Inelastic transition probabilities

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... 

Experimental data of Gibson and Sibener appears to confirm qualitatively these predictions at least for monolayers. The phonon linewidths were broadened around T up to half of the Brillouin zone. The hybridization splitting could not be resolved, but an increase of the inelastic transition probability centered around the crossing with the Rayleigh wave and extending up to 3/4 of the zone has been observed and attributed to a resonance between the adatom and substrate modes. 

Quantum IQS Transition Probabilities and Wavefunctions. For the purpose of characterizing resonance energies and widths, the most easily analyzed result of the lOS calculations Is the vlbratlonally Inelastic transition probability between Initial CO vibrational... 

Already in 1932, independently of each other, Landau, Zener and Stueckelberg proposed an expression for the inelastic transition probability... 

Stuckelberg did the most elaborate analysis (15). He applied the approximate complex WKB analysis to the fourth-order differential equation obtained from the original second-order coupled Schrodinger equations. In the complex / -plane he took into account the Stokes phenomenon associated with the asymptotic solutions in an approximate way, and finally derived not only the Landau-Zener transition probability p but also the total inelastic transition probability Pn as... 

The phase . (8) is called the Stokes phase. This Stokes phase correction .s is equal to tt/4(0) in the limit of zero (infinitely strong) diabatic coupling, i.e., in the limit of 8 = 0 (8 = °°). Thus the total inelastic transition probability Pn = S21 2 is given by... 

After the collision the wave function is projected onto the bound states of the diatomic molecule and the inelastic transition probabilities are obtained as... 

Marcus R A 1970 Extension of the WKB method to wave functions and transition probability amplitudes (S-matrix) for inelastic or reactive collisions Chem. Phys. Lett. 7 525-32... 

In [66], we have reported inelastic and reactive transition probabilities. Here, we only present the reactive case. Five different types of probabilities will be shown for each transition (a) Probabilities due to a full tri-state calculation carried out within the diabatic representation (b) Probabilities due to a two-state calculation (for which T] = 0) performed within the diabatic representation (c) Probabilities due to a single-state extended BO equation for the N = 3 case (to, = 2) (d) Probabilities due to a single-state extended BO equation for the N = 2 case (coy =1) (e) Probabilities due to a single-state ordinary BO equation when coy = 0. 

Table 2. Inelastic T R transition probabilities for full (M = 9) and truncated (M = 3) AD potentials as functions of relative translational energy...

Near one metal electrode the potential for a dipole oriented normal to the interface tends to add with that of its image in the metal electrode. The potentials of the dipole and its image tend to cancel for a dipole oriented parallel to the interface. The addition or cancellation of the dipole potential with its image makes it more probable for an inelastic transition to occur for a dipole oriented normal to the interface. Near the center of a tunneling junction, however, the reverse is true for two reasons. 1) The cancellation of potentials for the dipole oriented parallel is least important at the center. 2) The inelastic transition matrix element involves an integration over the barrier volume of the interaction potential times some nearly spatially homogeneous electronic wavefunction terms. If the dipole is oriented normal to the interfaces and located in the center of the barrier, the potential is an odd function in z ( if z defines the normal to the interfaces ), and integrates to... 

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. 

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 ... 

The interplay of the suppression and enhancement can be comparatively easily described for a quasielassical well when the inelastic transition and transmission across the step can be considered independently [14,15]. Under this condition, the suppression-enhancement factor S (k)=S (k) is expressed via the probability of transmission across the potential step T(k), and the phase (k), which is accumulated during the motion in the region of the well. The phase V(k) equals to a certain quasielassical phase integral complemented with a non-quasiclassical correction. The explicit expression for S (k) reads [15] ... 

Cohen and Alexander2 6A have applied the semiclassical theory to collinear collisions of D2 with H2/D2 and calculated transition probabilities for several inelastic processes of the type... 

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