quantum physics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...

, the scattering amplitude is the

probability amplitude In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The modulus squared of this quantity represents a probability density. Probability amplitudes provide a relationship between the quan ...

of the outgoing

spherical wave The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seis ...

relative to the incoming

plane wave In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, th ...

in a stationary-state

scattering process Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including ...

.''Quantum Mechanics: Concepts and Applications''
By Nouredine Zettili, 2nd edition, page 623. Paperback 688 pages January 2009 The plane wave is described by the

wavefunction A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...

\psi(\mathbf) = e^ + f(\theta)\frac \;,

where

\mathbf\equiv(x,y,z)

is the position vector;

r\equiv, \mathbf,

;

e^

is the incoming plane wave with the

wavenumber In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...

along the axis;

e^/r

is the outgoing spherical wave; is the scattering angle; and

f(\theta)

is the scattering amplitude. The

dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...

of the scattering amplitude is

length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...

. The scattering amplitude is a

; the differential cross-section as a function of scattering angle is given as its

modulus squared In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power 2, and is denoted by a superscript 2; for instance, the square ...

, :

\frac = , f(\theta), ^2 \;.

X-rays

The scattering length for X-rays is the

Thomson scattering length The classical electron radius is a combination of fundamental physical quantities that define a length scale for problems involving an electron interacting with electromagnetic radiation. It links the classical electrostatic self-interaction energ ...

classical electron radius The classical electron radius is a combination of fundamental physical quantities that define a length scale for problems involving an electron interacting with electromagnetic radiation. It links the classical electrostatic self-interaction energ ...

, ₀.

Neutrons

The nuclear

neutron scattering Neutron scattering, the irregular dispersal of free neutrons by matter, can refer to either the naturally occurring physical process itself or to the man-made experimental techniques that use the natural process for investigating materials. Th ...

process involves the coherent neutron scattering length, often described by .

Quantum mechanical formalism

A quantum mechanical approach is given by the

S matrix In physics, the ''S''-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT). More formal ...

formalism.

Measurement

The scattering amplitude can be determined by the

scattering length The scattering length in quantum mechanics describes low-energy scattering. For potentials that decay faster than 1/r^3 as r\to \infty, it is defined as the following low-energy limit (mathematics), limit: : \lim_ k\cot\delta(k) =- \frac\;, wher ...

in the low-energy regime.

References

Neutron X-rays Electron Scattering Diffraction {{quantum-stub

X-rays

Neutrons

Quantum mechanical formalism

Measurement

See also

References