First Born approximation

The calculation describes the differential cross sections and 2 P electron impact coherence parameters quite well. For the 2 P differential cross section it is contrasted with a variant of the distorted-wave Born approximation, first-order many-body theory, where the distorted waves are both calculated in the initial-state Hartree—Fock potential. 

Our particular interest lies in the calculation of the linear energy deposition, or stopping power, of swift ions in materials, 5o(v). In the first Born approximation, and for a fully stripped projectile, this quantity can be written [2-4]... 

Under the first order Born approximation, the solution to Equation (6) is... 

A possible computational strategy is to calculate Xs(r O first using the standard sum-over states formula (Equation 24.80). Equation 24.75 can be used next to generate successive Born approximations of the functions f (r). For instance, the first Bom approximation would be... 

From the middle of the fifties new experimental equipments have been built with high precision (e. g. and the use of fast electronic computers made their strong impact as well. With increasing accuracy of stmcture determination, the failures of some of the theoretical approximations have become apparent. Thus, e. g., attention turned to the failure of the first Born approximation, especially for molecules containing atoms with very different atomic numbers ... 

Quantum-Mechanical Treatment The First Born Approximation... 

In the first Bom approximation, the interaction between the photons and the scattering system is weak and no excited states are involved in the elastic scattering process. Furthermore, there is no rescattering of the scattered wave, that is, the single-scattering approximation is valid. In the Feynman diagrams (Fig. 1.2), there is only one point of interaction for first-Born-approximation processes. 

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... 

The first (and still the foremost) quantum theory of stopping, attributed to Bethe [19,20], considers the observables energy and momentum transfers as fundamental in the interaction of fast charged particles with atomic electrons. Taking the simplest case of a heavy, fast, yet nonrelativistic incident projectile, the excitation cross-section is developed in the first Born approximation that is, the incident particle is represented as a plane wave and the scattered particle as a slightly perturbed wave. Representing the Coulombic interaction as a Fourier integral over momentum transfer, Bethe derives the differential Born cross-section for excitation to the nth quantum state of the atom as follows. 

Despite the fact that Bohr s stopping power theory is useful for heavy charged particles such as fission fragments, Rutherford s collision cross section on which it is based is not accurate unless both the incident particle velocity and that of the ejected electron are much greater than that of the atomic electrons. The quantum mechanical theory of Bethe, with energy and momentum transfers as kinematic variables, is based on the first Born approximation and certain other approximations [1,2]. This theory also requires high incident velocity. At relatively moderate velocities certain modifications, shell corrections, can be made to extend the validity of the approximation. Other corrections for relativistic effects and polarization screening (density effects) are easily made. Nevertheless, the Bethe-Born approximation... 

As before, the three-particle contribution is split into two contributions, [/Jj and [/J2. Let us write [/Jj in the first Born approximation we get... 

If the interaction operator V can be regarded as a perturbation to the Hamiltonian H0 (this is the case for fast particles the velocity of which is much greater than those of atomic electrons), the function if>+ can be found using the perturbation theory. Such an approach was named the Born approximation. In the first Born approximation we replace the function ip+ by that of the initial state of the scattering system,48 that is, put i//+ = 0, and thereby do not have to solve Eq. (4.2). In this way, for the differential cross section of direct scattering, we get... 

A consecutive application of the Born approximation to the problem of calculating the scattering cross sections was first done by Bethe (a detailed discussion of his theory is presented in Refs. 104, 106, 113). Integrating over the coordinates of the incident particle we can obtain simple analytical formulas for the cross sections. 

If the velocity of the incident electron is comparable with those of molecular electrons, the former can be exchanged for one of the latter. Such processes are described by the exchange scattering amplitude, the form of which in the first Born approximation has been found by Oppenheimer.126 In the Born-Oppenheimer approximation the exchange amplitude (4.11) acquires the form... 

By tradition, the electrons with energy below 100 eV are called slow electrons. At such energies the analytical formulas we have obtained in Section IV.B in the first Born approximation are no longer valid, and we have to use some semiempirical expressions. Nevertheless, the principal channel of energy loss is still the ionization and excitation of molecules of the medium. 

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 preceding derivation has been performed for a free microparticle. However, the electron moves in a medium, and it is owing to its interaction with molecules that its energy acquires an uncertainty A . The conditions under which we can consider the electron as a free particle coincide with the conditions of applicability of the first Born approximation (within which we have derived all the main formulas for cross sections of interaction in Section IV), namely, it is necessary that the interaction energy be small in comparison with the kinetic energy of the particle, which is just the case for fast particles. 

Although the above argument is not rigorous, particularly in its neglect of electron exchange, it nevertheless provides a plausible explanation for the merging of the two total cross sections at much lower projectile energies than those for which the first Born approximation alone is valid. 

The total positronium formation cross section in the Ore gap, constructed from the addition of accurate variational results for the first three partial waves and the values given by the Born approximation for all partial waves with l > 2, is plotted in Figure 4.4. On the scale of the ordinate, the s-wave contribution is too small to be visible. A very small s-wave contribution is found to be a feature of the positronium formation cross section for several other atoms. 

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... 

Most other calculations of positronium formation in positron-helium scattering have employed much simpler methods of approximation, but results have usually been obtained over energy ranges extending well beyond the Ore gap. It must therefore be borne in mind that the experimental results include contributions from positronium formation into excited states as well as into the ground state. The Born approximation, used first by Massey and Moussa (1961) and subsequently by Mandal, Ghosh... 

Positronium formation into nPs = 2 excited states in positron H2 scattering was investigated in the first Born approximation by Ray, Ray and Saha (1980) and also by Biswas et al. (1991b). 

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