Bom approximation

In back-scattering, (n= - n ), and within the Bom approximation (mono-scattering), the asymptotic solution of (2) is ... 

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... 

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... 

The Bom approximation for the differential cross section provides the basis for the interpretation of many experimental observations. The discussion is often couched in temis of the generalized oscillator strength. 

Assuming the validity of the Bom approximation, an effective generalized oscillator strength can be derived in temis of experimentally accessible quantities ... 

This equation describes the Fourier transfonn of the scattering potential V r). It should be noted that, in the Bom approximation the scattering amplitude/(0) is a real quantity and the additional phase shift q(9) is zero. For atoms with high atomic number this is no longer tme. For a rigorous discussion on the effects of the different approximations see [2] or [5]. 

B-e collisions, then the Bom approximation for atom-atom collisions is also recovered for general scattering amplitudes. For slow atoms B, is dominated by s-wave elastic scattermg so thaty g = -a and cr g = 4ti... 

In the first Bom approximation [12], the scattering cross section for a beam of neutrons incident on a magnetic material, assuming form (3) for the interaction, is given by the square of the scattering amplitude, F(kf, ki), where... 

In this so called half range Bom approximation the upper limit is given by,... 

Equation (15) permits a straightforward analysis of dielectric continuum models of hydration that have become popular in recent decades. The dielectric model, also called the Bom approximation, for the hydration free energy of a spherical ion of radius R with a charge q at its center is... 

Nymeyer, H. Garcfa, A. E., Simulation of the folding equilibrium of o-helical peptides a comparison of the generalized Bom approximation with explicit solvent, Proc. Natl Acad. Sci. USA. Nov 2003,100,13934-13939. 

E-B. A model solution to the electrostatic problem, e.g., the Generalized Bom Approximation or a conductor-like screening solution. 

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... 

Giant dipole resonance. Isovector giant resonances contain information about the SE through the restoring force. In particular the excitation of the isovector giant dipole resonance (GDR) with isoscalar probes has been used to extract A R/R [32], In the distorted wave Bom approximation optical model analysis of the cross section the neutron and proton transition densities are needed as an input. For example, in the Goldhaber-Teller picture these are... 

Figure 8.3 The vibrational elastic and inelastic differential cross sections for electron scattering off LiF at = 5.44 eV (Alhassid and Shao, 1992b, where the source of the data is given). Solid lines with an improved dipole interaction [which breaks the 0(4) symmetry]. Long dashed lines the calculations by Bijker and Amado (1986). The short dashed lines are the Bom approximation.

The structure factor S(q as defined in Eq. (54) in terms of the Ising pseudospins Si, in the framework of the first Bom approximation describes elastic scattering of X-rays, neutrons, or electrons, from the adsorbed layer. SCq) is particularly interesting, since in the thermodynamic limit it allows to estimate both the order parameter amplitude tj/, the order parameter susceptibility X4, and correlati length since for q near the superstructure Bragg reflection q we have (k = q— q%)... 

It was a fairly common belief that such a theory should be a modification of the Bethe theory. Although the fact that the screening regime (uclassical theory must be more powerful than the Bom approximation in this regime was evidently not taken at face value. 

In order to calculate the GOS, one requires the excitation energies and the generalized transition moments. Oddershede and Sabin had already started in 1992 the investigation of the GOS and the stopping cross section in the first Bom approximation by means of the polarization propagator method [78]. 

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. 

This formula for the screened Coulomb field has been used for many applications it gives, for instance, a fairly satisfactory description of the residual resistance due to a small concentration of Zn, Ga, etc. in monovalent metals (Mott and Jones 1936, p. 293). Friedel (1956) points out, however, that the use of the Bom approximation gives too large a value of the scattering, and better values are obtained if one calculates the phase shifts exactly. 

Distorted Wave Bom Approximation. Quantum scattering calculations are sometimes made using the distorted wave Bom approximation (15). Such calculations have the advantage of almost always being feasible numerically. For simple cases, one can also obtain some results analytically (16). However, the accuracy of the results is generally poor, for most molecular collisions. A... 

---