Line profile Lorentzian

Application. In practice, three or more peak orders must be observed. The shape of the lines should be Gaussian or Lorentzian. On these premises it is promising to carry out line profile analysis. Corresponding graphical separation methods are readily derived from the breadth relations that have just been discussed, after that the breadths B ( (x)) and B (Hdq,) (s)) of a polycrystalline ensemble have been related to structure and substituted in Eq. (8.29) or Eq. (8.30). For the size term the structure relation is... 

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

The conventional Lorentzian, Gaussian, etc., profiles mentioned above are all symmetric, g(—v) = g(v). In contrast to this symmetry, the individual line profiles in collision-induced absorption have early been recognized as being quite asymmetric, roughly as [120, 215, 188]... 

Fig. 6.9. A Spontaneous Raman spectrum of d62-DPPC lipids and its decomposition into Lorentzian line profiles. B Normalized multiplex CARS spectra (dots) of a planar-supported bilayer and monolayer formed by d62-DPPC on a glass-water interface for parallel-polarized input beams, together with the fit using the center frequency and line width parameters extracted from the decomposition analysis in (A) (solid line). The spectrum exposure time was 0.64 s. Error bars indicate the shot-noise standard deviation (Copyright American Chemical Society [70])...

According to the expression used for the calculations ("Equation 15" in ( 6 )) the scattered intensity is characterized by a Lorentzian line profile. The atomic fractional coordinates reported for the crystal structure of ZSM-5 ( 4 ) were used as intrinsic atomic parameters for the pentasil layers as was done for the structure determination of H-BOR-D. 

For optical depth t 1 the observed interstellar molecular lines usually have a gaussian shape. This is to be expected, since collisional broadening, which causes the Lorentzian line profile, should be negligible and become important only at gas densities 1012 — 1014 cm- 3. If thermal motions of the molecules were the only source of line broadening the line half power width (i.e. the width between half power points) would be given by... 

The line shapes are described by Voigt functions, which reflect the Lorentzian line profiles due to natural line width and Gaussian profiles due to Doppler broadening. The instrumental broadening by the rocking curve of the crystal, de-focusing and the finite resolution of the detector is described well by a Voigt profile shape too [3[. 

Figure 9 The lower trace shows the experimental double-resonance spectrum of NO2 just above the dissociation threshold. The upper traces show enlargements including simulations (dotted lines) of the spectrum with multi-line fits (Lorentzian profiles). Courtesy of B. Abel.

In 1996, Nemet and Kozma showed the emission spectrometry of gold laser-produced plasma to be of interest for analytical purposes a delay time of 800-1000 ns was found to ensure nni/-thermal equilibrium and thorough atomization in the plasma. The line profiles obtained under such conditions (both resonant and Stark-broadened) were fitted to a symmetric Lorentzian curve [170]. Recently, LIBS was used in combination with effective chemometric tools to develop a determination method for gold in homogeneous samples that allows the characterization of jewellery products. The results confirmed the LIBS technique as an effective alternative to the hallmark official methods [143,144,171]. 

Fig. 7b shows representative (u /27rrf field strengths for the 12 ms adiabatic sequences (last points in Fig. 7a) under ideal conditions (solid line), 5% Lorentzian rf inhomogeneity (dashed line), 9.2% Gaussian rf inhomogeneity (dotted line). As mentioned earlier, these rf profiles match the conditions for a typical 4 mm tripleresonance probe. 

Figure J. 23. Lorentzian, Doppler, and Voigt spectral line profiles for approximately equal half widths and intensities. (From Andrews et al.., 1987).

Fig. III. 13. The effect of the longitudinal inhomogeneity of the magnetic field is illustrated in this calculated line profile of the JW = 2—M = 2 satellite of the 2i2—2gi rotational transition of ethyleneoxide. As described in the text, the line profile was calculated as a superposition of 21 Lorentzians corresponding to the different sections of the absorption cell. Half-widths of 20, 66, and 150 KHz (full width at half height) were used for the calculations which were based on a relative field distribution corresponding to the one shown in Fig. III.6. The insert gives an enlarged view of the center region. At a half-width of 150 KHz the peak frequency is shifted 6.8 kHz to lower frequencies (downfield) as compared to the position expected from a center field calculation...

Measurements by photographic photometry require careful calibration due to the nonlinear response of photographic plates saturation effects can lead to erroneous values. Line profiles can be recorded photoelectrically, if the stability of the source intensity and the wavelength scanning mechanism are adequate. Often individual rotational lines are composed of incompletely resolved spin or hyperfine multiplet components. The contribution to the linewidth from such unresolved components can vary with J (or TV). In order to obtain the FWHM of an individual component, it is necessary to construct a model for the observed lineshape that takes into account calculated level splitttings and transition intensities. An average of the widths for two lines corresponding to predissociated levels of the same parity and J -value (for example the P and R lines of a 1II — 1E+ transition) can minimize experimental uncertainties. A theoretical Lorentzian shape is assumed here for simplicity, but in some cases, as explained in Section 7.9, interference effects with the continuum can result in asymmetric Fano-type lineshapes. 

In the preceding sections, we have assumed that an absorption line has a Lorentzian shape. If this is not true, then the linewidth cannot be defined as the full width at half maximum intensity. Transitions from the ground state of a neutral molecule to an ionization continuum often have appreciable oscillator strength, in marked contrast to the situation for ground state to dissociative continuum transitions. The absorption cross-section near the peak of an auto-ionized line can be significantly affected by interference between two processes (1) direct ionization or dissociation, and (2) indirect ionization (autoionization) or indirect dissociation (predissociation). The line profile must be described by the Beutler-Fano formula (Fano, 1961) ... 

The total width, T, is the sum of partial widths, which can be calculated but not observed separately. Only the total width can be observed experimentally. This width does not depend on whether the line is observed in an absorption, photoionization, photodissociation, or emission spectrum because the width (or the lifetime) is characteristic of a given state (or resonance). In contrast, the peak profile can have different line shapes in different channels the line profile, q, is dependent on the excitation and decay mode (see Sections 7.9 and 8.9). For predissociation into H+CT, the transition moment from the X1E+ state to the 3n (or 3E+) predissociating state is zero, consequently q = oo and the lineshape is Lorentzian. In contrast, the ratio of the two transition moments for transitions to the XE+ continuum of the X2n state and to the (A2E+)1E+ Rydberg states leads to q 0 for the autoionized peaks (see Fig. 8.26) (Lefebvre-Brion and... 

Fig. 1.5 Lorentzian line profile a o)) of halfwidth y (FWHM) (a), with first (b), second (c), and third (d) derivatives...

In Vol. 1, Sect. 3.6 we saw that the saturation of homogeneously broadened transitions with Lorentzian line profiles results again in a Lorentzian profile with the... 

The change of absorption Aa is caused by those molecules within the velocity interval Av = 0ztys/k that simultaneously interact with the pump and probe waves. The line profile of Ao (saturation spectroscopy (Sect. 2.2) a Lorentzian profile... 

V sinO < y/k, the molecule after the collision is still in resonance with the standing light wave inside the laser resonator. Such soft collisions with deflection angles 0 < therefore do not appreciably change the absorption probability of a molecule. Because of their statistical phase jumps (Vol. 1, Sect. 3.3) they do, however, contribute to the linewidth. The line profile of the Lamb dip broadened by soft collisions remains Lorentzian. 

The combined effect of both kinds of collisions gives a line profile with a kernel that can be described by a Lorentzian profile slightly broadened by soft collisions. The wings, however, form a broad background caused by velocity-changing collisions. The whole profile cannot be described by a single Lorentzian function. In Fig. 8.4 such a line profile is shown for the Lamb peak in the laser output Pl(co) at... 

If the observation time extends from r = 0 to r = oo, a Fourier transformation of the measured intensity I(t) ex A t) yields, for the line profile of the fluorescence emitted by atoms at rest, the Lorentzian profile (Vol. 1, Sect. 3.1)... 

The resultant line profile is Lorentzian and forms the kernel of a broader background caused by velocity changing collisions. 

The output waveform of a stable, single-frequency laser far above the threshold of oscillation may be approximated by an almost perfect sine wave with nearly constant amplitude and frequency. For a laser operating in an ideal environment, the spectral purity is measured by a linewidth which is determined by frequency fluctuations caused by random walk of the oscillation phase under the Influence of spontaneous emission (quantum) noise. In their fundamental 1958 paper, Schawlow and Townes predicted that the quantum phase noise limited line profile will be a Lorentzian with a full width between the half power points (FVIHM) that may be approximated by ... 

This is a Lorentzian line profile (Fig. 2.19) with a full halfwidth yah = ya + yb-... 

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