Second Born approximation

The second term in Eq. (1.11) is the origin of the scattering in the first Born approximation. It leads to an amplitude for the scattering of photons with propagation vector k0 into photons with vector k equal to... 

Quantum-Mechanical Treatment The Second Born Approximation... 

Secondly one neglects in the exponent the term with aj f,. That means that powers of a beyond the second are omitted, just as in our (3.10). (This resembles the Born approximation in scattering theory.) One easily sees that the omitted terms are of relative order octc and of absolute order (x3r3. Thus... 

The Born approximation was used230 for calculating the cross sections of rotational excitation for multiatomic molecules of symmetrical and antisymmetrical top types. In the second study the authors have obtained very large cross sections of excitation of the first rotational levels at the energy of the electron of 0.01 eV, the cross section for H20 molecules was about 3 x 10 13 cm2, and about 10 12 cm2 for the H2CO molecule. The cross sections for diatomic molecules are much smaller for electron energies from the threshold to 0.01-0.1 eV, the cross sections for molecules H2, N2, 02, and CO are about 10 17-10-16 cm2. 230,231... 

The first Born approximation is known to provide a rather inaccurate description of positronium formation, even at high energies, because the process then becomes essentially two-stage this can be understood as follows. In order to form positronium, the positron and an electron must emerge from the target with very similar velocities, and the simplest way in which this can be achieved is via one or other of the processes represented in Figure 4.6. In both cases, first the positron scatters from the electron and then either the electron or the positron is scattered into the required final direction by the nucleus. It is therefore to be expected that the second Born approximation, with its quadratic... 

Dewangan, D.P. and Walters, H.R.J. (1977). The elastic scattering of electrons and positrons by helium and neon the distorted-wave second Born approximation. J. Phys. B At. Mol. Phys. 10 637-661. 

In applying the distorted-wave second Born approximation we have the same difficulty as in calculating the optical potential. We must calculate the spectrum of the Green s function of (6.87). The first iteration of (6.87) is written as... 

The direct amplitudes involving are analogous to the distorted-wave Born approximation and are calculated by (10.31). The T-matrix element in the second amplitude of (10.51), which has the observed resonances, is calculated by solving the problem of electron scattering on He" ". The solution consists of half-on-shell T-matrix elements at the quadrature points for the scattering integral equations (6.87). The same points are used for the k integration of (10.51). 

We have noticed that the background of the modified Born approximation and the new first order QL approximation is the same. The main difference is that in the case of the Born approximation the starting point (zero order approximation) for the iteration process is the zero anomalous field, while in the QL approach we start with the anomalous field proportional to the background field. In principle we can extend our approach to computing all iterations bj (9.149). In this case we will obtain a complete analog of the Born series. For example, the second order QL approximation is equal to... 

Distorted wave Born approximation resonance energies and widths were calculated numerically using Equation 17. for the reduced-dimensionality hamiltonian given by Equation 35. and employing Equation 36. for V (r,t) up to second order. The coefficients a(t) and b(t) were determined numerically from the ab imitio potential surface. Zero-order wavefimctions X,(t) Xj(t) were deter-... 

Substituting the first Born wavefunction into the right hand side of Eq. (4.4) gives the second Born approximation ... 

We have derived time-reversible, symplectic, and second-order multiple-time-stepping methods for the finite-dimensional QCMD model. Theoretical results for general symplectic methods imply that the methods conserve energy over exponentially long periods of time up to small fluctuations. Furthermore, in the limit m —> 0, the adiabatic invariants corresponding to the underlying Born-Oppenheimer approximation will be preserved as well. Finally, the phase shift observed for symmetric methods with a single update of the classical momenta p per macro-time-step At should be avoided by... 

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