Half-width Lorentzian

The next two temis (Lorentzians) arise from the mechanical part of the density fluctuations, the pressure fluctuations at constant entropy. These are the adiabatic sound modes (l/y)exp[-FA t ]cos[co(A) t ] with (D(k) = ck, and lead to the two spectral lines (Lorentzians) which are shifted in frequency by -ck (Stokes line) and +ck (anti-Stokes line). These are known as the Brillouin-Mandehtarn, doublet. The half-width at... 

The quasi-classical description of the Q-branch becomes valid as soon as its rotational structure is washed out. There is no doubt that at this point its contour is close to a static one, and, consequently, asymmetric to a large extent. It is also established [136] that after narrowing of the contour its shape in the liquid is Lorentzian even in the far wings where the intensity is four orders less than in the centre (see Fig. 3.3). In this case it is more convenient to compare observed contours with calculated ones by their characteristic parameters. These are the half width at half height Aa)i/2 and the shift of the spectrum maximum ftW—< > = 5a>+A, which is usually assumed to be a sum of the rotational shift 5larger scale A determined by vibrational dephasing. 

The shapes of both /w and 7hv lines are assumed to be represented by simple Lorentzians. For a totally symmetric vibration with a low polarization ratio as in the present case, the vibrational and reorientational relaxation times Tv and can be determined from the half-widths of the isotropic and anisotropic spectra. Since the value of /hv is much smaller than that of /w for the 1050 cm" line, the contribution of /gv to the isotropic intensity can be neglected ... 

In a Mdssbauer transmission experiment, the absorber containing the stable Mdssbauer isotope is placed between the source and the detector (cf. Fig. 2.6). For the absorber, we assume the same mean energy q between nuclear excited and ground states as for the source, but with an additional intrinsic shift A due to chemical influence. The absorption Une, or resonant absorption cross-section cr( ), has the same Lorentzian shape as the emission line and if we assume also the same half width , cr( ) can be expressed as ([1] in Chap. 1)... 

A plot of v vs. T2(a>o co) is shown in Figure 5.1. Equation (5.14) corresponds to the classical Lorentzian line shape function and the absorption curve of Figure 5.1 is a Lorentzian line . The half-width at half-height is easily found to be ... 

Membranes and vesicles were labeled at a DPH/lipid ratio of 1/400 and measured using phase-modulation fluorometry at ten frequencies between 5 and 90 MHz at 37°C. y2 values were calculated assuming errors of 0.2 and 0.002 in the phase and modulation, respectively, except where otherwise noted. /, 2, Fraction of Exponential term or Lorentzian t, 2, lifetime (ns) cu, center of Lorentzians (ns) w12, half-width of Lorentzians (ns). 

Two samples of the same phosphor crystal have quite different thicknesses, so that one of them has a peak optical density of 3 at a frequency of vo. while the other one has a peak optical density of 0.2 at vq. Assume a half width at half maximum of Av = IGHz and a peak wavelength of 600 nm, and draw the absorption spectra (optical density versus frequency) for both samples. Then show the absorbance and transmittance spectra that you expect to obtain for both samples and compare them with the corresponding absorption spectra. (To be more precise, you can suppose that both bands have a Lorentzian profile, and use expression (1.8), or a Gaussian line shape, and then use expression (1.9).)... 

FIGURE 14-8 (a) Meaning of equivalent width, W (b) Doppler and Lorentzian line-shapes for equivalent half-widths (c) transmission curves for an absorption line for a weak and strong absorber, respectively (adapted from Lenoble, 1993). 

Doppler broadening. Collisions broadening becomes less important at the lower pressures found at higher altitudes, so that the Lorentzian half-width and the Doppler half-width become comparable at altitudes of approximately 30-40 km. 

To consider gas molecules as isolated from interactions with their neighbors is often a useless approximation. When the gas has finite pressure, the molecules do in fact collide. When natural and collision broadening effects are combined, the line shape that results is also a lorentzian, but with a modified half-width at half maximum (HWHM). Twice the reciprocal of the mean time between collisions must be added to the sum of the natural lifetime reciprocals to obtain the new half-width. We may summarize by writing the probability per unit frequency of a transition at a frequency v for the combined natural and collision broadening of spectral lines for a gas under pressure ... 

When pressure broadening dominates, the situation is more complicated because the resulting Lorentzian profile contributes significant area far from the line center. A further complication in this case is that the Lorentzian half-width cannot be accurately calculated and must be measured in other experiments. If both Doppler and pressure broadening are present, however, and if the Lorentzian to Doppler half-width ratio is small, the correction necessitated by pressure broadening is small. In this situation an accurate value of the Lorentzian half-width may not be needed. Line strength in the case of combined Doppler and pressure broadening may be obtained from the equivalent width by the use of tables (Jansson and Korb, 1968). 

The Lorentzian shape of x-ray emission lines is well founded in quantum theory and has been substantiated experimentally (Hoyt, 1932). Siegbahn et al. (1967) discuss the aluminum anode x-ray source as applied to ESCA. Beatham and Orchard (1976) list doublet separations and half-widths derived from the literature and optimized by computer simulation. Kallne and Aberg (1975) and Senemaud (1968) also provide values. 

Plot the Gaussian and Lorentzian functions (3.89) and (3.86) against v — v0. Use the same axes for both plots and take the peak heights and half-widths to be equal. 

The spectrum consists of three components. The first term represents an unshifted line called the Rayleigh line, which is a Lorentzian with a halfwidth at half-maximum given by Acoc(q) = Dpq2. The next two terms represent a doublet called the Brillouin doublet. These are two Lorentzian lines shifted symmetrically from the origin by co = csq, each having half-width at half-maximum, AcoB(q) = Tq2. 

When Eq. (289) is substituted in Eq. (285), one finds a Lorentzian line shape where the half-width at half-maximum (HWHM) is t"1. Thus by assuming an exponential decay, as in Eq. (289), t can be obtained directly from either Eq. (282) or Eq. (290). However, (2(0)2(0) may be nonexponential, as opposed to exponential as assumed in Oxtoby s work, and, as shown later, may give rise to an overall subquadratic overtone dependence of the rate. 

Two methods used to find the area under the photometer trace are peak-height-times-half-width approximations and actual measurements with a polar planimeter. Both methods are time consuming and offer little increase in total accuracy over the peak center method. Another method involves computer fitting an assumed scattering function, usually a Gaussian or Lorentzian (though more exotic functions have also been used) to the scan data. The integrated area under the mathematical curve is then calculated. 

Eq. (4.28) implies that the absorption band-shape for m -> n is Lorentzian with half-width ynm. In thermal equilibrium, the Boltzmann distributions can be used for and In this case Eq. (4.28) becomes... 

Here kv is the absorption coefficient at frequency v, Nc is the number of absorbing centres per cubic centimeter, v is the frequency of absorption, and S(v) is the line shape function. For our estimates we shall assume that the line shape is Lorentzian having half width 6. If one evaluates the absorption cross section when the absorption is maximum the above expression takes the form... 

The SD Involving Indirect Damping in Terms of Lorentzian of Different Half-Widthes... 

As seen, the spectral density involving direct damping is the double sum over m and n of Lorentzians centered on 0) = o>° (n m)il — 2ao2Q and having the same half-width y°, but different intensities, given by c x" Am (a°) 2. 

Besides, examination of Eq. (201) shows that the spectral density is the sum of different components, each of them being a superposition of Lorentzians involving different half-widthes and intensities. Note that this result differs deeply from the situation without indirect damping given by Eq. (82), for which all the Lorentzians forming the line shape have the same half-width. This Ending may be verified by taking y = 0 in Eq. (201). 

InSe and GaSe crystals are characterized with a weak interaction of 3D Wannier excitons with homopolar optical A -phonons [18, 19]. Therefore, when calculating the exciton absorption spectra, we took into consideration effects of broadening the exciton states using the standard convolution procedure (see in [18]) for theoretical values of a(7jco) the absorption coefficient in the Elliott s model [20] with y /io>) — 77 [n(E 2+/ 2)] the Lorentzian function in the Toyozawa s model [21], where r is the half-width at half-maximum which is usually associated with the lifetime tl/2r. 

Av then becomes the half-width at half-peak height. As the gas pressure is increased, however, the second term becomes significant and the line shape departs from the simple Lorentzian. 

Figure J A simulated H pulsed NMR spectrum of an idealized chemical species H Hx, nominally taken at fixed carrier frequency v0=500.0... MHz. Here the Zeeman field B is deemed to be kept fixed at 11.74334 T, and Fourier transformation yields the spectrum. There are 6 (A) and 6 (X) distinct lines of appreciable intensity (12 and 14 transitions, respectively), with each individual line depicted as a Lorentzian with half-width 0.1 Hz.

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