Line shape half-width

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 ... 

Except for the last term, which is usually difficult to observe, the light-scattering spectrum is a sum of Lorentzian line shapes. The first term represents an unshifted line called the Rayleigh or central line which is a Lorentzian line with half-width at half maximum... 

The exponential decay of the A population corresponds to a Lorentzian line shape for the absorption (or emission) cross section, a, as a fiinction of energy E. The lineshape is centred around its maximum at E. The fiill-width at half-maximum (F) is proportional to... 

Figure 2.5 shows, for a sample in the gas phase, a typical absorption line with a HWHM (half-width at half-maximum) of Av and a characteristic line shape. The line is not infinitely narrow even if we assume that the instmment used for observation has not imposed any broadening of its own. We shall consider three important factors that may contribute to the line width and shape. 

RAIRS spectra contain absorption band structures related to electronic transitions and vibrations of the bulk, the surface, or adsorbed molecules. In reflectance spectroscopy the ahsorhance is usually determined hy calculating -log(Rs/Ro), where Rs represents the reflectance from the adsorhate-covered substrate and Rq is the reflectance from the bare substrate. For thin films with strong dipole oscillators, the Berre-man effect, which can lead to an additional feature in the reflectance spectrum, must also be considered (Sect. 4.9 Ellipsometry). The frequencies, intensities, full widths at half maximum, and band line-shapes in the absorption spectrum yield information about adsorption states, chemical environment, ordering effects, and vibrational coupling. 

The bracketed term in Eq. (4-60b) describes a Lorentzian line shape for the NMR absorption band. The maximum in the band occurs at the resonance frequency, wq. Expressed in units of X0W0T2/2, the maximum value of x" s 1 at one-half this maximum peak height we find, by substitution, that (wq — w) = IIT. Using w = 2 ttv to convert to frequency (in Hz) gives (vq — v) = 3-7 T 2. However, the peak width is twice this, or... 

In the fast exchange limit ft2 A2, a Lorentzian at the centre d>, is observed with full width at half height 2/T + A2/Q. In the ultrafast exchange limit discussed in the previous section, A2/fi 2/T, the line shape becomes independent of the exchange... 

The simplest definition of sensitivity is the signal-to-noise ratio. One criterion for judging the sensitivity of an NMR spectrometer or an NMR experiment is to measure the height of a peak under standard conditions and to compare it with the noise level in the same spectrum. Resolution is the extent to which the line shape deviates from an ideal Lorentzian line. Resolution is generally determined by measuring the width of a signal at half-height, in hertz. 

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 ... 

The emission line is centered at the mean energy Eq of the transition (Fig. 2.2). One can immediately see that I E) = 1/2 I Eq) for E = Eq E/2, which renders r the full width of the spectral line at half maximum. F is called the natural width of the nuclear excited state. The emission line is normalized so that the integral is one f l(E)dE = 1. The probability distribution for the corresponding absorption process, the absorption line, has the same shape as the emission line for reasons of time-reversal invariance. 

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)... 

FIGURE 4.4 Line shapes. Lorentzian (broken lines) and Gaussian (solid lines) line shapes and their first derivatives are given. The outermost vertical lines delimit full width at half height (FWHH) of the absorption lines. 

Much worse than the oscillator strength is the line shape. The calculated absorption spectra has no similarity with what is experimentally seen. The calculated half-width is always smaller, typically by a factor of 2 the exact reasons for this are only speculated. It is common knowledge that a photodetachment process is capable of giving a very broad absorption spectrum, but a satisfactory method has not been developed to adopt this with the bound-bound transition of the semicontinuum models. Higher excited states (3p, 4p, etc.) have been proposed for the solvated electron, but they have never been identified in the absorption spectrum. 

Treating vibrational excitations in lattice systems of adsorbed molecules in terms of bound harmonic oscillators (as presented in Chapter III and also in Appendix 1) provides only a general notion of basic spectroscopic characteristics of an adsorbate, viz. spectral line frequencies and integral intensities. This approach, however, fails to account for line shapes and manipulates spectral lines as shapeless infinitely narrow and infinitely high images described by the Dirac -functions. In simplest cases, the shape of symmetric spectral lines can be characterized by their maximum positions and full width at half maximum (FWHM). These parameters are very sensitive to various perturbations and changes in temperature and can therefore provide additional evidence on the state of an adsorbate and its binding to a surface. 

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).)... 

Although this equation only applies when the coalescence point is reached, rate constants for the exchange between two or more exchanging sites are accessible by analysis of line widths at half height, Avip, and shift differences, Ar, in Hz. The comparison between the experimental spectrum and the spectrum calculated by use of a simulation package for line shapes provides the mechanism for determining the rate constant of exchange [161, 162]. 

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). 

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 ... 

Let us establish the required relationships more precisely. Consider a narrow idealized rectangular absorption line AT(x) = rect(x/2 AxL) having half-width AxL and centered at x = 0. Its variance is easily found to be <7l = (2 Axl/3)2. Its area is 2 AxL. Now, let us assume that this line is being used to measure an instrument response function exp( —x2/2cr2) that has Gaussian shape and variance ... 

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. 

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. 

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