Discrete and Continuous Spectra

The spectral distribution of the radiant flux from a source is called its spectrum. The thermal radiation discussed in Sect.2.2 has a continuous spectral distribution described by its spectral energy density (2.13). Discrete spectra where the radiant flux has distinct maxima at certain frequencies v l, are generated by transitions of atoms or molecules between two bound states, a higher energy state and a lower state E. with the relation 

Examples of discrete absorption lines are the Fraunhofer lines in the spectrum of the sun which appear as dark lines in the bright continuous spectrum (see Fig.2.12). They are produced by atoms in the sun s atmosphere which absorb at their specific eigenfrequencies the continuous blackbody radiation from the sun s photosphere. 

The absorbed power is proportional to the density N. of molecules in level E. . The absorption lines are only measurable if the absorbed power is sufficiently high, which means that the density N. or else the path length through the probe must be large enough. In a gas at thermal equilibrium, the Boltzmann relation (2.18) 

Schematic diagram to illustrate the origin of discrete and continuous absorption spectra for (a) atomic and (b) molecular absorption 

Examples of such pumping mechanisms are absorption of light. 

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Analysis of dynamics in terms of eigenstates, both for discrete and continuous spectra. Time dependent perturbation theory. 

These are produced by autoionization transitions from highly excited atoms with an inner vacancy. In many cases it is the main process of spontaneous de-excitation of atoms with a vacancy. Let us recall that the wave function of the autoionizing state (33.1) is the superposition of wave functions of discrete and continuous spectra. Mixing of discrete state with continuum is conditioned by the matrix element of the Hamiltonian (actually, of electrostatic interaction between electrons) with respect to these functions. One electron fills in the vacancy, whereas the energy (in the form of a virtual photon) of its transition is transferred by the above mentioned interaction to the other electron, which leaves the atom as a free Auger electron. Its energy a equals the difference in the energies of the ion in initial and final states ... 

Here AEdiscr and A on are, respectively, the average energies of all the transitions into the discrete and continuous spectra E is the energy of the transition into the continuous spectrum and is the first ionization potential. The sudden-perturbation theory Eqs. (68) and (69) especially convenient for calculating A and A , since they involve only the ground-state wave function, allowing one to avoid the calculation of the spectrum of exited states. 

Fig. 4.1. The geometrical construction by which df/dE is plotted as a function of energy, so that the discrete and continuous spectra join at the threshold tangents are drawn to the curve of linestrength s against energy at the positions of the lines, and their crossing points determine the bases of boxes , whose surface area is made equal to the line strength for each transition, (after A.R.P. Rao [133]).

The Seaton-Cooper minima of alkali spectra are the best known. Because of the continuity between discrete and continuous spectra noted in the previous subsection, if a Seaton-Cooper minimum drops below the threshold, it will turn into a minimum in the course of intensities of the corresponding Rydberg series, in such a manner that continuity of the df/dE plot is preserved (see above). In the discrete part of the spectrum, one may find a minimum rather than a zero, because it is not necessary that a transition should exist at precisely the energy where cancellation occurs, i.e. the Seaton-Cooper minimum may very well fall between two members of a Rydberg series. However, the anomaly in the course of intensities in the series will be apparent. 

Figure 8.2 Differential cross sections for p—>d and p—>s transitions in both discrete and continuous spectra (from Cooper, 1962).

Valence-Rydberg-continuum mixing and a unified treatment of discrete and continuous spectra in polyelectronic atoms... 

With the same recipe, when two operators are involved and their discrete and continuous spectra of eigen-values... 

Fig. 2.2. Schematic illustration of a bound-bound (k = 0 —> k — 1) and a bound-free (k = 0 —> k = 2) transition where k labels the electronic states involved. The right-hand side depicts the corresponding discrete and continuous absorption spectra. if(Ef) is the continuum wavefunction and Eexcess is the total energy available for distribution among the various degrees of freedom. Note, that the arrow labelled tua2 also starts at the ground electronic state.

The rheological consequences of the Maxwell model are apparent in stress relaxation phenomena. In an ideal solid, the stress required to maintain a constant deformation is constant and does not alter as a function of time. However, in a Maxwellian body, the stress required to maintain a constant deformation decreases (relaxes) as a function of time. The relaxation process is due to the mobility of the dashpot, which in turn releases the stress on the spring. Using dynamic oscillatory methods, the rheological behavior of many pharmaceutical and biological systems may be conveniently described by the Maxwell model (for example, Reference 7, Reference 17, References 20 to 22). In practice, the rheological behavior of materials of pharmaceutical and biomedical significance is more appropriately described by not one, but a finite or infinite number of Maxwell elements. Therefore, associated with these are either discrete or continuous spectra of relaxation times, respectively (15,18). 

The theory and computation of state-specific wavefuncfions for discrete and continuous atomic spectra are discussed in Section 5. These are constructed in terms of numerical as well as of analytic orbitals, each set... 

The affinity spectra or constant distribution is a matter that has attracted several researchers, especially in the case of HS (Koopal and Vos 1993 Borkovec and Koper 1994a Manunza et al. 1995 Borkovec et al. 1996 Lin and Rayson 1998 Avena, Koopal, and van Riemsdijk 1999 Lin, Drake, and Rayson 2002 Garces, Mas, and Puy 2004 Orsetti, Andrade, and Molina 2009 David et al. 2010). The proton affinity spectrum is treated separately from the spectra of other ions. The reason is that, as H" " binding is almost always present, for all other ions (either metal cations or anions), there is always competition with protons thus, competitive adsorption must be considered. Here, discrete and continuous affinity spectra are discussed, both for noncompetitive and for competitive cases, and in Section 11.3, methods to extract them from experimental data are presented. 

FIGURE 11.6 Sorption isotherms of dodecylpiridinium on a soil material (EPA-12). (a) Points experimental data line fitted isotherms (almost indistinguishable) with discrete and continuous affinity spectra (b) discrete affinity spectrum, found using regularization for a small number of sites (c) continuous affinity spectrum, after regularizing for smoothness note that the smaller peaks are magnified by a factor of 30. (Reprinted with permission from Cernik, M. et al.. Environ. Sci. TechnoL, 29,2,413-425. Copyright 1995 American Chemical Society.)... 

In the modeling of ion binding to HSs, both discrete and continuous types of affinity spectra have been used. It should be noted that here we are not using the term discrete in the same sense of Chapter 12, where each binding constant is expected to correspond to an actual adsorption reaction, but instead they are separate representative values of all the binding constants in the HSs. It should be recalled here that, from a mathematical point of view, these two types are equivalent for describing titration curves, as discussed in Section 1.3.3 (see also Borkovec et al. 1996). 

Fig.2.13a,b. Discrete and continuous emission spectra and the corresponding energy level diagrams... 

In the ultraviolet SnCl2 showed a continuous absorption with a maximum intensity at about 21,044 cm"1 (3220 A). The absence of discrete bands is probably due to overlapping of closely spaced diffuse bands. For PbCl2 three regions of continuous absorption were observed. These had maximum intensities at 3600, 3200 and below 2916 A. The SnCl2 and PbCl2 spectra were interpreted as being due to 1A2 - 1B2 transitions 119 ... 

Interaction-induced absorption by the vibrational or rotational motion of an atom, ion, or molecule trapped within a Ceo cage, so-called endohedral buckmin-sterfullerene, has excited considerable interest, especially in astrophysics. The induced bands of such species are unusual in the sense that they are discrete, not continuous they may also be quite intense [127]. Other carbon structures, such as endohedral carbon nanotubes, giant fullerenes, etc., should have similar induced band spectra [128], but current theoretical and computational research is very much in flux while little seems to be presently known from actual spectroscopic measurements of such induced bands. 

In this case C(t) does not have any long-time limit. If the spectrum is entirely continuous, then it follows from the lemma of Riemann-Lebesque that C(t) vanishes as t-> oo. A system is irreversible if and only if all time correlation functions of properties t) (with zero mean) vanish as t-+ ao. Consequently, irreversible systems must have continuous spectra. In finite isolated systems, the spectrum is discrete and... 

In diatomic spectra, one distinguishes between individual bands each corresponding to a definite pair of quantum numbers v, v", and band systems, each composed of an ensemble of bands associated with a particular electronic transition. In polyatomic spectra, often (a), the individual bands of an electronic transition are so numerous and strongly overlapping that it is difficult or impossible to distinguish them individually, or (b), the electronic transition gives rise only to continuous absorption in both these situations the entire spectrum of an electronic transition is commonly called a band. IT IS RECOMMENDED (REC. 39) that the word band be reserved for definite individual bands, and that electronic transition or transition be used for the entire spectrum, whether discrete, pseudo-continuous, or strictly continuous, associated with an electronic transition or band system if the spectrum consists of discrete bands. ... 

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