Scattering inelastic

Inelastic scattering of atoms and molecules is clearly of more physical interest than elastic scattering. This includes excitation or relaxation of rotational and vibrational degrees of freedom and also of electronic states. Rotational/vibrational inelasticity is important in understanding classical transport phenomena in gases, and the vibrational relaxation of highly excited molecules is of crucial importance for describing unimolecular reactions in, e.g. the Lindeman mechanism [68]. 

There is much effort devoted to learning more quantitatively about vibrational de-activation of highly excited vibrational states [69]. Electronically inelastic collisions are important in many gas laser systems, in the upper atmosphere, and in plasmas. Many of these applications involve one of the collision partners being an ion. 

The quantum mechanical description of an inelastic scattering process is straightforward. Leaving aside details involving angular momentum, the wavefunction for the generic situation is expanded as 

The theoretical task, therefore, is to solve (2.10) with the boundary conditions of (2.12) to obtain the S-matrix, in terms of which the inelastic cross 

Using a simple rigid rotor-harmonic oscillator approximation for the energy levels, 

Coherent Inelastic Scattering.—Inelastic neutron collisions with the solid can excite phonon modes (collective vibrations) and if the coherently scattered component can be detected variation with direction within the solid, i.e. the phonon dispersion curve, can be determined. This technique is well established for bulk solids and has been used recently to examine the properties of small particles (carbon black). 

Howard, T. C. Waddington, and C. J. Wright, in Neutron Inelastic Scattering , 1977, Internat. Atomic Energy Agency, Vienna, 1978, p. 499. 

Similar experiments with adsorbed layers are possible in principle but are limited at present to a few cases, such as 3 Ar where Uch incoh- Future improvements in source intensities and spin-polarization experiments, however, should increase the range of systems accessible.  

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Studies of inelastic scattering are of considerable interest in heterogeneous catalysis. The degree to which molecules are scattered specularly gives information about their residence time on the surface. Often new chemical species appear, whose trajectory from the surface correlates to some degree with that of the incident beam of molecules. The study of such reactive scattering gives mechanistic information about surface reactions. 

Electrons interact with solid surfaces by elastic and inelastic scattering, and these interactions are employed in electron spectroscopy. For example, electrons that elastically scatter will diffract from a single-crystal lattice. The diffraction pattern can be used as a means of stnictural detenuination, as in FEED. Electrons scatter inelastically by inducing electronic and vibrational excitations in the surface region. These losses fonu the basis of electron energy loss spectroscopy (EELS). An incident electron can also knock out an iimer-shell, or core, electron from an atom in the solid that will, in turn, initiate an Auger process. Electrons can also be used to induce stimulated desorption, as described in section Al.7.5.6. 

Radiation probes such as neutrons, x-rays and visible light are used to see the structure of physical systems tlirough elastic scattering experunents. Inelastic scattering experiments measure both the structural and dynamical correlations that exist in a physical system. For a system which is in thennodynamic equilibrium, the molecular dynamics create spatio-temporal correlations which are the manifestation of themial fluctuations around the equilibrium state. For a condensed phase system, dynamical correlations are intimately linked to its structure. For systems in equilibrium, linear response tiieory is an appropriate framework to use to inquire on the spatio-temporal correlations resulting from thennodynamic fluctuations. Appropriate response and correlation functions emerge naturally in this framework, and the role of theory is to understand these correlation fiinctions from first principles. This is the subject of section A3.3.2. 

To generalize what we have just done to reactive and inelastic scattering, one needs to calculate numerically integrated trajectories for motions in many degrees of freedom. This is most convenient to develop in space-fixed Cartesian coordinates. In this case, the classical equations of motion (Hamilton s equations) are given... 

Manolopoulos D E 1986 An improved log derivative method for inelastic scattering J. Chem. Phys. 85 6425-9... 

Quack M and Troe J 1975 Complex formation in reactive and inelastic scattering statistical adiabatic channel model of unimolecular processes III Ber. Bunsenges. Phys. Chem. 79 170-83... 

The ratio of elastically to inelastically scattered electrons and, thus, their importance for imaging or analytical work, can be calculated from basic physical principles consider the differential elastic scattering cross section... 

Inelastic scattering processes are not used for structural studies in TEM and STEM. Instead, the signal from inelastic scattering is used to probe the electron-chemical environment by interpreting the specific excitation of core electrons or valence electrons. Therefore, inelastic excitation spectra are exploited for analytical EM. 

Specimens for (S)TEM have to be transparent to the electron beam. In order to get good contrast and resolution, they have to be thin enough to minimize inelastic scattering. The required thin sections of organic materials can be obtained by ultramicrotomy eitlier after embedding into suitable resms (mostly epoxy- or methacrylate resins [H]) or directly at low temperatures by cryo-ultramicrotomy [12]. 

Perhaps the best known and most used optical spectroscopy which relies on the use of lasers is Raman spectroscopy. Because Raman spectroscopy is based on the inelastic scattering of photons, the signals are usually weak, and are often masked by fluorescence and/or Rayleigh scattering processes. The interest in usmg Raman for the vibrational characterization of surfaces arises from the fact that the teclmique can be used in situ under non-vacuum enviromnents, and also because it follows selection rules that complement those of IR spectroscopy. 

This chapter deals with qnantal and semiclassical theory of heavy-particle and electron-atom collisions. Basic and nsefnl fonnnlae for cross sections, rates and associated quantities are presented. A consistent description of the mathematics and vocabnlary of scattering is provided. Topics covered inclnde collisions, rate coefficients, qnantal transition rates and cross sections. Bom cross sections, qnantal potential scattering, collisions between identical particles, qnantal inelastic heavy-particle collisions, electron-atom inelastic collisions, semiclassical inelastic scattering and long-range interactions. 

Inelastic scattering produces a pennanent change in the internal energy and angrilar momentum state of one or both structured collision partners A and B, which retain their original identity after tire collision. For inelastic = (a, P) — /= (a, P ) collisional transitions, tlie energy = 1 War 17 of relative motion, before ( ) and after... 

Both conventions are identical only for direct collisions A (a) + B((3) A(a )+B(P ). This nonnalization is customary [5] for elastic and inelastic scattering processes. 

The cross section for inelastic scattering of beam of particles by potential V(r, R) is... 

Here the distortion (diagonal) and back coupling matrix elements in the two-level equations (section B2.2.8.4) are ignored so that = exp(ik.-R) remains an imdistorted plane wave. The asymptotic solution for ij-when compared with the asymptotic boundary condition then provides the Bom elastic ( =f) or inelastic scattering amplitudes... 

In tenns of the phase shifts h associated with potential scattering by U, tlie amplitudes for elastic and inelastic scattering are then... 

The basic assumption here is the existence over the inelastic scattering region of a connnon classical trajectory R(t) for the relative motion under an appropriately averaged central potential y[R(t)]. The interaction V r, / (t)] between A and B may then be considered as time-dependent. The system wavefiinction therefore satisfies... 

Parker G A and Pack R T 1978 Rotationally and vibrationally inelastic scattering in the rotational lOS approximation. Ultra-simple calculation of total (differential, integral and transport) cross sections for nonspherical molecules J. Chem. Phys. 68 1585... 

Hossenlopp J M, Anderson D T, Todd M W and Lester M I 1998 State-to-state inelastic scattering from vibrationally excited OH-Hj complexes J. Chem. Phys. 109 10 707-18... 

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