Symmetry of line profiles

If bound state effects are suppressed, the classical profile peaks at zero frequency where it has a zero slope the classical profile is symmetric in frequency. The quantum profile, on the other hand, peaks at somewhat higher frequencies and has a logarithmic slope of h/2kT near zero frequency. At positive frequencies, the quantum profile is more intense than the classical profile, but at not too small negative frequencies the opposite is true. These facts are related to the different symmetries of these profiles, which we examine in the next subsection. We note that various procedures have been proposed to correct classical profiles somehow so that these simulate the symmetry of quantum profiles. 

Detailed balance. If in the lower part of Eq. 2.86 we interchange the arbitrary subscripts i and /, with the help of Eq. 2.82, we have 

The condition, Eq. 5.73, is also often quoted using frequency shifts instead of absolute frequencies in that case, the symmetry may be altered, see Chapter 6. 

Welsh and his associates have pointed out early on that the observed spectral profiles are strikingly asymmetric [422]. Of course, line shapes computed on the basis of a quantum formalism will always have the proper asymmetry so that measurement and theory may be directly compared. Problems may arise, however, if classical profiles are employed for analysis of a measurement, or if classical expressions for computation of spectral moments are used for a comparison with the measurement. 

It was widely believed that the main defect of classical line shape can approximately be corrected with the help of one of the various desym-metrization procedures proposed in the literature that formally satisfy Eq. 5. 73. However, it has been pointed out that the various procedures give rise to profiles that differ greatly in the wings [70]. While they are sufficient to generate the asymmetry, Eq. 5.73, the resulting desym- 

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M. Moraldi, A. Borysow, and L. Frommhold. Rotovibrational collision induced absorption by nonpolar gases and mixtures (H2-He pairs) About the symmetry of line profiles. Phys. Rev. A 38 1839, 1988. 

Due to symmetry only half of the period is shown.) The curve indicated by the thick solid line is the profile after x = 131072 updates per surface site Also shown are the sine function and the predicted profile from the Lan on-Villain theory, chosen in such a way that the slopes at the steepest part of the profile agree with one another. It, is seen that the Lan on-Villain curve agrees quite well with the measured profile apart from the top one or two layers, where the latter has more rounded shape The rounding tendency at the top of the groove is more pronounced (in relative terms) for smaller systems, in agreement with earlier observations made by Selke and Duxbury (1995). 

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). 

Symmetry. For the translational and rototranslational bands, Eq. 5.73 is usually a good description of the observed asymmetry. In that case, the use of the BC and K0 profiles is straightforward because these have the same symmetry, Eq. 5.73. In the next Chapter we will see that profiles of the rotovibrational bands are of a symmetry which is different from that relationship [295], In that case, new line profiles must be constructed that satisfy the correct symmetry relationship [295, 62, 48], These will be discussed next Chapter. 

As an example, Fig. 6.20 below compares the Ai AL = 0001 and 2023 line profiles at 195 K which were computed with and without (solid and dashed curves, respectively) accounting for the vibrational dependences of the interaction potential. The correct profiles (solid curves) are more intense in the blue wing, and less intense in the red wing by up to 25% relative to the approximation (dashed), over the range of frequencies shown. Whereas the dashed profiles satisfy the detailed balance relation, Eq. 6.59, if a> is taken to be the frequency shift relative to the line center, the exact profiles deviate by up to a factor of 2 from that equation over the range of frequencies shown. In a comparison of theory and measurement the different symmetries are quite striking use of the correct symmetry clearly improves the quality of the fits attainable. 

On pp. 31 Iff., a preliminary discussion of the symmetry of induced line profiles was given. The spectral lines encountered in collision-induced absorption show a striking asymmetry which is described roughly by a Boltzmann factor, Eq. 6.59. However, it is clear that at any fixed frequency shift, the intensity ratio of red and blue wings is not always given exactly by a Boltzmann factor, for example if dimer structures of like pairs shape the profile, or more generally in the vibrational bands. We will next consider the latter case in some detail. 

Fig. 6.20. Symmetry of the line profiles of the two main components of unmixed hydrogen in the fundamental band at 297 K. Higher curve isotropic overlap component (A1A2AL = 0001) lower curve quadrupole-induced component (A1A2AL = 0223). For comparison, the dashed curves represent the profiles computed without accounting for the vibrational dependence of the interaction potential [281].

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. 

The following provides an outline of a geometrical design that enables self-cleaning profiles to be produced using elementary geometry. As a first step we determine the center M of the profile and draw the symmetry lines 1 and 2. 

If Z= 1 (single-flighted profile), the lines of symmetry 1 and 2 merge at the center of the profile. These two lines are the profile s lines of symmetry. The profile diameter reaches its largest point at line 1 and its smallest point at line 2. 

The design of the profile can fail if we make the internal diameter too small. This is the case if point K is outside the lines of symmetry, i. e., the screw can no longer reach the external diameter. 

The Raman intensity plotted against the exciting laser wavelength is called an excitation profile. Excitation profiles such as that shown in Fig. 3-21 of Chap. 3 provide important information about electronic excited states as well as symmetry of molecular vibrations. The intensity of a Raman line is maximized if strict resonance conditions are met (Section 1.15). When constructing excitation profiles, the frequency dependence of /(v) is of interest. It is difficult, however, to determine the / dependence on v from intensity changes because K and A also vary with v. 

Profiles comprise cross sections that are not a circle, annulus, or wide sheet. Like pipe and tubing lines, profile extrusion lines consist of an extruder, profile die, calibration device, cooling system, puller, and a cut-off saw and stacker or wind-up unit. The main differences are the dies and calibration units. Due to lack of symmetry, obtaining a correct cross section in a profile die is difficult. Differential flow resistance in different parts of the cross section alters the flow rate for these parts of the die. In addition, die swell may vary due to the differences in flow. Consequently, the extrudate may bend as it exits the die. To equalize flow, the die land length is varied or restricting plates are used in channels where the flow is too rapid. Many profile dies are split into sections, with the die sliced perpendicular to the major axis. Thus, sections can be altered in the process of die development. Flow simulation software is particularly useful in profile die design. 

To analyze the recorded spectra, the spectrometer needs to be calibrated. The three main calibration parameters are the velocity scale, the center point of the spectrum and the nonlinearity of the velocity/time profile of the oscillation compared to a standard reference. The calibration is performed using a spectrum recorded from an a-iron foil at room temperature using the well defined line positions of the sextet from a-iron, which occur at 5.312mms , 3.076mms , and 0.840mms The center of this a-iron spectrum at room temperature is taken as the reference point (0.0 nun s ) for isomer shift values of sample spectra. The typical Mossbauer spectrum of the 14.4 keV transition of Fe in natural iron (Fig. 4.10) represents a simple example of pure nuclear Zeeman effect. Because of the cubic symmetry of the iron lattice, there is no quadrupole shift of the nuclear energy levels. The relative intensities of the six magnetic dipole transitions are... 

The stability of the system with WL-DDTS is studied with symmetric electrode pair (SEP) analysis [44] and it is compared with stability of the W-DDTS-based system. The SEP analysis shows that the profiles of the boundary obtained for all the current projections in opposite [44] and common-ground [44] methods are found symmetric, which proved the stability of the system [44]. The potential of the SEPs (Figure 30.13a and b) shows that the experimental phantom is almost symmetric with respect to the axis of symmetry of the current flux line. 

This is the historic hyperbolic tangent profile, implicit in van der Waals and explicit in Landau and Lifshitz and in Cahn and Hilliard, " of which wc have already had an intimation in (5.103) and which we shall encounter again in Chapter 9. By the obvious symmetry of (8.50) and (8.51), the profile of the density p in the afi or Py interfooe far from the three-phase line is also given by (8.54), with p or p replaced by p, and with z then the distance from the mid-plane of the aB or ary interface. 

The profile of the powder spectrum is determined by several parameters, including the symmetry of the tensor, the actual values of its components, and the line shape and the line width of the resonance. Concerning the symmetry of the g tensor, three possible cases can be identified. 

Figure 8 Powder EPR spectrum of a paramagnetic species with / = 0 in axial symmetry. (A) absorption profile, (B) first derivative profile. The dotted lines have been calculated by assuming a zero line width whereas the solid lines correspond to a finite line width.

In this section we extend previous studies of surfaces with sinusoidal concentric grooves to the more general case of an axially or cylindrically symmetric rough surface of arbitrary profile. Gravity is neglected. The entire system has the same symmetry as the solid surface and in the case of cylindrical symmetry there is a plane of symmetry such that the triple line contacts identical regions of the solid on the two sides. 

This velocity profile is commonly called drag flow. It is used to model the flow of lubricant between sliding metal surfaces or the flow of polymer in extruders. A pressure-driven flow—typically in the opposite direction—is sometimes superimposed on the drag flow, but we will avoid this complication. Equation (8.51) also represents a limiting case of Couette flow (which is flow between coaxial cylinders, one of which is rotating) when the gap width is small. Equation (8.38) continues to govern convective diffusion in the flat-plate geometry, but the boundary conditions are different. The zero-flux condition applies at both walls, but there is no line of symmetry. Calculations must be made over the entire channel width and not just the half-width. 

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