Intensities of vibrational transitions

To compare the relative populations of vibrational levels, the intensities of vibrational transitions out of these levels are compared. Figure B2.3.10 displays typical potential energy curves of the ground and an excited electronic state of a diatomic molecule. The intensity of a (v, v ) vibrational transition can be written as... 

The shapes of the absorption band associated with the intensities of vibrational transitions, are sensitive functions of the equilibrium bond length, about which approximately harmonic vibrational oscillations occur. Potential energy curves for a diatomic molecule (Figure 4.2), are commonly represented by Morse equation,... 

Figure 52. Ratios of intensities of vibrational transitions in N2+ first negative system as function of projectile-ion velocity (laboratory) for collisions of various ions with N2. Intensity ratios and relative populations of upper vibrational states are indicated by left- and right-hand vertical axes, respectively. Arrows on vertical axes indicate relative populations and corresponding intensity ratios predicted by Franck-Condon principle.154... 

The intensities of vibrational transitions in an electronic band system are proportional to the square of the matrix elements... 

The first term vanishes unless V = V" because of the orthogonality of the functions i//y. Therefore, the permanent electric moment /x has no influence on the intensity of vibrational transitions it does, however, determine the intensity of the pure rotation spectrum. The integral in the second term can be split up into factors as shown below ... 

In calculations quantum-mechanical model allows to calculate energies and relative intensities of vibrational transitions, that is, Dirac 5 functions. The band shapes mimicking experimental spectra were obtained by superimposing Gaussian bands with the same half-width on the 5 functions to visualize the comparison between the experimental and calculated spectra. 

In principle, the derivatives required for setting up eq. (II-l) could be obtained from the intensities of vibrational transitions. Actually, even in the most favorable cases only the first derivatives have been evaluated, generally with some ambiguity of sign. For this reason, among others, the writer (20) has treated polyatomic molecules, in an admittedly crude way, as if each were a set of independent oscillators, much like an aggregate of diatomic molecules. For a diatomic molecule, eq. (II-l) reduces to ... 

If the spectmm is observed in absorption, as it usually is, and at normal temperatures the intensities of the transitions decrease rapidly as v" increases, since the population of the uth vibrational level is related to Nq by the Boltzmann factor... 

For films on non-metallic substrates (semiconductors, dielectrics) the situation is much more complex. In contrast with metallic surfaces both parallel and perpendicular vibrational components of the adsorbate can be detected. The sign and intensity of RAIRS-bands depend heavily on the angle of incidence, on the polarization of the radiation, and on the orientation of vibrational transition moments [4.267]. 

One of the most familiar uses of dipole derivatives is the calculation of infrared intensities. To relate the intensity of a transition between states with vibrational wavefunctions i/r and jfyi it is necessary to evaluate the transition dipole moment... 

To determine the vibrational structure of electronic transitions of polyatomics, we can make the same approximation [Equation (7.21)] as for diatomics of replacing Ptl by some sort of average Pel, which is independent of the Qi s. The intensities of vibrational bands in an electronic transition then depend on the vibrational overlap integral, which is like... 

Hence vibrational transitions are allowed between the (12°3) level and 2 and Ug levels. The only fundamental level that combines with (12°3) is (10°0). [Group theory gives only qualitative and not quantitative information. It tells us that an IR transition between the (12°3) and (10°0) levels is allowed, but it says nothing about the intensity of the transition. Since this transition involves a change of 2 in v2 and a change of 3 in t 3, the considerations of Section 6.5 tell us that it will be extremely weak.]... 

The intensities of the /<-/ transitions of the tripositive lanthanoid element spectral bands are a good deal lower than those of the transition metals, presumably because the coupling between the electronic and vibrational wave functions is smaller on account of the shielding from the valence electrons. Value of/in the order of KT7 are observed.47-49 109 However, certain of the elements in the divalent state show d -f transitions which, as expected, are much more intense and blanket the /<-/bands.47-49... 

Fig. 22 shows the results of photometry of plates similar to that illustrated in Fig. 21. The relative intensities of suitable transitions were determined from the asymptotic limit at long time delays when the system attains equilibrium. (These resemble, but are not identical to, the relative/ values because of the usual instrumental effects which depend on line width.) The time variation of the relative concentrations is shown in Fig. 23 the upper four levels attain Boltzmann equilibrium amongst themselves after 100 /isec, to form a coupled (by collision) system overpopulated with respect to the 5DA state. The equilibration of the upper four levels causes the initial rise (Fig. 22) in the population of Fe(a5D3). Thus relaxation amongst the sub-levels is formally similar to vibrational relaxation in most polyatomic molecules, in which excitation to the first vibrational level is the rate determining step. In both cases, this result is due to the translational overlap term, for example, in the simple form of equation (14) of Section 3.

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