Vibrational Broadening

As will be clear in Chapter 7, the absolute value of the transition dipole moment vector ( ix, Py, Pz) determines the intensity of a transition. The x-component can be written as 

dx is an integration element for both electron and nuclear coordinates. The nuclear part of the transition moment should be included, but since the electronic states are orthogonal (zero overlap), this contribution will vanish. 

The total wave functions for the states involved in the excitation are 

The transition moment thns depends on vibrational overlap. Since u is usually the lowest vibrational state (u = 0), it may be written as (HoolHi a The square of the latter is called the Franck-Condon integral. The width of the spectrum depends on how many vibrational levels in the ground state have overlapped with the vibrational levels in the excited state. If the excitation takes place in the gas phase and there are no obvious broadening mechanisms, the spectrum will be resolved in sharp vibration levels. Equation 4.80 is the quantum mechanical expression for the Franck-Condon principle. If the overlap is calculated in Eqnation 4.80, one obtains the highest intensity for the almost vertical transitions. 

One may also conclnde from Equation 4.80 that many final vibrational states are populated after the transition. This is, in fact, the main reason for the dynamics that takes place after excitation. The excited state as well as the ground state may involve many atoms. The excited states may be charge transfer states or exciton states, as will be discussed in Chapter 12. 

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Vibrational broadening in [162] was taken into account under the conventional assumption that contributions of vibrational dephasing and rotational relaxation to contour width are additive as in Eq. (3.49). This approximation provides the largest error at low densities, when the contour is significantly asymmetric and the perturbation theory does not work. In the frame of impact theory these relaxation processes may be separated more correctly under assumption of their statistical independence. Inclusion of dephasing causes appearance of a factor... 

Fig. 3.15, The CARS spectrum rotational width versus methane density for various values of parameter y (1) y = 0, (2) y = 0.3, (3) y = 0.5, (4) y = 0.7, (5) y = 0.75, (6) y = 0.9, (7) y = 0.95, (8) y = 1. Curves (4) and (6) are obtained by subtraction of the dephasing contribution from the line width calculated taking account of vibrational broadening. The other dependences are found assuming purely rotational broadening (vibrational relaxation neglected).

Valiev-Ivanov model 219, 275 vibrational broadening 123 vibrational dephasing 111, 113-15, 123 vibrational relaxation, and angular momentum relaxation 92 vibrational transition, adiabatic dephasing 92... 

Chromium activated ruby was the first laser material and its luminescence properties are carefully studied. It is a classical example of Cr + in octahedral crystal field. Here Cr + substitutes the AP ions, while such a possibihty can be rationalized by an excellent chemical fit of Cr in place of Al. Ruby is a high crystal field material and thus the T2g state Hes above the E2g level. Pumping is accomplished by a spin-allowed transition into the state, while emission occurs from the level without vibrational broadening and almost all excited... 

Garnet activated by trivalent Cr is a promising system for tunable laser appUcations and those systems have been well studied. Cr + replaces Ap" in octahedral sites with a weak crystal field. The transition involved in laser action is T2- A2, a vibrationally broadened band. At room temperature it has a maximum in the 715-825 nm range with a decay time in the 100-250 ps range depending on AE between the E and T2 levels. When the AE is maximal, narrow fines also appear from the E level. At low temperatures, when thermal activation of the T2 level is difficult, J -lines luminescence becomes dominant with the main fine at 687 nm (Monteil at al. 1988). We studied pyrope artificially activated by Cr and also found the two emission types described above (Fig. 5.26). 

The decay widths are in meV, citations are given in square brackets. Experimental value for ammonia is lacking because of the vibrational broadening in the Auger electron spectrum of ammonia [65], See Ref. [44] for the details of the Fano-ADC computation. 

Thermally-induced network vibrations broaden the absorption edge and shift the band gap of semiconductors. The thermal disorder couples to the optical transition through the deformation potential, which describes how the electronic energy varies with the displacement of the atoms. The bond strain in an amorphous material is also a displacement of atoms from their ideal position, and can be described by a similar approach. The description of static disorder in terms of frozen phonons is a helpful concept which goes back 20 years. Amorphous materials, of course, also have the additional disordering of the real phonon vibrations. 

Such bands are attributed to Cr3+ which occur with low ligand field strength where the T2 level lies below the first doublet 2E. In all low field sites the room temperature Cr3+ emission consists of a broad structureless band in the near infra red due to a vibrationally broadened T2 4 2 transition. Figure 7 presents a typical absorption spectrum of... 

The sharp doublet has been described as due to an ionization from the central sulfur atom, the broad doublet as due to an ionization from the outer sulfur centers. The broadening (vibrational broadening) is explained by assuming a U-shaped potential for the ground state emd that ionic state (5i) which corresponds to ejection of an electron from the 2p shell of the central sulfur atom. The ejection of an electron out of the 2p shell on the outer sulfur atoms will cause a change in the geometry in the corresponding state (Sg). 

For the small cluster sizes (n = 3-9), single, well-separated resonances are observed, which could be well fitted by Gaussians. As discussed above (see Section 5.3.1), the peaks are interpreted as vibrationally broadened electronic transitions of an Na " " molecule [17, 30, 40]. 

Spectral narrowing simply means that an absorption in the spectrum is surprisingly narrow. In Chapter 6, we mentioned that some spin-forbidden transitions can be seen as very sharp absorptions in transition metal systans. The reason in that case is that the same spatial orbitals are occupied as in the ground state and therefore, the bond lengths are almost the same in the ground state and in the excited state. The Stokes shift is close to zero and therefore, there is no vibrational broadening. 

H) have been reported, and discussed in regard to the effect of changes of the 6a-atom on all other non-hydrogen atoms in the molecule. The observed vibrational broadening of the oxygen Is lines is related to the observed bond lengths, within a simple model. 

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