Transition Probability and Cross Section

The main purpose of the scattering theory is to calculate the probability for transition from one to another state resulting from a collision between two particles. We may use, instead, the notion of a cross section , which is a convenient measure of that probability. 

For simplicity we consider first the elastic scattering of two particles with masses m and m2t in which only the directions of motion of the particles change during the collision. In a center-of-mass coordinate system the scattering process is described by the motion of a mass point with an effective (reduced) mass p = m m2/(m-j+m2) in a potential field V(r) depending on the interparticular distance r. For a given initial relative velocity, the scattering direction, i.e., the direction of the final velocity relative to 1, is de- 

The differential cross section cess (i—i f) is defined by the equation 

The number of scattered particles may be expressed as a product of the radial current density and the surface element ds = r dn, i.e., dN = j r dn. so that 

The definitions (30,11) and (31.11) also hold for the cross sections of inelastic and reactive scattering in which the particles before and after the collision have different internal states and/or different compositions. Then, the letters i and j may be used to represent two sets of qusmttmi numbers for the initial state (separated reactants) and for the final state (separated products), respectively, in which also the reduced masses i and j ) and the relative velocities (v and v ) are generally different. 

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To describe the shifts and intensities of the m-photon assisted collisional resonances with the microwave field Pillet et al. developed a picture based on dressed molecular states,3 and we follow that development here. As in the previous chapter, we break the Hamiltonian into an unperturbed Hamiltonian H(h and a perturbation V. The difference from our previous treatment of resonant collisions is that now H0 describes the isolated, noninteracting, atoms in both static and microwave fields. Each of the two atoms is described by a dressed atomic state, and we construct the dressed molecular state as a direct product of the two atomic states. The dipole-dipole interaction Vis still given by Eq. (14.12), and using it we can calculate the transition probabilities and cross sections for the radiatively assisted collisions. 

Eigure 11 shows the product state distributions after decay of the Ai and A2 resonances at 4.41 and 4.49 eV respectively. In both cases, H -F H2 decay products have significant internal energy for the Ai symmetry, 41% of the available energy appears as rovibrational energy, and 51 % for the A2 case. Thus, these resonances decay exclusively into excited rovibrational states and were not observed on previously computed reactive scattering transitions probabilities and cross-sections... 

Photoabsorption transition probabilities and cross-sections for the two categories contain different satellite peaks due to the presence (or absence) of different zero-order Fermi-sea SACs in initial and final sfafes, for example, Ref. [101]. This is in accordance wifh FOTOS, where, as explained in Section 4, the essential features of the transition probability and the related phenomena are explained by using in the zero-order transition matrix element the Fermi-sea multiconfigurational wavefunctions of the initial and the final states of the transition [26b, 45]. 

Another important application of all-orders in aZ atomic QED is the theory of the multicharged ions. Nowadays all elements of the Periodic Table up to Uranium (Z=92) can be observed in the laboratory as H-like, He-like etc ions. The recent achievements of the QED theory of the highly charged ions (HCI) are summarized in [11], [12]. In principle, the QED theory of atoms includes the evaluation of the QED corrections to the energy levels and corrections to the hyperfine structure intervals, as well as the QED corrections to the transition probabilities and cross-sections of the different atomic processes photon and electron scattering, photoionization, electron capture etc. QED corrections can be evaluated also to the different atomic properties in the external fields bound electron -factors and polarizabilities. In this review we will concentrate mainly on the corrections to the energy levels which are usually called the Lamb Shift (here the Lamb Shift should be understood in a more broad sense than the 2s, 2p level shift in a hydrogen). 

One of the most Important questions to be addressed In this study Is the Importance of quantum effects In determining vibrational excitation probabilities and cross sections. We will study this here by comparing the calculated quantum lOS transition probabilities with the corresponding classical lOS probabilities Integral cross sections for vibrational excitation will also be compared The extreme complexity of the y and l dependence of transition probabilities has prevented us from obtaining rotational and angular distributions, but qualitative Information about these distributions will be Inferred from the results that we do have ... 

The conclusion which seems to be emerging from these examples, therefore, is that individual S-matrix elements, and thus the transition probability between a complete set of initial and final quantum numbers, cannot be described accurately without proper inclusion28 of the interference terms provided by the classical S-matrix approach. If the transition probabilities or cross sections of interest are summed and/or averaged over some of the final or initial quantum numbers, however, the interference terms tend to average to zero so that the completely classical treatment becomes adequate. 

Abstract A consistent relativistic energy approach to the calculation of probabilities of cooperative electron-gamma-nuclear processes is developed. The nuclear excitation by electron transition (NEET) effect is studied. The NEET process probability and cross section are determined within the S-matrix Gell-Mann and Low formalism (energy approach) combined with the relativistic many-body perturbation theory (PT). Summary of the experimental and theoretical works on the NEET effect is presented. The calculation results of the NEET probabilities for the y Os, yy Ir, and yg Au atoms are presented and compared with available experimental and alternative theoretical data. The theoretical and experimental study of the cooperative electron-gamma-nuclear process such as the NEET effect is expected to allow the determination of nuclear transition energies and the study of atomic vacancy effects on nuclear lifetime and population mechanisms of excited nuclear levels. 

We have considered in some detail several exact and approximate methods for calculation of transition probabilities (or cross sections) in colinear collisions. There exists now a great variety of other methods proposed during the last few years. Some of them will be mentioned here only briefly. 

Most of the classical, semiclassical and quantal calculations of transition probabilities (or cross sections) refer to colinear atom-diatom gas phase reactions. However, a consideration of the nonlinear collisions seems to be very important for an adequate description of the chemical elementary processes in physical space. Quite recently, encouraging progress in this direction has been made / /. 

The END equations are integrated to yield the time evolution of the wave function parameters for reactive processes from an initial state of the system. The solution is propagated until such a time that the system has clearly reached the final products. Then, the evolved state vector may be projected against a number of different possible final product states to yield coiresponding transition probability amplitudes. Details of the END dynamics can be depicted and cross-section cross-sections and rate coefficients calculated. 

We shall have instances of this procedure in the ensuing sections. Now, we wish only to find the relationship between the transition probability and the differential scattering cross-section for a particular example. 

The fundamental parameter of the cooperative NEET process is a probability NEET (cross section) of the nuclear excitation by electron transition. In fact, it can be defined as the probability that the decay of the initial excited atomic state will result to the excitation of and subsequent decay from the corresponding nuclear state. Within the energy approach, the decay probability is connected with an imaginary part of energy shift for the system (nuclear subsystem plus electron subsystem) excited state. An imaginary part of the excited state / energy shift in the lowest PT order can be in general form written as [18,26]... 

Phenomenological parameters such as lifetimes and cross-sections are linked to quantum mechanical microscopic properties of the doping ions by the so-called f-numbers. These are dimensionless numbers often used in spectroscopy to characterize the strength of a transition. They are pure numbers < 1. The f-numbers are linked to transition probabilities... 

The transition probability and the differential and total cross sections for reactive collisions... 

In the analogous quantum theory an internal state of the molecule is defined by a set of quantum numbers, which can be represented by a single S5mibol such as i or j. Then the probability that a molecule initially in state i will make a transition to state j when it suffers a collision characterized by appropriately defined parameters p can be symbolized by q ip), and cross sections exactly analogous to those of Eqs. (2) are familiar quantities in quantum mechanical scattering theory. 

If UV radiation (photon energies typically of 20-40 cV) were to be used rather than X-rays to excite emission from the valence band, the resultant spectra might be mote intense, but since the photoelectrons would have much lower kinetic energies the spectra would correspond to emission from the outer surface layers only. The process would therefore be particularly sensitive to the presence of impurities and contamination on the surface. In addition, UV excitation involves transition probabilities and ionization cross-sections different from those appearing in X-ray excitation, which has the consequence that spectral analysis by calculation has to include both ground and virtual states, and is therefore more difficult. 

Radiative transition probabilities and laser cross sections of rare... 

The previously discussed radiative transition probabilities and line strengths are connected to the peak-stimulated emission cross sections of the rare-earth ion in a glass by... 

The statistical adiabatic channel model (SACM) " is one realization of the laiger class of statistical theories of chemical reactions. Its goal is to describe, with feasible computational implementation, average reaction rate constants, cross sections, and transition probabilities and lifetimes at a detailed level, to a substantial extent with state selection , for bimolecular reactive or inelastic collisions with intermediate complex formation (symbolic sets of quantum numbers v, j, E,J. ..)... 

In the higher atmosphere the aerosol density decreases rapidly with altitude and other detection schemes may become more advantageous. Raman spectroscopy or detection of laser-induced fluorescence excited by frequency-doubled pulsed lasers has been utilized [14.22]. Both Raman and fluorescence intensities excited by the laser at a location x are proportional to the density n. (x) of scattering particles. However, because of the high pressure (p latm) the fluorescence is quenched if the collisional deactivation na v becomes faster than the spontaneous decay A. = 1/t. (see Sect. 12.2). Transition probabilities and quenching cross sections must therefore be known if quantitative results are to be obtained from measurements of the fluorescence intensity. 

We commence by deriving the absorption cross-section of a classical electric dipole oscillator.. The result should be similar to that obtained on the basis of the quantum theory and is of further interest since the frequency dependence of the cross-section is predicted in a simple way. Next we obtain the relations between the spontaneous emission transition probability, and the... 

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