Lorentz band

To obtain the absorbances at 910 and 967 cm. 1, it was necessary to correct the observed band intensities for the overlapping of adjacent bands. The band at 910 cm."1 for the vinyl group was corrected for the absorbance from the wing of the 967-cm."1 frarw-vinylene band,. and the latter band was corrected for the vinyl band at 995 cm. 1. The Lorentz band shape equation was used to calculate the absorbance in the wings, and in the thicker specimens, successive approximations were necessary. This treatment gave the four equations below, which yielded the concentrations of trans and vinyl groups for the emulsion and sodium polybutadienes listed in Table I. Implicit in these equations is the assumption that the absorptivities are independent of concentration. 

Malkmus, W., Random Lorentz band model with exponential-tailed S-l line intensity distribution function. J Opt Soc Amer 57, 323, 1967. 

The most simple, but general, model to describe the interaction of optical radiation with solids is a classical model, due to Lorentz, in which it is assumed that the valence electrons are bound to specific atoms in the solid by harmonic forces. These harmonic forces are the Coulomb forces that tend to restore the valence electrons into specific orbits around the atomic nuclei. Therefore, the solid is considered as a collection of atomic oscillators, each one with its characteristic natural frequency. We presume that if we excite one of these atomic oscillators with its natural frequency (the resonance frequency), a resonant process will be produced. From the quantum viewpoint, these frequencies correspond to those needed to produce valence band to conduction band transitions. In the first approach we consider only a unique resonant frequency, >o in other words, the solid consists of a collection of equivalent atomic oscillators. In this approach, coq would correspond to the gap frequency. 

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. 

The interpretation of band progressions by the time dependent procedure is therefore identical with the Franck-Condon analysis and, in the low temperature limit, to the method of molecular distributions as well. The line shape function obtained on the basis of Eq. (52) (for E = hv) differs under this condition from that of Eq. (12) only in the line shape function of each vibrational member in the progression which in Eq. (52) is the delta function and in Eq. (12) has a Lorentz type distribution. 

Fig. 3. Emission spectrum of Cs2SeBr6 (powder) at 10 K due to F4/T( S0) transition obtained from argon laser excitation at 363.8 nm and band analysis by deconvolution into Lorentz...

The latter is determined by the oscillation frequency, decaying coefficient, and vibration lifetime. This nonrigid dipole moment stipulates a Lorentz-like addition to the correlation function. As a result, the form of the calculated R-band substantially changes, if to compare it with this band described in terms of the pure hat-curved model. Application to ordinary and heavy water of the so-corrected hat-curved model is shown to improve description (given in terms of a simple analytical theory) of the far-infra red spectrum comprising superposition of the R- and librational bands. 

The effect of the anharmonicity on At follows from estimation (54) of the frequency xm. The right-hand part of this formula is actually the result of gathering of the Lorentz lines in (51), each being determined by a relevant pair h, 1 of arbitrary constants. Evidently, the steeper the dependence of the period on these variables—that is, the greater the (S >-1/0/i) and (0absorption curve, the wider the absorption band. Thus,... 

In the case of weak interactions with the electronic levels of the electrode (wide band approximation) Eq. (27) has the form of two Lorentz distributions centred at the energies of the bonding and antibonding states (see Fig. 12). The integral of Eq. (27) up to the Fermi level gives the occupation of both, bonding and antibonding orbitals and in the case of the wide band approximation it is an analytical expression ... 

Figure 47 shows taken from Equation 20 versus Vj. It shows that S. is quite sensitive to Vp and is therefore a good means to evaluate v, with the numerical values of Fig. 47. It can be estimated that the tensile modulus E of the bulk PMMA is not affected by the very low pressure toluene gas environment during the short duration of the experiment. The optical craze index in PMMA in air without load is known as n = 1.32, which corresponds to v = 0.6. From the optical interferometry, it is known that the craze just before breakage is twice as thick as unloaded, (v, = 0.3) and hence using Lorentz-Lorenz equation its optical index is n = 1.15. From Figs. 46 and 47 it can be concluded that the bulk modulus around the propagating crack is about 4400 MPa, which is a somewhat high value, in view of the strain rates at a propagating crack tip (10 to s" ). Using the scatter displayed in Fig. 46, it can be concluded from Fig. 47 that the fibril volume fraction is constant, v = 0.3, within a scatter band of 0.08, and is therefore not sensitive to the toluene gas.

The spectrum emitted by the lamp corresponds to the superimposition of radiation emitted by the cathode and by the gaseous atmosphere within the lamp. The width of the emission lines, which depends upon different effects (Doppler, Stark (ionization) and Lorentz (pressure)), is narrower than the corresponding absorption band. The monochromator enables the elimination of a large part of the stray light due to the filling gas, and the selection of the most intense spectral line in order to obtain a better sensitivity (Figure 13.8), except for cases of interference caused by other elements. 

Narrow band models parameterize the transmission for wavenumber intervals Sv of typically 5 to 20 cm-1. The narrow band model of Elsasser (1942) represents the spectrum by a series of regularly spaced Lorentz lines of the same size and intensity. This model is best applied to linear triatomic molecules such as CO2 and N2O. The model by Goody (1964) is based on the idea that the lines are randomly spaced over a particular wavelength interval, with some exponential distribution of line strength. This model can readily be applied to water vapor and to carbon dioxide. If an exponential distribution of the line intensities... 

The b and c vibrations, generated by the harmonic potentials, reveal themselves as the Lorentz lines, respectively, in T- and V-bands. These are... 

In Section VI we study in detail two fast short-lived vibration mechanisms b and c, which concern item 2. The dielectric response to the elastic rotational vibrations of hydrogen-bonded (HB) polar molecules and to translational vibrations of charges, formed on these molecules, is revealed in terms of two interrelated Lorentz lines. A proper force constant corresponds to each line. The effect of these constants on the spectra of the complex susceptibility is considered. The dielectric response of the H-bonded molecules to elastic vibrations is shown to arise in the far IR region. Namely, the translational band (T-band) at the frequency v about 200 cm-1 is caused by vibration of charges, while the neighboring V-band at v about 150 cm-1 arises due to elastic rigid-dipole reorientations. In the case of water these bands overlap, and in the case of ice they are resolved due to longer vibration lifetime. 

In Sections III, IV, and V, these results allowed us calculation of ice/water elastic-vibration spectra, which are located in the translational band and in the nearby placed V-band. Each band is represented in terms of one Lorentz line. For this calculation the simple formulas (148) were used. 

For liquids and solutions, p and a in the preceding equations are replaced by M (molar concentration) and e (molar absorption coefficient), respectively. However, the extrapolation method just described is not applicable, since experimental errors in determining B values become too large at low concentration or at small cell length. The true integrated absorption coefficient of a liquid can be calculated if we assume that the shape of an absorption band is represented by the Lorentz equation and that the sht function is triangular [96]. 

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