Integrated absorption intensities equation

Section BT1.2 provides a brief summary of experimental methods and instmmentation, including definitions of some of the standard measured spectroscopic quantities. Section BT1.3 reviews some of the theory of spectroscopic transitions, especially the relationships between transition moments calculated from wavefiinctions and integrated absorption intensities or radiative rate constants. Because units can be so confusing, numerical factors with their units are included in some of the equations to make them easier to use. Vibrational effects, die Franck-Condon principle and selection mles are also discussed briefly. In the final section, BT1.4. a few applications are mentioned to particular aspects of electronic spectroscopy. 

In order to make quantitative measurements. Miles (1958f)) has determined integrated infrared absorption intensities for some nucleosides, nucleotides, and polynucleotides in DjO solution (Fig. 12.8 and Table 12.2) by application of Ramsay s method I (Ramsay, 1952). When there is a well-resolved band, the application of Ramsay s method encounters no difficulty, but when overlapping bands occur, as in uridine and its derivatives, some uncertainty exists in determining the halfband width of a particular band. (The half-band width is a factor in Ramsay s equation and is defined as, Av, 2 = the width of the band in cm" at half-maximum intensity.) The equation as used by Miles to obtain A, the true integrated absorption intensity... 

Einstein derived the relationship between spontaneous emission rate and the absorption intensity or stimulated emission rate in 1917 using a thennodynamic argument [13]. Both absorption intensity and emission rate depend on the transition moment integral of equation (B 1.1.1). so that gives us a way to relate them. The symbol A is often used for the rate constant for emission it is sometimes called the Einstein A coefficient. For emission in the gas phase from a state to a lower state j we can write... 

The nature and the distribution of different types of Fe species in calcined (C) and steamed (S) samples were investigated by means of UV-vis spectroscopy. UV-vis spectra of Fe species were monitored on UV-vis spectrometer GBS CINTRA 303 equipped with a diffuse reflectance attachment with an integrating sphere coated with BaS04 and BaS04 as a reference. The absorption intensity was expressed using the Schuster-Kubelka-Munk equation. 

UV-VIS-NIR diffuse reflectance (DR) spectra were measured using a Perkin-Elmer UV-VIS-NIR spectrometer Lambda 19 equipped with a diffuse reflectance attachment with an integrating sphere coated by BaS04. Spectra of sample in 5 mm thick silica cell were recorded in a differential mode with the parent zeolite treated at the same conditions as a reference. For details see Ref. [5], The absorption intensity was calculated from the Schuster-Kubelka-Munk equation F(R ,) = (l-R< )2/2Roo, where R is the diffuse reflectance from a semi-infinite layer and F(R00) is proportional to the absorption coefficient. 

Net integrated absorption band intensities are usually characterized by one of the quantities A, 5, 5, F, or Gnet as defined in the table. The relation between these quantities is given by the (approximate) equations... 

The equality resulting from Kirchhoff s law between the directional spectral absorptivity and the emissivity, aA = eA, suggests that investigation of whether the other three (integrated) absorptivities aA, a and a can be calculated from the corresponding emissivities sx, s and e should be carried out. This will be impossible without additional assumptions, as the absorptivities ax, a and a are not alone material properties of the absorbing body, they also depend on the incident spectral intensity Kx of the incident radiation, see Table 5.1. The emissivities sx, s and s are, in contrast, purely material properties. An accurate test is therefore required to see whether, and under what conditions, the equations analogous to (5.69), ax = sx, a = s and a = e are valid. 

The concentration of base sites was measured by the integral intensity of CD band in the range of 2190-2255 cm-i. The integral absorption coefficient was calculated from the correlation equations (Paukshtis, 1992) ... 

Here, is the intensity of the fundamental vibration absorption band. Note that Equation 2.10 can be derived by assuming a smooth dipole moment curve, by which bonds have no strong vibrational coupling with the other bonds in the molecule. Figure 2.4 shows the relative integral band intensities EJE- for different C-X bonds as a function of the absorption peak wavelengths, which were calculated using the reported values for Xe> - i Equations 2.9 and 2.10... 

If the experiment to which the calculation refers were actually attempted, several differences would appear. A lost important, the x-ray power would be expended over a vide spectrum. The intensity in Equation-4-10 would be the integrated intensity from the short-wavelength limit to the critical absorption wavelength. /Also, gmax and wka would need to be replaced by values that reflect the wavelength range of the integrated intensity. The net effect of all these differences would be to reduce /k below the value of Equation 4-16, perhaps by as much as ten-fold. 

Equation (3.19) gives a first approximation to the temperature structure of an atmosphere in radiative equilibrium, and departures from greyness can also be treated approximately by defining a suitable mean absorption coefficient (see Chapter 5). The emergent monochromatic intensity at an angle 9 to the normal (relevant to some point on the solar disk) is also found by integrating the equation of transfer (3.11) ... 

Table 1 gives a comparison of Raman and pmr results for a series of copolymers. In the pmr data of Figure the CHg absorption of the polymer backbone at 6O.8 to 3.0 partially overlaps with the CH doublet centered at S2.h and this reduces the accuracy of the integrated intensity of the ester moiety to no better than 25. On the other hand, the accuracy of the Raman data is on the order of 3%, so the two techniques do agree within experimental error. The error associated with the Raman method could be reduced if calibration curves were employed. The weight percent feed and polymer compositions were converted to mole percent and reactivity ratios for MMA and OM were calculated by the Yezrielev, Erokhina and Riskin (YBR) method (9). The following equation, derived from the copolymer... 

If 70 represents the incident intensity of the radiation before it passes through a medium of thickness / that has an absorption coefficient k, the transmitted intensity /will be given by the integrated form, equation (11.6), of the previous equation ... 

In order to explore this point, it is necessary to have estimates for the magnitude of V. One approach to estimating V is based on the integrated intensity of absorption bands for optically induced electron transfer as for the transition illustrated in equation (9).16 From this type of analysis V a 120 cm-1 for the particular case shown in equation (9).15 A second approach to estimating V, which is purely theoretical and based on orbital overlap calculations, typically gives values in the range 100-200 cm-1 for outer-sphere reactions.12 36 37... 

With V in hand from the integrated intensity of the absorption band and a from the band maximum, all of the quantities that appear in equation (30) which are needed to calculate ket in the classical limit are available from the properties of an IT band. Of course, this remarkable conclusion must be tempered by the fact that if V is appreciable, vet may be dictated by timescales arising from the trapping vibrations or solvent motions and not by F.65b As noted below, there are additional complicating features that may limit the validity of equations like (70) and (72). 

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... 

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