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The scattering length in
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
describes low-energy
scattering In physics, scattering is 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 particles and radiat ...
. For potentials that decay faster than 1/r^3 as r\to \infty, it is defined as the following low-energy limit: : \lim_ k\cot\delta(k) =- \frac\;, where a is the scattering length, k is the wave number, and \delta(k) is the
phase shift In physics and mathematics, the phase (symbol φ or ϕ) of a wave or other periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is expressed in such a s ...
of the outgoing spherical wave. The elastic cross section, \sigma_e, at low energies is determined solely by the scattering length: : \lim_ \sigma_e = 4\pi a^2\;.


General concept

When a slow particle scatters off a short ranged scatterer (e.g. an impurity in a solid or a heavy particle) it cannot resolve the structure of the object since its de Broglie wavelength is very long. The idea is that then it should not be important what precise
potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...
V(r) one scatters off, but only how the potential looks at long length scales. The formal way to solve this problem is to do a partial wave expansion (somewhat analogous to the
multipole expansion A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, \R^3. Multipo ...
in classical electrodynamics), where one expands in the
angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
components of the outgoing wave. At very low energy the incoming particle does not see any structure, therefore to lowest order one has only a spherical outgoing wave, called the s-wave in analogy with the
atomic orbital In quantum mechanics, an atomic orbital () is a Function (mathematics), function describing the location and Matter wave, wave-like behavior of an electron in an atom. This function describes an electron's Charge density, charge distribution a ...
at angular momentum quantum number ''l''=0. At higher energies one also needs to consider p and d-wave (''l''=1,2) scattering and so on. The idea of describing low energy properties in terms of a few parameters and symmetries is very powerful, and is also behind the concept of
renormalization Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that is used to treat infinities arising in calculated quantities by altering values of the ...
. The concept of the scattering length can also be extended to potentials that decay slower than 1/r^3 as r\to \infty. A famous example, relevant for proton-proton scattering, is the Coulomb-modified scattering length.


Example

As an example on how to compute the s-wave (i.e. angular momentum l=0) scattering length for a given potential we look at the infinitely repulsive spherical
potential well A potential well is the region surrounding a local minimum of potential energy. Energy captured in a potential well is unable to convert to another type of energy ( kinetic energy in the case of a gravitational potential well) because it is cap ...
of radius r_0 in 3 dimensions. The radial
Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
(l=0) outside of the well is just the same as for a free particle: :-\frac u''(r)=E u(r), where the hard core potential requires that the
wave function In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
u(r) vanishes at r=r_0, u(r_0)=0. The solution is readily found: :u(r)=A \sin(k r+\delta_s). Here k=\sqrt/\hbar and \delta_s=-k \cdot r_0 is the s-wave
phase shift In physics and mathematics, the phase (symbol φ or ϕ) of a wave or other periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is expressed in such a s ...
(the phase difference between incoming and outgoing wave), which is fixed by the boundary condition u(r_0)=0; A is an arbitrary normalization constant. One can show that in general \delta_s(k)\approx-k \cdot a_s +O(k^2) for small k (i.e. low energy scattering). The parameter a_s of dimension length is defined as the scattering length. For our potential we have therefore a=r_0, in other words the scattering length for a hard sphere is just the radius. (Alternatively one could say that an arbitrary potential with s-wave scattering length a_s has the same low energy scattering properties as a hard sphere of radius a_s.) To relate the scattering length to physical observables that can be measured in a scattering experiment we need to compute the cross section \sigma. In
scattering theory In physics, scattering is 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 particles and radiat ...
one writes the asymptotic wavefunction as (we assume there is a finite ranged scatterer at the origin and there is an incoming plane wave along the z-axis): :\psi(r,\theta)=e^+f(\theta) \frac where f is the scattering amplitude. According to the probability interpretation of quantum mechanics the differential cross section is given by d\sigma/d\Omega=, f(\theta), ^2 (the probability per unit time to scatter into the direction \mathbf). If we consider only s-wave scattering the differential cross section does not depend on the angle \theta, and the total scattering cross section is just \sigma=4 \pi , f, ^2. The s-wave part of the wavefunction \psi(r,\theta) is projected out by using the standard expansion of a plane wave in terms of spherical waves and
Legendre polynomials In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and t ...
P_l(\cos \theta): :e^\approx\frac\sum_^(2l+1)P_l(\cos \theta)\left (-1)^e^ + e^\right By matching the l=0 component of \psi(r,\theta) to the s-wave solution \psi(r)=A \sin(k r+\delta_s)/r (where we normalize A such that the incoming wave e^ has a prefactor of unity) one has: :f=\frac(e^-1)\approx \delta_s/k \approx - a_s This gives: \sigma= \frac \sin^2 \delta_s =4 \pi a_s^2


See also

* Fermi pseudopotential * Neutron scattering length


References

*{{cite book , first=L. D. , last=Landau , first2=E. M. , last2=Lifshitz , year=2003 , title=Quantum Mechanics: Non-relativistic Theory , location=Amsterdam , publisher=Butterworth-Heinemann , isbn=0-7506-3539-8 Quantum mechanics Scattering theory