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In the last few years there has been an increasing interest in the visual representation of mathematical concepts. The fact that computers can help us perform graphical tasks very easily has been translated into an increasing interest in diagrammatic representations in general. Several experiments have shown that diagrammatic reasoning plays a main role in the way in which experts in several areas solve problems (Gobert and Freferiksen [1992] and Kindfield [1992]). Two kinds of explanations have been given for the advantages of visual representations over linguistic ones. The first kind of explanation is psychological. It has been argued that visual representations are easier to use because they resemble the mental models hurnans build to solve problems Stenning and Oberlander [1991], Johnson-Laird and Byrne [1991], arid Tverski [1991]. The second kind of explanation is related to computational efficiency. Larkin and Simon [1987] have argued that diagrammatic representations are computationally more efficient than sentential representations because the location of each element in the diagram corresponds to the spatial or topological properties of the objects they represent. However, the efficiency of the use of diagrams is not enough justification for their use in analytical areas of knowledge. Mathematical discoveries often have been made using visual reasoning, but those very same discoveries were not justified by the visual reasoning. Diagrams are associated with intuitions and illustrations, not with rigorous proofs. Visual representations are allowed in the context of discovery, not in the context of justification. Many authors have considered diagrams in opposition to deductive systems. Lindsay [1988], for instance, has claimed that the main feature of visual representations is that they correspond to a non-deductive kind of inference system. Koedinger and Anderson [1991] have related diagrammatic reasoning in geometry to informal, inductive strategies to solve problems. Thus, though we have an empirical justification for the use of diagrams in mathematics (people use them and they work!) we do not usually have an analytical justification. In fact, the history of mathematics, and especially the history of geometry, is full of mistakes related to the use of diagrams.

Diagrammatic reasoning is reasoning whose task is partially taken over by operations on diagrams. It consists of two kinds of activities: (i) physical operations, such as drawing and erasing lines, curves, figures, patterns, symbols, through which diagrams come to encode new information (or discard old information), and (ii) extractions of information diagrams, such as interpreting Venn diagrams, statistical graphs, and geographical maps. Given particular tasks of reasoning, different types of diagrams show different degrees of suitedness. For example, Euler diagrams are superior in handling certain problems concerning inclusion and membership among classes and individuals, but they cannot be generally applied to such problems without special provisos. Diagrams make many proofs in geometry shorter and more intuitive, while they take certain precautions of the reasoner’s to be used validly. Tables with particular configurations are better suited than other tables to reason about the train schedule of a station. Different types of geographical maps support different tasks of reasoning about a single mountain area. Mathematicians experience that coming up with the “right” sorts of diagrams is more than half-way to the solution of most complicated problems. Perhaps many of these phenomena are explained with reference to aspect (ii) of diagrammatic reasoning because some types of diagrams lets a reasoner retrieve a kind of information that others do not. or lets the reasoner retrieve it more “easily” than others. In fact, this is the approach that psychologists have traditionally taken. In this chapter, we take a different path and focus on aspect (i) of diagrammatic reasoning. Namely, we look closely at the process in which a reasoner applies operations to diagrams and in which diagrams come to encode new information through these operations. It seems that this process is different in some crucial points from one type of diagrams to another, and that these differences partially explain why some types of diagrams are better suited than others to particular tasks of reasoning.

In this book we are concerned primarily with disequilibrium chemistry, of which the sun is the principal driving force. The sun is not, however, the only source of disequilibrium chemistry in the solar system. We briefly discuss other minor energy sources such as the solar wind, starlight, precipitation of energetic particles, and lightning. Note that these sources are not independent. For example, the ultimate energy source of the magnetospheric particles is the solar wind and planetary rotation; the energy source for lightning is atmospheric winds powered by solar irradiance. Only starlight and galactic cosmic rays are completely independent of the sun. While the sun is the energy source, the atoms and molecules in the planetary atmospheres are the receivers of this energy. For atoms the interaction with radiation results in three possibilities: (a) resonance scattering, (b) absorption followed by fluorescence, and (c) ionization. lonization usually requires photons in the extreme ultraviolet. The interaction between molecules and the radiation field is more complicated. In addition to the above (including Rayleigh and Raman scattering) we can have (d) dissociation, (e) intramolecular conversion, and (f) vibrational and rotational excitation. Note that processes (a)-(e) involve electronic excitation; process (f) usually involves infrared radiation that is not energetic enough to cause electronic excitation. The last process is important for the thermal budget of the atmosphere, a subject that is not pursued in this book. Scattering and fluorescence are a source of airglow and aurorae and provide valuable tools for monitoring detailed atomic and molecular processes in the atmosphere. Processes (c) and (d) are most important for determining the chemical composition of planetary atmospheres. Interesting chemical reactions are initiated when the absorption of solar energy leads to ionization or the breaking of chemical bonds. In this chapter we provide a survey of the absorption cross sections of selected atoms and molecules. The selection is based on the likely importance of these species in planetary atmospheres.

In common with astrophysical usage the word intensity will denote specific intensity of radiation, i.e., the flux of energy in a given direction per second per unit frequency (or wavelength) range per unit solid angle per unit area perpendicular to the given direction. In Fig. 2.1 the point P is surrounded by a small element of area dπs, perpendicular to the direction of the unit vector s. From each point on dπs a cone of solid angle dωs is drawn about the s vector. The bundle of rays, originating on dπs, and contained within dωs, transports in time dt and in the frequency range v to v + dv, the energy . . . Ev = Iv(P,S) dπs dωs dv dt, (2.1). . . where Iv(P, s) is the specific intensity at the point P in the s-direction. If Iv is not a function of direction the intensity field is said to be isotropic ; if Iv is not a function of position the field is said to be homogeneous.

The formal theory developed in Chapter 2 assumed the Stokes parameters to be additive. The sufficient condition for additivity is that the radiation fluxes in the atmosphere shall have no phase coherence. Thermal emission from independently excited molecules is necessarily incoherent with respect to phase. Atmospheric scattering centers are widely and randomly spaced, and they can be treated as independent and incoherent scatterers. The situation differs, however, when we consider details of the scattering process within a single particle, and in order to derive the extinction coefficient and the scattering matrix (see § 2.1.3) we must make use of a theoretical framework that involves the phase explicitly. The problem of the interaction between an electromagnetic wave and a dielectric particle can be precisely formulated using Maxwell’s equations. For a plane wave and a spherical particle, Mie’s theory provides a complete solution (see §7.6). But the general problem is complicated and our understanding is rendered more difficult by preconceptions based on the approximations of elementary optics. This chapter provides a brief survey of the important results and the underlying concepts. The geometry of the problem is illustrated in Fig. 7.1. An isolated particle is irradiated by an incident, plane electromagnetic wave. The plane wave preserves its character only if it propagates through a homogeneous medium; the presence of the scattering particle, with electric and magnetic properties differing from those of the surrounding medium, distorts the wave front. The disturbance has two aspects: first, the plane wave is diminished in amplitude; second, at distances from the particle that are large compared with the wavelength and particle size, there is an additional, outward-traveling spherical wave. The energy carried by this spherical wave is the scattered energy; the total energy lost by the plane wave corresponds to extinction; the difference is the absorption. The properties of the spherical wave in one particular direction (the line of sight) will be considered. This direction can be specified by the scattering angle 6 (see Fig. 7.1) in a plane containing both the incident and scattered wave normals (the plane of reference), and the azimuth angle ϕ) between the plane of reference and a plane fixed in space.

In this chapter the same basic topics are addressed as in the previous chapter, but now in the presence of matter. This greatly complicates the description of optical propagation and continues to be the primary topic of the remaining chapters. A formal structure is developed to handle absorption and scattering phenomena in general. The modeling of optical propagation is reduced to having to know the complex index of refraction of the medium. A macroscopic description represents the large-scale observable character of optical propagation. At this level, many models are phenomenological, but lead to important general properties, definitions, formulas, and the establishment of basic concepts. Because microscopic models to be presented in future chapters contain considerable detail, this section is an important prerequisite to the remaining text. Again, plane waves are a useful tool for the description of optical propagation. The Poynting vector, causality, and Poynting’s theorem are used to develop and derive quantities and relationships concerning radiometry and the flow of electromagnetic power at optical frequencies. Consider Maxwell’s equations again, but in the presence of linear isotropic matter. Now the constitutive relations will play a more important role and are the foundation of classical dispersion theory.

Although the primarily phenomenological theory of absorption and refraction of light by matter, based on classical models as presented in Chapter 4, is very useful, it is incomplete and often inadequate. A more complete and accurate picture of electrodynamics is given by the theory of quantum optics, and that is the topic of this chapter. The models developed in this chapter are more detailed and therefore more complicated than the phenomenological models of Chapter 4. The most robust models, which are applied in Part II, are presented in this chapter. The quantum models accurately represent experimental data and allow extrapolation and interpolation of such data. Many practical computer based models concerning optical propagation are based on this theory. The theory of elastic scatter as presented in Chapter 4 is consistent with quantum optics and is not presented again. (However, inelastic scatter must address the quantum nature of the scattering medium.) Quantum optics is not completely covered in this chapter. Entire textbooks are devoted to this diverse and comprehensive topic covering optics (see Refs. 5.1–5.3). The emphasis of this book is on absorption and reflection spectroscopy. Now details of internal structure of the medium impacting light–matter interaction are examined. The classical oscillator model is upgraded by semiclassical radiation theory and a quantum oscillator model is developed. Semiclassical radiation theory is based on a quantized medium coupled to a classical field. It is often applied to laser theory, where near-line-center stimulated emission dominates. The quantum oscillator model again utilizes the quantized medium and classical field, but with more attention to detailed balance between absorption and emission. It satisfies causality and the fundamental symmetry relationships established in Chapter 2. These quantum optics models are more complete formalisms and provide solutions to the shortcomings of classical electrodynamics. Of particular interest to propagation in gaseous media is the line shape in the far wing. To achieve long path lengths, propagation near line center of a resonance must be avoided. Line shape models in quantum optics accurately represent much of the frequency and temperature dependence observed in experimental data.

The first step in a unimolecular reaction involves energizing the reactant molecule above its decomposition threshold. An accurate description of the ensuing unimolecular reaction requires an understanding of the state prepared by this energization process. In the first part of this chapter experimental procedures for energizing a reactant molecule are reviewed. This is followed by a description of the vibrational/rotational states prepared for both small and large molecules. For many experimental situations a superposition state is prepared, so that intramolecular vibrational energy redistribution (IVR) may occur (Parmenter, 1982). IVR is first discussed quantum mechanically from both time-dependent and time-independent perspectives. The chapter ends with a discussion of classical trajectory studies of IVR. A number of different experimental methods have been used to energize a unimolecular reactant. Energization can take place by transfer of energy in a bimolecular collision, as in . . . C2H6 + Ar → C2H6* + Ar . . . . . . (4.1) . . . Another method which involves molecular collisions is chemical activation. Here the excited unimolecular reactant is prepared by the potential energy released in a reactive collision such as . . . F + C2H4 → C2H4F* . . . . . . (4.2) . . . The excited C2H4F molecule can redissociate to the reactants F + C2H4 or form the new products H + C2H3F. Vibrationally excited molecules can also be prepared by absorption of electromagnetic radiation. A widely used method involves initial electronic excitation by absorption of one photon of visible or ultraviolet radiation. After this excitation, many molecules undergo rapid radiationless transitions (i.e., intersystem crossing or internal conversion) to the ground electronic state, which converts the energy of the absorbed photon into vibrational energy. Such an energization scheme is depicted in figure 4.1 for formaldehyde, where the complete excitation/decomposition mechanism is . . . H2CO(S0) + hν → H2CO(S1) → H2CO*(S0) → H2 + CO . . . . . . (4.3) . . . Here, S0 and S1 represent the ground and first excited singlet states.