Born approximation
Generally in scattering theory and in particular in quantum mechanics, the Born approximation consists of taking the incident field in place of the total field as the driving field at each point in the scatterer. The Born approximation is named after Max Born who proposed this approximation in early days of quantum theory development. It is the perturbation method applied to scattering by an extended body. It is accurate if the scattered field is small compared to the incident field on the scatterer. For example, the scattering of radio waves by a light styrofoam column can be approximated  by assuming that each part of the plastic is polarized by the same electric field that would be present at that point without the column, and then calculating the scattering as a radiation integral over that polarization distribution. Born approximation to the Lippmann–Schwinger equation. The Lippmann–Schwinger equation for the scattering state formula_1 with a momentum p and out-going (+) or in-going (−) boundary conditions is where formula_3 is the free particle Green's function, formula_4 is a positive infinitesimal quantity, and formula_5 the interaction potential. formula_6 is the corresponding free scattering solution sometimes called the incident  field. The factor formula_1 on the right hand side is sometimes called the "driving field". Within the Born approximation, the above equation is expressed as which is much easier to solve since the right hand side no longer depends on the unknown state formula_1. The obtained solution is the starting point of the Born series. Born approximation to the scattering amplitude. Using the outgoing free Green's function for a particle with mass formula_10 in coordinate space, one can extract the Born approximation to the scattering amplitude from the Born approximation to the Lippmann–Schwinger equation above, where formula_13 is  the transferred momentum. Applications. The Born approximation is used in several different physical contexts. In neutron scattering, the first-order Born approximation is almost always adequate, except for neutron optical phenomena like internal total reflection in a neutron guide, or grazing-incidence small-angle scattering. The Born approximation has also been used to calculate conductivity in bilayer graphene and to approximate the propagation of long-wavelength waves in elastic media. The same ideas have also been applied to studying the movements of seismic waves through the Earth. Distorted-wave Born approximation. The Born approximation is simplest when the incident waves  formula_6 are plane waves. That is, the scatterer is treated as a perturbation to free space or to a homogeneous medium. In the distorted-wave Born approximation (DWBA), the incident waves are solutions formula_15 to a part formula_16 of the problem formula_17 that is treated by some other method, either analytical or numerical. The interaction of interest formula_5 is treated as a perturbation formula_19 to some system formula_16 that can be solved by some other method. For nuclear reactions, numerical optical model waves are used. For scattering of charged particles by charged particles, analytic solutions for coulomb scattering are used . This gives the non-Born preliminary equation and the Born approximation Other applications include bremsstrahlung and the photoelectric effect. For a charged-particle-induced direct nuclear reaction, the procedure is used twice. There are similar methods that do not use the Born approximations. In condensed-matter research, DWBA is used to analyze grazing-incidence small-angle scattering.