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 the 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.

Approximate scattering amplitude

Starting with a physical model based on the Schrodinger wave equation, calculating the scattering amplitude,This is mathematically equivalent to the Fourier transform: between the input and output directions:

For the Lippmann–Schwinger equation

The Lippmann–Schwinger equation and out-going (+) or in-going () boundary conditions is

is the free particle Green's function is a positive infinitesimal on the right hand side is sometimes called the driving field.

The Born approximation sets Within the Born approximation, the above equation is expressed as

.

The obtained solution is the starting point of a perturbation series known as the Born series.

Scattering amplitude

in coordinate space,

one can extract the Born approximation to the scattering amplitude from the Born approximation to the Lippmann–Schwinger equation above,

is the transferred momentum. The Born scattering amplitude is proportional to the Fourier transform of the potential.

, the scattering amplitude becomes

.

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 is the same as the Fourier transform of the scattering potential . Using this concept, the electronic analogue of Fourier optics has been theoretically studied in monolayer graphene. 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

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 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

Applications include bremsstrahlung and the photoelectric effect. For a charged-particle-induced direct nuclear reaction, the procedure is used twice. In condensed-matter research, DWBA is used to analyze grazing-incidence small-angle scattering.

See also

• Born series

• Lippmann–Schwinger equation

• Dyson series

• Partial-wave analysis

• Rayleigh–Gans approximation

References

Further reading

• Wu and Ohmura, Quantum Theory of Scattering, Prentice Hall, 1962