuous spectrum to be used for x-ray excitation. The efficiency of x-ray production,... 

Let us consider that particles are adsorbed on surface sites whose activation energies of desorption form a continuous spectrum between certain limits. The problem now consists of finding the distribution of initial surface populations ne0i according to the energies EA<-... 

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... 

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