Model line profiles

Despite the obvious power of quantum line shape calculations for the analysis of measured collision-induced absorption spectra, a need persists for simple but accurate model line profiles, especially for extrapolating experimental spectra to both low and high frequencies for an accurate determination of the spectral moments. Reliable model profiles are also useful for line shape analyses, i.e., for representing complex spectra as a superposition of lines (where this is possible). 

The LB model. Levine and Birnbaum have developed a classical line shape theory, assuming straight paths for the molecular encounters and a dipole model of the form [232] 

The factor y R was included to simulate the repulsive intermolecular inter- 

K2(z) is the second-order, modified Bessel function of the second kind. For h — 0, this expression for a reduces to the classical result [232], 

The BC model. Birnbaum and Cohen start from a two-parameter time correlation function [36, 38] 

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We note that line shape calculations may be reduced to near trivial proportions if the basic line profile is approximated by one of the better model profiles mentioned in Chapters 5 and 6. This is generally possible, even advantageous, if the van der Waals bound-bound and bound-free transitions do not shape the spectra significantly. In such a case, the spectra are constructed by a simple superposition of the model line profiles, which is done in seconds even if small (desktop type) computers are used. The simplified line shape calculation has been used successfully on many occasions. Early examples are shown in Figs. 3.11, 3.13, and 3.33, but many more are known [58],... 

Instead of representing the model line profile T(center frequency, an ad hoc correlation function model is constructed whose Fourier transform suggests a representation of the line profiles by three functions which are shifted relative to the center frequency,... 

Feshbach or compound resonances. These latter systems are bound rotovibra-tional supramolecular states that are coupled to the dissociation continuum in some way so that they have a finite lifetime these states will dissociate on their own, even in the absence of third-body collisions, unless they undergo a radiative transition first into some other pair state. The free-to-free state transitions are associated with broad profiles, which may often be approximated quite closely by certain model line profiles, Section 5.2, p. 270 If bound states are involved, the resulting spectra show more or less striking structures pressure broadened rotovibrational bands of bound-to-bound transitions, e.g., the sharp lines shown in Fig. 3.41 on p. 120, and more or less diffuse structures arising from bound-to-free and free-to-bound transitions which are also noticeable in that figure and in Figs. 6.5 and 6.19. At low spectroscopic resolution or at high pressures, these structures flatten, often to the point of disappearance. Spectral contributions of bound dimer states show absorption dips at the various monomer Raman lines, as in Fig. 6.5. 

Figure 3.6 A comparison of an experimentally obtained STM image and line profile (f) with those calculated15 from different Si(l 11)7x7 models. In the line profiles underneath the image the dotted lines are the experimentally obtained data from (f) and the solid lines are the equivalent profiles from different structural models (a) Binnig et al. 3 (b) Chadi 44 (c) Snyder 45 (d) McRae and Petroff 46 and (e) Takayanagi et al.47 Very good agreement is obtained with Takayanagi et al. s model. (Adapted from Tromp et al.15).

Spectra like the ones shown in Fig. 3.10 may be readily decomposed into their line profiles. As an example, we show that the low-temperature measurement may be accurately represented by three identical profiles. Using the so-called BC model profile with three adjustable parameters and centering one at zero frequency (the Qo(l) line), another one at 354 cm-1 (the H2 So(0) line) and the third one at 587 cm-1 (the So(l) line), one may fit the measurement using least mean squares techniques, Fig. 3.11. The superposition (heavy line type) of the three profiles (thin... 

In conventional spectroscopy, analytical models of line profiles have been of great utility. Specifically, we mention the Lorentzian shape,... 

From the beginnings, attempts to model the line shapes of collision-induced absorption spectra were based on the assumption that the various rotational lines of induced spectra, Figs. 3.10 through 3.14, are superpositions of scaled and shifted line profiles, g(C)(v), °f a small number of different, e.g., overlap- and quadrupole-induced, types [313, 404],... 

We start with the basic relationships ( Ansatz ) of collision-induced spectra (Section 5.1). Next we consider spectral moments and their virial expansions (Section 5.2) two- and three-body moments of low order will be discussed in some detail. An analogous virial expansion of the line shape follows (Section 5.3). Quantum and classical computations of binary line shapes are presented in Sections 5.4 and 5.5, which are followed by a discussion of the symmetry of the spectral profiles (Section 5.6). Many-body effects on line shape are discussed in Sections 5.7 and 5.8, particularly the intercollisional dip. We conclude this Chapter with a brief discussion of model line shapes (Section 5.10). 

The K0 line profile. The K0 model is a three-parameter analytical expression which has been shown to approximate closely a wide selection of profiles associated with overlap induction [70, 69]. We start with the correlation function... 

Related to this near absence of logarithmic curvature of the wings is the fact that the KO model is superior in describing line profiles resulting from overlap induction. The BC shape, on the other hand, shows more curvature in the wing, as this is needed for the modeling of profiles generated by low-order multipolar induction. Purely quadrupole-induced components are closely modeled by the BC shape. 

Fig. 5.8. Root mean square relative errors of model line shapes fitted to a quantum profile, the quadrupole-induced (XL = 23) component [69], The abscissa gives the ratio of peak intensity and wing intensity of the fitted portion of the exact profile. The superiority of the BC model (lower set of data points) over the desymmetrized Lorentzian (upper set) is evident.

Second moments have also been computed, both from first principles and on the basis of the classical multipole-induction model. These are found to be in close agreement with measurements where these exist. Second moments are of a special interest in connection with modeling of three-parameter line profiles from three spectral moments [52]. In analyses based on classical expressions, the second moment is expressible in terms of the first moment specified above, multiplied by 2kT/h. 

For the rotovibrational spectra, certain model profiles, such as the Lo-rentzian (preferably modified to satisfy Eq. 6.72) and the BC profiles have previously been used successfully [422, 342], However, significant improvements are possible if model profiles are chosen that mimic the symmetry of the rotovibrational line profiles, Eq. 6.73. 

Improved vibrational line profile. In some of the modeling attempts based on the X and Y functions introduced above, numerical problems have been encountered which required special attention. Although in all practical cases simple solutions to these problems could be found, an alternative approach is now preferred because it is easy to use [48],... 

The approach does not aim to satisfy the condition Eq. 6.73 exactly. The model functions consist of a sum of three functions whose parameters are related to three -independent and three -dependent terms of the quantum spectral moments, Eqs. 6.31 through 6.34 v is the vibrational quantum number of the final states which differs from v, the initial vibrational state. As a result, the line profile consists of a core which is the same as for rototranslational spectra, and a i/-dependent correction . It converges to the standard solution for potentials that do not depend on the vibrational excitation. The models are six parameter functions which are defined by the lowest three spectral moments [48, 65],... 

The model-input parameters are T, R and M the informations provided by the calculations which can be compared with the observations are the emergent line profiles and the continuum fluxes. 

Figure 2. Comparison of calculated helium line profiles to the observed spectrum of the WN5+abs-A star WR138. These lines were not employed in the spectral analyses of this star, but are reproduced automatically by the final model obtained only by... 

Figure 3. Rectified spectra of WR46, WN3pec-A, and WR18, WN5-B. Superimposed are theoretical line profiles of models with mass-loss rates log (M/M /yr) = -4.1 (top) and -4.9 (botom) and terminal velocities of 2100 and 2500 km/s c, respectively. Otherwise, both models have the same parameters T 90kK and R = 2.5R... 

Figure 1. Theoretical line profiles of the final model with 8=0.5 superimposed on the observations. The profiles are rotationally broadend by 450 km/s... 

Three important parameters enter the model calculations the mass loss rate M, the stellar radius R and the temperature parameter T . The fit of the helium line profiles of HD 50896 requires a final wind velocity of 1700 km/s. Hence, we now calculate a small grid of models in the appropriate range of R and T and with the specially adapted v. The mass-loss rate is kept at log (M/(M /yr)) = -4.4 as an arbitrary choice. The results are presented in the form of contour lines in the log T, -log R,-plane. Those of the contours which match the observed equivalent widths or peak intensities are extracted and yield a "fit diagram". We obtain a well-defined intersection region centered about R = 2.6 R, T = 60 kK (hereafter quoted as "model B"). °... 

In total, we feel that the agreement between theoretical and observed line profiles is satisfactory and internally consistent. We take this as a confirmation of the basic physical input of our models, and as evidence that the results of the analysis are reliable. 

Figure 2. Observed H alpha line profiles from CTIO data fit by a simple model.

Ha line profile. The blue edge of the feature, which samples the highest velocity matter, is distinctly rounded in the observations. The brightest models reproduce this aspect rather well, whereas the dimmer models give sharper, abrupt blue edges. Other features sensitive to the luminosity are the Ca II lines, at H and K and the infrared triplet. These are not observed at two days, but appear in many of the theoretical spectra which are sufficiently dim or extended that appreciable Ca II exists. Satisfactory models must have the Ca II ionized away at this epoch. 

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