The absorption cross section

2 Light absorption and photodissociation 2.2 The absorption cross section 

The microscopic theory derived in the previous section describes the evolution of the molecular system under the influence of the electric field of the light beam. In this section, following Loudon (1983 ch.l), we use (2.22) to deduce an expression for the phenomenological absorption cross section cr(u ) defined in (2.1). See Loudon for a more detailed discussion. 

The beam energy in the slice is S = AdzW. As a result of the molecular transition from state iq) to state Ff) it decreases by 

Equation (2.25) governs the time dependence of the energy density W. However, we are searching for an expression which describes the spatial dependence of the intensity I. Using the relations (Loudon 1983 ch.l) dW/dt = dl/dz and W = I/c, where c is the velocity of light, we obtain 

Integration of (2.26) readily yields the desired Equation (2.1). We note that d 2/eo has units of energy times volume and with that it is easy to show that the cross section has units of area. 

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The strength of a fluorescence signal is directly related to the absorption cross section CJ ... 

Local surface structure and coordination numbers of neighbouring atoms can be extracted from the analysis of extended X-ray absorption fine structures (EXAFS). The essential feature of the method22 is the excitation of a core-hole by monoenergetic photons modulation of the absorption cross-section with energy above the excitation threshold provides information on the distances between neighbouring atoms. A more surface-sensitive version (SEXAFS) monitors the photoemitted or Auger electrons, where the electron escape depth is small ( 1 nm) and discriminates in favour of surface atoms over those within the bulk solid. Model compounds, where bond distances and atomic environments are known, are required as standards. 

The overall OD vibrational distribution from the HOD photodissociation resembles that from the D2O photodissociation. Similarly, the OH vibrational distribution from the HOD photodissociation is similar to that from the H2O photodissociation. There are, however, notable differences for the OD products from HOD and D2O, similarly for the OH products from HOD and H2O. It is also clear that rotational temperatures are all quite cold for all OH (OD) products. From the above experimental results, the branching ratio of the H and D product channels from the HOD photodissociation can be estimated, since the mixed sample of H2O and D2O with 1 1 ratio can quickly reach equilibrium with the exact ratios of H2O, HOD and D2O known to be 1 2 1. Because the absorption spectrum of H2O at 157nm is a broadband transition, we can reasonably assume that the absorption cross-sections are the same for the three water isotopomer molecules. It is also quite obvious that the quantum yield of these molecules at 157 nm excitation should be unity since the A1B surface is purely repulsive and is not coupled to any other electronic surfaces. From the above measurement of the H-atom products from the mixed sample, the ratio of the H-atom products from HOD and H2O is determined to be 1.27. If we assume the quantum yield for H2O at 157 is unity, the quantum yield for the H production should be 0.64 (i.e. 1.27 divided by 2) since the HOD concentration is twice that of H2O in the mixed sample. Similarly, from the above measurement of the D-atom product from the mixed sample, we can actually determine the ratio of the D-atom products from HOD and D2O to be 0.52. Using the same assumption that the quantum yield of the D2O photodissociation at 157 nm is unity, the quantum yield of the D-atom production from the HOD photodissociation at 157 nm is determined to be 0.26. Therefore the total quantum yield for the H and D products from HOD is 0.64 + 0.26 = 0.90. This is a little bit smaller ( 10%) than 1 since the total quantum yield of the H and D productions from the HOD photodissociation should be unity because no other dissociation channel is present for the HOD photodissociation other than the H and D atom elimination processes. There are a couple of sources of error, however, in this estimation (a) the assumption that the absorption cross-sections of all three water isotopomers at 157 nm are exactly the same, and (b) the accuracy of the volume mixture in the... 

H2O and D2O mixed sample used in the experiment. For the absorption cross-sections, there are probably some small differences among the three isotopomers in reality. Nevertheless, this estimation should be quite realistic. The estimated branching ratios of the H and D productions from HOD at 157 nm excitation should be 2.46 with about 15% estimated error bar. More accurate measurement on the branching ratio should be possible with the experimentally measured cross-section values for H2O and D2O. 

Fig. 14. Plot of the quantum efficiency for the formation of C>2(v < 26) as a function of wavelength. The value at 193 nm is taken from Stranges et aJ.44 A 10% branching ratio into the triplet channel is assumed. The absorption cross-section for ozone is plotted as the dashed curve on the right-hand axis.

Absorption of X-ray radiation of energy well above the threshold for an X-ray transition will result in the ejection of a photoelectron since the initial unoccupied band stale to which the transition takes place will be above the vacuum level. The Kronig fine structure is due to oscillations induced in the absorption cross-section of the absorbing atom as a result of interference... 

The rate of photolysis, J, depends on the absorption cross-section, a, the number density, the scale height and the angle, all of which are unique properties of a planetary atmosphere. For the Earth and the Chapman mechanism for ozone the O3 concentration maximum is 5 x 1012 molecules cm-3 and this occurs at 25 km, shown in Figure 7.12, and forms the Chapman layer structure. 

As a second example of the application of ion-beam analysis techniques to semiconductors, we take the calibration of IR absorption measurements of the hydrogen content of sputtered amorphous silicon and silicon nitride. In early measurements, the hydrogen content of glow-discharge a-Si H deduced from IR absorption measurements, using ablsinitio calculations of the absorption cross section of the Si—H IR absorption bands, was com-... 

Given that the absorption cross-section of 07+ at the photo-ionization threshold is 10-19 cm2 and that the oxygen abundance is 10-2 by mass, find the bound-free opacity (cm2 gm-1) due to oxygen at that frequency, and compare it to the electron scattering opacity. 

That is, by measuring the optical density and the sample thickness, the absorption coefficient can be determined. According to Equation (1.6) we can now determine the absorption cross section a if the density of centers is known. This means that, by a... 

Let N(j,Ni,N2, and Nj, be the equilibrium population densities of the states 0, 1,2, and 3, respectively (reached under continuous wave excitation intensity Iq), and let N = NQ + Ni+N2 + N3he the total density of optical absorbing centers. The up-converted luminescence intensity ho (corresponding to the transition 2 0) depends on both N2 and on the radiative emission probability of level 2, A2. This magnitude, which is dehned below, is proportional to the cross section a20 (called the emission cross section and equal to the absorption cross section ao2, as shown in Chapter 5). Thus we can write... 

As a result of the second step process, the population of the emitting state 2 (N3 0, due to the feeding of level 2 by level 3), increases with the population of level 1, with ctu (the absorption cross section for the transition 1 3), and... 

To conduct such an experiment using electron beam radiation is more tedious, since the absorption cross section of organic materials to electrons is... 

Again the radiative association kinetics described above allow a direct comparison for some realistic values of k and k. For most chemically activated systems at the threshold for unimolecular dissociation, the observed radiative rate constants are of the order of 10-100 s and hence are much below the values expected for k of about 10 s . Therefore, the first limit is most likely to be valid, with the interesting conclusion that the observed unimolecular dissociation rate constant will depend only on the photon density and the absorption cross section (rate constant) at a given wavelength. 

The absorption cross sections of the Sn Si transition are determined from the kinetic analyses based on the two-photon absorption measurement. In contrast, tetraphenylporphin scarcely show the S2 state fluorescence. In order to elucidate the effect of the imino hydrogens on the radiationless transitions, deuterium isotope effect on the relaxation processes from the lowest excited singlet state of metal free porphyrins has been investigated. 

Ito et al. [20] studied the mechanism to produce strand breakage by phosphorus photoabsorption using a dinucleotide as a sample, and revealed that even with photoabsorption by phosphorus strand breakage occurs through the destruction of deoxyribose of the 5 end, leaving adenine and 5 -AMP as products. Yamada et al. [21] used penta-deoxythymidylic acid (d(pT)5) as a sample, and analyzed the products quantitatively with on-resonance (2153 eV) and off-resonance (2147 eV) x-rays. They found that the distribution of the products was independent of the x-ray energy, and that the yields of the products were proportional to the absorption cross section of the sample. Unexpectedly, they could not find any evidence of Auger-specific products, neither qualitatively nor quantitatively. 

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