Absorption coefficient profiles

Development of the W — fN H relation, the CoG, beyond the weak-line limit depends on the profile of the absorption coefficient <)>(AA). An extreme form for the profile is effective at illustrating this point. Suppose (AA) = a for AA = AXD and 0 for AA > AAj> Normalization of provides the relation connecting the constant a, the width AXd, and the /-value - the derivation is left as an exercise for the student With increasing fN H, I(AX)/Iq falls within the line to its minimum value of zero. At which point, the equivalent width has saturated at W = 2AAd- Note that, unlike W in the weak-line limit, the CoG beyond the weak-line portion depends on the shape of the line absorption coefficient - here, the width AAd. This dependence means that conversion of a measured W to JNlH for realistic absorption coefficient profiles demands observational or theoretical knowledge of the absorption coefficient s profile. This requirement plus the reduced sensitivity of W to /NlH make this part of the CoG less attractive for abundance determinations. This stretch of the CoG is variously referred to as the flat, Doppler or saturated part. 

In this expression, z is the distance from the surface into the sample, a(z) is the absorption coefficient, and S, the depth of penetration, is given by Eq. 2. A depth profile can be obtained for a given functional group by determining a(z), which is the inverse Laplace transform of A(S), for an absorption band characteristic of that functional group. 

It would appear that measurement of the integrated absorption coefficient should furnish an ideal method of quantitative analysis. In practice, however, the absolute measurement of the absorption coefficients of atomic spectral lines is extremely difficult. The natural line width of an atomic spectral line is about 10 5 nm, but owing to the influence of Doppler and pressure effects, the line is broadened to about 0.002 nm at flame temperatures of2000-3000 K. To measure the absorption coefficient of a line thus broadened would require a spectrometer with a resolving power of 500000. This difficulty was overcome by Walsh,41 who used a source of sharp emission lines with a much smaller half width than the absorption line, and the radiation frequency of which is centred on the absorption frequency. In this way, the absorption coefficient at the centre of the line, Kmax, may be measured. If the profile of the absorption line is assumed to be due only to Doppler broadening, then there is a relationship between Kmax and N0. Thus the only requirement of the spectrometer is that it shall be capable of isolating the required resonance line from all other lines emitted by the source. 

Fenton. The maximum performance is usually observed at pH slightly below 3 because of two reasons (i) colloids that begin to precipitate at pH above 3 via the binuclear species (Q 2,2) are suppressed and (ii) the concentration of Fe(OH) ( i) is close to its maximum. Fe(OH ) species possess a high absorption coefficient under irradiation and maximize the oxidation yield. This holds for diluted systems. When the total Fe concentration is increased, the binuclear species become dominant and precipitation is favored. Figure 6.3b clarifies this aspect by showing the Fe2(OH)2 + concentration profiles for increasing Fe concentrations. In those cases, lowering the pH to about 2 is favorable. 

The depth-profile of photon absorption is analogous to that for UV-visible light, i.e. I = Io exp(-Ad), where the mass energy absorption coefficient, u/g is used instead of the extinction coefficient. Particulate energy absorption can be described by relative stopping powers. 

Accurate measurements of low order structure factors are based on the refinement technique described in section 4. Using the small electron probe, a region of perfect crystal is selected for study. The measurements are made by comparing experimental intensity profiles across CBED disks (rocking curves) with calculations, as illustrated in fig. 5. The intensity was calculated using the Bloch wave method, with structure factors, absorption coefficients, the beam direction and thickness treated as refinement parameters. 

Several other laser lines have already been used in a similar way 2-90) to measure absorption coefficients and absolute wavelengths of absorption lines, as well as the line profiles and possible fine structures of many polyatomic molecules. By substituting different isotopes, the isotopic shift of absorption lines has been studied Table 2 shows some of the molecules investigated by the laser line absorption technique, together with the exciting laser wavelengths and some remarks about the kind of measurements performed. 

Since the single-mode laser linewidth is small compared to the absorption linewidth, one can probe the absorption profile by tuning the laser line across it, getting more information than by measuring the absorption coefficient averaged over the whole doppler width 6). 

In this case only one group with velocity components = 0 Avz is available for absorption of the laser line. The absorption coefficient therefore has a minimum at the center of the inhomogeneous molecular absorption profile (see Fig. 14 b), and the laser intensity will... 

For most treatments, the spectral density, J(a>), Eq. 2.86, also referred to as the spectral profile or line shape, is considered, since it is more directly related to physical quantities than the absorption coefficient a. The latter contains frequency-dependent factors that account for stimulated emission. For absorption, the transition frequencies ojp are positive. The spectral density may also be defined for negative frequencies which correspond to emission. 

We note that in a classical formula Planck s constant does not appear. Indeed, the zeroth moment Mo of the spectral density, J (o), does not depend on h, as the combination of Eqs. 5.35 and 5.38 shows. On the other hand, the classical moment y of the absorption profile, a(cu), is proportional to /h because the absorption coefficient a depends on Planck s constant see the discussions of the classical line shape below, p. 246. In a discussion of classical moments it is best to focus on the moments Mn of the spectral density, J co), instead of the moments, yn, of the spectral profile. 

The classical emission profile, Eq. 5.72, may be converted to an absorption profile with the help of Kirchhoff s law, Eq. 2.70, which relates the absorption coefficient a to the emitted power per unit frequency interval per unit volume, with the help of Planck s law, Eq. 2.71, according to... 

In other words, for tetrahedral molecules, these relationships differ from the ones used for the linear molecules, especially Eq. 4.18. As a consequence, we must rederive the relationships for the spectral line shape and spectral moments. If the intermolecular interaction potential may be assumed to be isotropic, the line shape function Vg(a> T), Eq. 6.49, which appears in the expression for the absorption coefficient a, Eq. 6.50, may still be written as a superposition of individual profiles,... 

In the framework of the impact approximation of pressure broadening, the shape of an ordinary, allowed line is a Lorentzian. At low gas densities the profile would be sharp. With increasing pressure, the peak decreases linearly with density and the Lorentzian broadens in such a way that the area under the curve remains constant. This is more or less what we see in Fig. 3.36 at low enough density. Above a certain density, the l i(0) line shows an anomalous dispersion shape and finally turns upside down. The asymmetry of the profile increases with increasing density [258, 264, 345]. Besides the Ri(j) lines, we see of course also a purely collision-induced background, which arises from the other induced dipole components which do not interfere with the allowed lines its intensity varies as density squared in the low-density limit. In the Qi(j) lines, the intercollisional dip of absorption is clearly seen at low densities, it may be thought to arise from three-body collisional processes. The spectral moments and the integrated absorption coefficient thus show terms of a linear, quadratic and cubic density dependence,... 

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