Operators Raman transitions

We present here a summary of recent work with light-pulse interferometer based inertial sensors. We first outline the general principles of operation of light-pulse interferometers. This atomic interferometer (Borde et al., 1992 Borde et al., 1989) uses two-photon velocity selective Raman transitions (Kasevich et al., 1991), to manipulate atoms while keeping them in long-lived ground states. 

In addition to energy eigenvalues it is of interest to calculate intensities of infrared and Raman transitions. Although a complete treatment of these quantities requires the solution of the full rotation-vibration problem in three dimensions (to be described), it is of interest to discuss transitions between the quantum states characterized by N, m >. As mentioned, the transition operator must be a function of the operators of the algebra (here Fx, Fy, F7). Since we want to go from one state to another, it is convenient to introduce the shift operators F+, F [Eq. (2.26)]. The action of these operators on the basis IN, m > is determined, using the commutation relations (2.27), to be... 

As in the previous case of infrared transitions, one wants to calculate the line strengths S(v,J —> v, J ) defined in Eq. (2.127). For Raman transitions there are two contributions, as discussed in Chapter 1. The so-called trace scattering is induced by the monopole operator... 

These results apply specifically to Rayleigh, or elastic, scattering. For Raman, or inelastic, scattering the same basic CID expressions apply but with the molecular property tensors replaced by corresponding vibrational Raman transition tensors between the initial and final vibrational states nv and rn . In this way a s are replaced by (mv aap(Q) nv), where aQ/3(<3) s are effective polarizability and optical activity operators that depend parametrically on the normal vibrational coordinates Q such that, within the Placzek polarizability theory of the Raman effect [23], ROA intensity depends on products such as (daaf3 / dQ)0 dG af3 / dQ) and (daaf3 / dQ)0 eajS dAlSf / dQ)0. 

In these equations, Hw(qa, q, 0) and jls(qa, qfo, 0) are the dipole moment operators at initial time belonging, respectively, to the irreducible representations B and A of the C2 symmetry group that transforms themselves, the first one, according to x and v, and the last one, according to x2 and y2 (allowed Raman transition), where x and y and the Cartesian coordinates that are perpendicular to the C2 symmetry axis. Here, we prefer the notations g and u in place of A and B of group theory. 

In these equations, pu(0) and p ( 0) are, respectively, the IR and Raman transition moment operators sat initial time and at time f, whereas pu(f) and p (t) are the corresponding moments at time t. They are respectively... 

In order to calculate the IR and Raman transition probabilities per unit time between the vibration state, i, to the vibration state, j, it is necessary to express in quantum mechanical terms the operator describing the interaction between the molecule and the electromagnetic radiation, which is given by... 

The bond polarizability theory of conventional Raman intensity is well-established 46,47). The starting point is Placzek s approximation for the vibrational Raman transition polarizability at transparent frequencies48 . On expanding the effective polarizability operator aotp(Q) in the normal vibrational coordinates Qp, the transition polarizability becomes... 

One of the most important aspects in both experimental and theoretieal studies in molecular spectroscopy is, undoubtedly, the characterization of intensities induced by electromagnetic radiation. We are, of course, interested in obtaining information concerning infrared and Raman transitions which are driven by electric dipole and quadrupole operators, respectively. These transitions can be represented as... 

The one-dimensional model is also more suitable for the direct computation of infrared and Raman transition intensities. They are obtained through known vibrational wavefunctions and through simple calculations involving linear combinations of exponential operators. If we adopt the approximate formulas discussed in Section III.D, these calculations can be carried out within the same program used for the vibrational energies. 

Most practical combustion devices operate at pressures considerably well above atmospheric. The CARS signatures, used to extract temperature and species concentration Information, Figs. 3 and 4, are quite pressure sensitive for most of the heavy diatomic and trlatomlc molecules due to the phenomenon of colllslonal narrowing. Even In the absence of such band narrowing, the signatures would and do exhibit pressure sensitivity due to constructive Interferences among neighboring Raman transitions. 

The Raman transition selection rules are available the same way as the electric dipole selection rules, but the transition moment operator has the symmetry of the second order firnctions x, y, z, xy, yz, and xz. If we think of the Raman transition represented in Fig. 6.17 as a dual process—absorption and then emission—then this makes sense the probability of the Raman transition depends on the transition moment for reaching the virtual state (when the incident photon hits the molecule)... 

One way to directly measure rotational transitions in non-polar molecules such as these is by rotational Raman spectroscopy, which operates on the same principle as other Raman techniques (see Section 6.3). A rotational Raman transition connects initial and final rotational levels within the same vibrational state, so only the rotational quantum number changes. However, this technique is limited in precision by the uncertainties in the photon energies of the incident and scattered light. The scattering intensity increases dramatically with photon energy. 

Many of the characterization techniques described in this chapter require ambient or vacuum conditions, which may or may not be translatable to operational conditions. In situ or in opemndo characterization avoids such issues and can provide insight and information under more realistic conditions. Such approaches are becoming more common in X-ray adsorption spectroscopy (XAS) methods ofXANES and EXAFS, in NMR and in transmission electron microscopy where environmental instruments and cells are becoming common. In situ MAS NMR has been used to characterize reaction intermediates, organic deposits, surface complexes and the nature of transition state and reaction pathways. The formation of alkoxy species on zeolites upon adsorption of olefins or alcohols have been observed by C in situ and ex situ NMR [253]. Sensitivity enhancement techniques play an important role in the progress of this area. In operando infrared and RAMAN is becoming more widely used. In situ RAMAN spectroscopy has been used to online monitor synthesis of zeolites in pressurized reactors [254]. Such techniques will become commonplace. 

The fact that the order parameter vanishes above does not mean that Nature does not have an inkling of things to come well below (or above) T. Such indicators are indeed found in many instances in terms of the behaviour of certain vibrational modes. As early as 1940, Raman and Nedungadi discovered that the a-) transition of quartz was accompanied by a decrease in the frequency of a totally symmetric optic mode as the temperature approached the phase transition temperature from below. Historically, this is the first observation of a soft mode. Operationally, a soft mode is a collective excitation whose frequency decreases anomalously as the transition point is reached. In Fig. 4.4, we show the temperature dependence of the soft-mode frequency. While in a second-order transition the soft-mode frequency goes to zero at T, in a first-order transition the change of phase occurs before the mode frequency is able to go to zero. 

Raman spectra of adsorbed species, when obtainable, are of great importance because of the very different intensity distributions among the observable modes (e.g., the skeletal breathing frequency of benzene) compared with those observed by infrared spectroscopy and because Raman spectra of species on oxide-supported metals have a much wider metal oxide-transparent wavenumber range than infrared spectra. Such unenhanced spectra remain extremely weak for species on single-crystal surfaces, but renewed efforts should be made with finely divided catalysts, possibly involving pulsed-laser operation to minimize adsorbate decomposition. Renewed efforts should be made to obtain SER and normal Raman spectra characterizing adsorption on surfaces of the transition metals such as Ni, Pd, or Pt, by use of controlled particle sizes or UV excitation, respectively. 

Recall that homonuclear diatomic molecules have no vibration-rotation or pure-rotation spectra due to the vanishing of the permanent electric dipole moment. For electronic transitions, the transition-moment integral (7.4) does not involve the dipole moment d hence electric-dipole electronic transitions are allowed for homonuclear diatomic molecules, subject to the above selection rules, of course. [The electric dipole moment d is given by (1.289), and should be distinguished from the electric dipole-moment operator d, which is given by (1.286).] Analysis of the vibrational and rotational structure of an electronic transition in a homonuclear diatomic molecule allows the determination of the vibrational and rotational constants of the electronic states involved, which is information that cannot be provided by IR or microwave spectroscopy. (Raman spectroscopy can also furnish information on the constants of the ground electronic state of a homonuclear diatomic molecule.)... 

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