Lorentz shape

The spectral function thus has a Lorentz shape with a halfwidth at the half distribution height equal to the average reorientation frequency w. If expressed in spectroscopic units (cm 1), the halfwidth Avv2 amounts to cd2nca (c0 designates the velocity of light in vacuo). 

The complex spectral structure from 750 to 650 cm."1 was resolved mathematically by Binder into a series of overlapping bands of the theoretical Lorentz shape (3). It was shown that only the band at 740 cm."1 originated in the cis-olefin group. However, measurements on... 

If the absorption band in question is overlapped by neighboring bands, it is clear that the determination of the true values of Dn (v0) and Da(v0) can be ambiguous. In such cases it is necessary to resolve the band of interest from the complex absorption of which it is a part. This is usually done by a graphical analysis, which assumes symmetrical bands of a Lorentz shape centered at the absorption maxima and simple summation of these to give the observed spectrum. The results of such a resolution are of course subject to the uncertainty of the true band shape and width, so that an indeterminate, if nevertheless small, error can be introduced in this manner. 

Curves 2 (Ei E) and 5 (Ei L E) in Fig. 4.15 refer to the uoj Jj/T 1 scale, in which the excited state Hanle effect manifests itself (the ground state Hanle effect is already fully developed and does not manifest itself in this scale). The signal is of Lorentz shape ... 

Equations 1.16 and 1.17 are based on the assumption that all the molecules undergoing the transition m —> do so at the same frequency v. In reality they will have slightly different transition frequencies centred around the centre frequency Vq due predominately to collisional interactions between molecules. Doppler broadening also makes a small contribution giving a Gaussian shape to the line (Figure 1.4), but the overall result is a profile approximated by the Lorentz shape function 5 (v) ... 

Fig. 9.1 (a) Gauss, Lorentz and Voigt (ABl/ABq = 1.0) lines as 1st derivatives adjusted to the same peak-peak amplitudes, (b) Integrated area of derivative lines with the same amplitude as function of the ABl/ABq ratio of a Voigt Une. For ABl/ABq 1 the line approaches a Lorentz shape with an area which is 3.51 times larger than that of a Gaussian with the same amplitude [13]... 

The line-shape of an experimental spectrum can in principle be determined by the procedure illustrated in Fig, 9.2. The 2nd derivative of the resonance line is then recorded. For a Gauss line the ratio hi/h2 between the minimum and maximum amplitudes of the 2nd derivative (Fig. 9.2(a)) equals 2.24 [18], while for a Lorentz shape it approaches the value 4. The hi/h2 ratio for a Voigt profile varies... 

Fig. 9.13 Schematic shapes of saturation curves for homogeneous, inhomogeneous and Voigt ESR-lines. A homogeneous line has the Lorentz shape usually occurring in liquids. An inhomogeneous line is an envelope of narrow homogeneous lines, with the envelope usutilly approximated by a Gaussian, wWle the Voigt line is an envelope of homogeneous lines with an appreciable line-width...

The absorption component in particular is seen to have a Lorentz shape-function dependence on the frequency variable v. If Hi is small (i.e., if y H TiTi is small) the resonance shape is independent of the longitudinal time T, but does depend on the transverse time Tj. In this case, if the expression for the absorption component eq. (18.60) is written as... 

This imcertainty in frequency, which is inversely proportional to the lifetime, generates a line profile of Lorentz shape, centered at vq, with a width Using the relation AAn = (A /c) Ai/N the so-called natural line width AAn is obtained and the corresponding wavelength-dependant intensity distribution /n(A) of the area-normalized profile is given by ... 

Figure 2.14 Normalized, shot-noise determined, minimnm detectable absorbance for Gauss- and Lorentz-shaped absorption lines as function of AAinst rectangular instrument profile assumed...

Now let s consider the part played by the slit width in determining the shape and intensity of an absorption band, i,e, its effect. We shall assume that the sample band is of the Lorentz curve shape and has a half band width of 8 cm (Ai ). We shall also assume that the slit width is 1 cm (Ay/) and follows a triangular function. In this case the band as seen by the instrument will have a half band width (Ayi/ ) approximately equal to Ay, and the band will be of the Lorentz shape. Here the spectrophotometer accurately sees the band shape (see Fig. 9). 

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