Integrals of the absorption coefficient

Instead of the moments M of the spectral function, integrals of the absorption coefficient a(v) are often obtained experimentally that are closely related to the M through Eq. 3.2,... 

To obtain the necessary parameters for (lIIB-33, 34) we look at the absorption spectrum of the molecule. The positions of the absorption maxima immediately give us the vs. The ji s are related to the integral of the absorption coefficient. We can define an average oscillator strength /j related to the average absorption coefficient by... 

Equation (2,64) shows that the integral of the absorption coefficient is a constant independent of the line-broadening process. With a. = (47t/X)k. , the integrand of (2.64) can be replaced by (2.58). This yields... 

It would appear that measurement of the integrated absorption coefficient should furnish an ideal method of quantitative analysis. In practice, however, the absolute measurement of the absorption coefficients of atomic spectral lines is extremely difficult. The natural line width of an atomic spectral line is about 10 5 nm, but owing to the influence of Doppler and pressure effects, the line is broadened to about 0.002 nm at flame temperatures of2000-3000 K. To measure the absorption coefficient of a line thus broadened would require a spectrometer with a resolving power of 500000. This difficulty was overcome by Walsh,41 who used a source of sharp emission lines with a much smaller half width than the absorption line, and the radiation frequency of which is centred on the absorption frequency. In this way, the absorption coefficient at the centre of the line, Kmax, may be measured. If the profile of the absorption line is assumed to be due only to Doppler broadening, then there is a relationship between Kmax and N0. Thus the only requirement of the spectrometer is that it shall be capable of isolating the required resonance line from all other lines emitted by the source. 

The molar absorption coefficient, e(2), expresses the ability of a molecule to absorb light in a given solvent. In the classical theory, molecular absorption of light can be described by considering the molecule as an oscillating dipole, which allows us to introduce a quantity called the oscillator strength, which is directly related to the integral of the absorption band as follows ... 

An approach widely used by atmospheric scientists is to infer the imaginary part of the refractive index k from measurements of the absorption coefficient a of particulate samples. Diffuse reflection, the photoacoustic effect, and integrating plates have been used for determining absorption even in the presence of considerable scattering these methods are discussed briefly in the following section. The relation (2.52) between a and k, a - 4nk/, is, of course, strictly valid only for homogeneous media. But under some circum-... 

The intensity of the fluorescence, I(v), must be measured in relative number of quanta at each frequency. An approximated expression of Equation 6.68 is obtained with a substitution of the term in angle brackets by the frequency of the fluorescence maximum, Eemmax, and Jeddn v,m) by a product, s maxAEem, that is, the extinction coefficient at the maximum of the absorption band times the bandwidth of the emission band. Electronic transitions fulfilling electric dipole restrictions will make je d(ln vem) (the integral of the extinction coefficient e over the emission band) to take a sizable value. Due to simplifications in the derivation of Equation 6.68, it is expected to fail when the electronic transition is between electronic states... 

An important parameter in RTE that needs modeling is the calculation of the absorption coefficient k(x,A), which depends on the local gas and soot concentrations. In a typical fire CFD, a gray gas is assumed, which means that the spectral dependency of k is not considered instead kx is replaced with an integrated value of k that is obtained by integrating over the entire wavelength... 

Radiative Recombination. The rate of radiative recombination follows from the relations (4.17) and (4.11) for a 2-level system. Since we have expressed the spontaneous emission rate in terms of the absorption coefficient, integration over all transitions involving identical photon energies is already taken into account by using the absorption coefficient for the 2-band system. Integration over all photon energies occurring in transitions between the conduction band and the valence band yields the rate of radiative recombination ... 

By carrying out one experiment, changes of monomer and polymer and also of the initiator (azobisisobutyronitrile) are reflected in the NIR and the IR, respectively. Quantitative analysis is largely facilitated by the appearance of characteristic non-overlapping bands. On the other hand, computer programs for band separation are available. In addition, the integration of molar absorption coefficients to yield concentrations need not extend over entire bands, but may be performed over half bands or even over suitable band sections. 

The surface concentration of Bronsted and Lewis sites was evaluated by integration of the absorption bands at 1545 and 1455 cm due to adsorbed pyridinium ion and pyridine, with integrated molar extinction coefficients values 813 = 1-67 and Sij, = 2.22 cm pmol" [15]. Although absolute concentration of surface acid sites obtained by this method is affected by a 10 - 15% error [12], reliable and valuable information can usually be obtained on a relative scale. Moreover, in the present case pyridine is an ideal probe molecule. Indeed, being very similar in structure to the reactant involved in the present reaction, it is able to titrate exactly those acidic sites which can be reached also by HEP. 

The integral of the extinction coefficient at 3330 cm was shown to vary approximately linearly with hydrogen content [94], and the absorption at 840 cm depends linearly on the value of x in SiN [95,96] ... 

The Integrated Absorption Coefficient. Although experimental difficulties impede the straightforward evaluation of the absorption coefficient, methods have been devised to overcome the experimental problems, and it is possible to measure the integrated absorption coefficient defined by... 

If the mirror-image law (Eq. 5.13) holds, an alternative approach is to recast Eq. (5.19) to give the rate constant for fluorescence at frequency Va — S as a function of the absorption coefficient at frequency v a + S, and then to integrate over the absorption spectmm instead of emission. Letting the fluorescence frequency be v = 2vqo — V, this gives ... 

However, of greater theoretical significance than the maximal molecular extinction coefficient is the integrated absorption intensity A. (This can be referred to as the true integrated intensity.) This is the area under the absorption curve and is most commonly defined in terms of the absorption coefficient o... 

It should be noted that no such difficulty appears with the integral of an absorption spectrum because the absorption coefficient is proportional to the logarithm of a ratio of intensities, so that e(A) = e(v). For instance, in the calculation of an oscillator strength (defined in Chapter 2), integration can be done either in the wavelength scale or in the wavenumber scale. 

It is difficult to measure the oscillator strengths of molecules embedded in a matrix. Despite this, good values of can be determined as a function of the temperature. A procedure we have used to extract the information from excitation spectra was to set the maximum of the excitation spectrum measured at room temperature equal to the extinction coefficient at the absorption maximum in solution. The integrals of the excitation spectra were then normalized to the integral of the corresponding spectmm at room temperature, which is reasonable because the oscillator strength / of a transition n <— m does not depend on the temperature. 

For indirect-gap materials, all of the occupied states in the valence band can be connected to all the empty states in the conduction band. In this case, the absorption coefficient is proportional to the product of the densities of initial states and final states (see Eqnation (4.27)), bnt integrated over all the possible combinations of states separated by bro being the energy of the phonon involved). This... 

Equation (8) therefore allows us to use directly tabulated subshell photo-ionization cross sections (p) instead of mass absorption coefficients (a). F is the integrated photoelectron signal from an appropriate subshell of the monolayer adatom Yg the integrated signal from the relevant subshell of the substrate which is not simply the area of the core-level peak p and p. 

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