Scattering transition probability

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... 

J. Z. H. Zhang, Y. Zhang, D. J. Kouri, B. C. Garrett, K. Haug, D. W. Schwenke, and D. G. Truhlar, Calculations of accurate quantal-dynamical reactive scattering transition probabilities and their use to test semiclassical applications, Faraday Discuss. Chem. Soc. 84 3711 (1987). 

In this section the electron-scattering transition probability amplitude through an open QD, t( ), has been studied for a real-space 2D model Hamiltonian. A sharp change of the phase of t E) by tt occurs when t E) intersects the origin. It implies that two conditions should be satisfied in order to observe a sharp drop of the phase by tt in the tail of the resonant peak. One condition is t Eo) = 0, whereas the second condition is dt E)/dE EQ 7 0. We have shown that this phase drop is a resonance interference phenomenon that happens even within the framework of an one electron effective QD potential. The fact that the QD has at least 2D is a crucial point in the mechanism we have presented here. Our explanation of a sharp phase change is based on the destructive interference between neighboring resonances and thus differs from the mechanism based on the Fano resonance (see, for example. Refs [22,25]). 

This is the one dimensional version of what is usually called the Bom approximation in scattering theory. The transition probability obtained from equation A3.11.43() is... 

There are two basic physical phenomena which govern atomic collisions in the keV range. First, repulsive interatomic interactions, described by the laws of classical mechanics, control the scattering and recoiling trajectories. Second, electronic transition probabilities, described by the laws of quantum mechanics, control the ion-surface charge exchange process. 

Another related issue is the computation of the intensities of the peaks in the spectrum. Peak intensities depend on the probability that a particular wavelength photon will be absorbed or Raman-scattered. These probabilities can be computed from the wave function by computing the transition dipole moments. This gives relative peak intensities since the calculation does not include the density of the substance. Some types of transitions turn out to have a zero probability due to the molecules symmetry or the spin of the electrons. This is where spectroscopic selection rules come from. Ah initio methods are the preferred way of computing intensities. Although intensities can be computed using semiempirical methods, they tend to give rather poor accuracy results for many chemical systems. 

All ab initio applications of multiple scattering theory in dilute substitutional alloys rely on the one-to-one correspondence configuration. This holds both for the calculation of transition probabilities [7], represented by Eq. (7), and the electronic structure [8], represented by the Green s function equation [9]... 

Under some circumstances the rotationally anisotropy may be even further simplified for T-R energy transfer of polar molecules like HF (41). To explore this quantitatively we performed additional rigid-rotator calculations in which we retained only the spherically symmetric and dipole-dipole terms of the AD potential, which yields M = 3 (see Figures 1, 3, and 4). These calculations converge more rapidly with increasing N and usually yield even less rotationally inelastic scattering. For example Table 2 compares the converged inelastic transition probabilities... 

Fig. 9.24 Theoretical calculations of nuclear forward scattering for the relaxation rates as indicated for a system with electron spin S = 1/2, hyperfine parameters A y jg fi = 50 T, and AF.q = 2 mm s in an external field of 75 mT applied perpendicular to k and O . The transition probabilities co in ((9.8a) and (9.8b)) are expressed in units of mm s , with 1 mm corresponding to 7.3 10 s. (Taken Ifom [30])...

Transition probability, non-adiabatic coupling, Longuet-Higgins phase-based treatment, two-dimensional two-surface system, scattering calculation, 152-155 Triangular phase diagram, geometric phase... 

The energy and state resolved transition probabilities are the ratio of two quantities obtained by projecting the initial wave function on incoming plane waves (/) and the scattered wave function on outgoing plane waves (F)... 

Figure 41. Transition probabilities for vibrational excitation as function of center-of-mass scattering angle for collisions of 10-eV H+.307...

The most serious problem associated with the use of neutron scattering for nuclear spectroscopy comes from the fact that the resolution for neutron detection is typically rather poor, and the sensitivity to small transition probabilities is also poor when neutron detection is being employed. These difficulties can be alleviated by observing the y rays which de-excite the excited levels rather than the inelastically scattered neutrons. 

The transition probabilities per unit time and per scattering center in Eqs (B.ll) and (B.12) are for a collision process, often written in a more explicit form to emphasize the particular transition considered ... 

In the chemically interesting case that the excited state is a continuum leading asymptotically to a product channel S at total energy E, the one-photon transition probability, integrated over all product scattering angles, k, is given by... 

To second order in Hey, the transition probability of the scattering event y(k )->y(k2)—the phonon system passing from state R [energy ER and population p(R)] to state R, while conserving the energy of the total system is... 

Utilization of both ion and neutral beams for such studies has been reported. Toennies [150] has performed measurements on the inelastic collision cross section for transitions between specified rotational states using a molecular beam apparatus. T1F molecules in the state (J, M) were separated out of a beam traversing an electrostatic four-pole field by virtue of the second-order Stark effect, and were directed into a noble-gas-filled scattering chamber. Molecules which were scattered by less than were then collected in a second four-pole field, and were analyzed for their final rotational state. The beam originated in an effusive oven source and was chopped to obtain a velocity resolution Avjv of about 7 %. The velocity change due to the inelastic encounters was about 0.3 %. Transition probabilities were calculated using time-dependent perturbation theory and the straight-line trajectory approximation. The interaction potential was taken to be purely attractive ... 

Experimental studies on the K0/Ka x-ray intensity ratio for 3d elements have shown [18-23] that this ratio changes under influence of the chemical environment of the 3d atom. Brunner et al. [22] explained their experimental results due to the change in screening of 3p electrons by 3d valence electrons as well as the polarization effect. Band et al. [24] used the scattered-wave (SW) Xa MO method [25] and calculated the chemical effect on the K0/Ka ratios for 3d elements. They performed the SW-Xa MO calculations for different chemical compounds of Cr and Mn. The spherically averaged self-consistent-field (SCF) potential and the total charge of valence electrons in the central atom, obtained by the MO calculations, were used to solve the Dirac equation for the central atom and the x-ray transition probabilities were calculated. 

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