Born approximations

For the moment let us restrict our investigation of the Bom approximation to an elastic collision between two particles without internal stracture as in Section IV-B (2). Then, and in the time-independent equation, Eq. (112c), are nonexistent, and that equation may be rewritten in the present notation as  

Here we know that W/ is given by Eq. (135), and our purpose is to determine the detailed asymptotic behavior of y, so that we may check our previous result for the differential cross-section. An appropriate general form to assume for as5mptotic y j, neglecting normalization constants, is 

The first part of the above wave function is an incident plane wave, and the second part is the scattered wave. 

The Born approximation neglects the term i7(r)A,-. This is justified if A is small compared with or if U r) is small compared with k. These conditions are often concurrent and will generally be valid when high-velocity incident particles are involved. 

In Eqs. (148) and (149) K has the same meaning as in Eq. (134), where k. is the vector of the initial plane wave along the z axis, and k,- is the vector in the direction (6, ). Since the number of particles scattered into d 2 per unit time is easily seen to be N f d, f ) rfQ, we may apply the relation 

Consider a beam of electrons incident along the + z direction. The wave function is of the form exp(/ k, z), and the current is given by26 

The cross section oe](0,fj ,E,Eo) is measured using the detectors, and since we expect it to be proportional to both the solid angle subtended by the detector and to the incident current, we have 

Thus to calculate the cross section, we need only calculate the asymptotic part of the scattered electron wave function. This is straightforward, at least for first-order perturbation theory. Provided that the time scale during which the perturbation of the molecule by the impact electron occurs is small compared to the time scale for electronic motion, we find26 

In addition to the assumptions underlying the use of first-order perturbation theory, a number of other assumptions underlie equation (11.25). 

In effect, the equation has been derived under the assumption of infinite nuclear mass. This is accurate enough for electron scattering, but for proton and atom scattering a coordinate transformation is needed, the details of which are given in Mott and Massey.26 

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In Fig. 3a,b are shown respectively the modulus of the measured magnetic induction and the computed one. In Fig. 3c,d we compare the modulus and the Lissajous curves on a line j/ = 0. The results show a good agreement between simulated data and experimental data for the modulus. We can see a difference between the two curves in Fig. 3d this one can issue from the Born approximation. These results would be improved if we take into account the angle of inclination of the sensor. This work, which is one of our future developpements, makes necessary to calculate the radial component of the magnetic field due to the presence of flaw. This implies the calculation of a new Green s function. 

A number of improvements to the Bom approximation are possible, including higher order Born approximations (obtained by inserting lower order approximations to i jJ into equation (A3.11.40). then the result into (A3.11.41) and (A3.11.42)), and the distorted wave Bom approximation (obtained by replacing the free particle approximation for the solution to a Sclirodinger equation that includes part of the interaction potential). For chemical physics... 

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]... 

Totrov [31] developed a model to estimate electrostatic solvation transfer energy AGd" in Eq. (1) based on the Generalized Born approximation, which considers the electrostatic contribution to the free energy of solvation as ... 

The structure of the ions, where the bulky phenyl groups surround the central ion in a tetrahedron, lends validity to the assumption that the interaction of the shell of the ions with the environment is van der Waals in nature and identical for both ions, while the interaction of the ionic charge with the environment can be described by the Born approximation (see Section 1.2), leading to identical solvation energies for the anion and cation. 

Ruland and Smarsly [9,84,240] can analyze their recorded data in the classical Born approximation, but have to correct for the special geometry of the grazing incidence experiment. They propose not to carry out the necessary corrections in a... 

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

In general Rytov approximation provides better reconstruction than Born approximation [8]. Here, MBLL is approximated locally at each sampling volume and the effective path length is estimated using the Rytov approximation as... 

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... 

For 1, de Broglie s wavelength is small enough compared to the classical collision radius b so that a wave packet can be constructed which, approximately, follows the classical Coulomb trajectory [3]. The opposite limit, where the Sommerfeld parameter Zie hv<, denotes the case of weak Coulomb interaction where the Born approximation may be expected to be valid. 

In Bethe theory the shell correction ALsheii is conveniently defined as the difference between the stopping number LBom in the Born approximation and the Bethe logarithm LBethe —in (2mv /I). Fano [12] wrote the leading correction in the form... 

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 ... 

The magnitude of the effect has been roughly estimated from the Born approximation of the energy of an ion in solution and the ratio of the loii-pati oi Ion exchange cquitibilnni constant in a given eluent to that in a roference eluent, has been calculated 34, 129). The effect of in-... 

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 apparent similarity of the Bohr and the Bethe stopping power formulae, the conditions of their validity are rather complimentary than the same. Bloch [23] pointed out that Born approximation requires the incident particle velocity v ze jh, the speed of a Is electron around the incident electron while the requirement of Bohr s classical theory is exactly the opposite. For heavy, slow particles, for example, fission fragments penetrating light media, Bohr s formula has an inherent advantage, although the typical transition energy has to be taken as an adjustable parameter. 

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