# Scattering length

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{{Short description|Concept in quantum mechanics}}
The '''scattering length''' in [quantum mechanics](/source/quantum_mechanics) describes low-energy [scattering](/source/scattering). For potentials that decay faster than <math>1/r^3</math> as <math>r\to \infty</math>,  it is defined as the following low-energy [limit](/source/limit_(mathematics)):

:<math>
\lim_{k\to 0} k\cot\delta(k) =- \frac{1}{a}\;,
</math>

where <math>a</math> is the scattering length, <math>k</math> is the [wave number](/source/wave_number), and <math>\delta(k)</math> is the [phase shift](/source/phase_shift) of the outgoing spherical wave. The elastic [cross section](/source/Cross_section_(physics)), <math>\sigma_e</math>, at low energies is determined solely by the scattering length:

:<math>
\lim_{k\to 0} \sigma_e = 4\pi a^2\;.
</math>

== 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](/source/de_Broglie_wavelength) is very long. The idea is that then it should not be important what precise [potential](/source/Scalar_potential) <math>V(r)</math> 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](/source/partial_wave_analysis) (somewhat analogous to the [multipole expansion](/source/multipole_expansion) in [classical electrodynamics](/source/Classical_electromagnetism)), where one expands in the [angular momentum](/source/angular_momentum) 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](/source/atomic_orbital) 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](/source/renormalization).

The concept of the scattering length can also be extended to potentials that decay slower than <math>1/r^3</math> as <math>r\to \infty</math>. 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 <math>l=0</math>) scattering length for a given potential we look at the infinitely repulsive spherical [potential well](/source/potential_well) of radius <math>r_0</math> in 3 dimensions. The radial [Schrödinger equation](/source/Schr%C3%B6dinger_equation) (<math>l=0</math>) outside of the well is just the same as for a [free particle](/source/free_particle):

:<math>-\frac{\hbar^2}{2m} u''(r)=E u(r),</math>

where the hard core potential requires that the [wave function](/source/wave_function) <math>u(r)</math> vanishes at <math>r=r_0</math>, <math>u(r_0)=0</math>.
The solution is readily found:

:<math>u(r)=A \sin(k r+\delta_s)</math>.

Here <math>k=\sqrt{2m E}/\hbar</math> and <math>\delta_s=-k \cdot r_0</math> is the s-wave [phase shift](/source/phase_shift) (the phase difference between incoming and outgoing wave), which is fixed by the boundary condition <math>u(r_0)=0</math>; <math>A</math> is an arbitrary normalization constant.

One can show that in general <math>\delta_s(k)\approx-k \cdot a_s +O(k^2)</math> for small <math>k</math> (i.e. low energy scattering). The parameter <math>a_s</math> of dimension length is defined as the '''scattering length'''. For our potential we have therefore <math>a=r_0</math>, in other words the scattering length for a [hard sphere](/source/Hard_spheres) is just the radius. (Alternatively one could say that an arbitrary potential with s-wave scattering length <math>a_s</math> has the same low energy scattering properties as a hard sphere of radius <math>a_s</math>.)
To relate the scattering length to physical observables that can be measured in a scattering experiment we need to compute the [cross section](/source/Cross_section_(physics)) <math>\sigma</math>. In [scattering theory](/source/scattering_theory) one writes the asymptotic wavefunction as (we assume there is a finite ranged scatterer at the origin and there is an incoming [plane wave](/source/plane_wave) along the <math>z</math>-axis):

:<math>\psi(r,\theta)=e^{i k z}+f(\theta) \frac{e^{i k r}}{r}</math>

where <math>f</math> is the [scattering amplitude](/source/scattering_amplitude). According to the probability interpretation of quantum mechanics the [differential cross section](/source/differential_cross_section) is given by <math>d\sigma/d\Omega=|f(\theta)|^2</math> (the probability per unit time to scatter into the direction <math>\mathbf{k}</math>). If we consider only s-wave scattering the differential cross section does not depend on the angle <math>\theta</math>, and the total [scattering cross section](/source/scattering_cross_section) is just <math>\sigma=4 \pi |f|^2</math>. The s-wave part of the wavefunction <math>\psi(r,\theta)</math> is projected out by using the standard expansion of a plane wave in terms of spherical waves and [Legendre polynomials](/source/Legendre_polynomials) <math>P_l(\cos \theta)</math>:

:<math>e^{i k z}\approx\frac{1}{2 i k r}\sum_{l=0}^{\infty}(2l+1)P_l(\cos \theta)\left[ (-1)^{l+1}e^{-i k r} + e^{i k r}\right] </math>

By matching the <math>l=0</math> component of <math>\psi(r,\theta)</math> to the s-wave solution <math>\psi(r)=A \sin(k r+\delta_s)/r</math> (where we normalize <math>A</math> such that the incoming wave <math>e^{i k z}</math> has a prefactor of unity) one has:

:<math>f=\frac{1}{2 i k}(e^{2 i \delta_s}-1)\approx \delta_s/k \approx -  a_s</math>

This gives:

<math>\sigma= \frac{4 \pi}{k^2} \sin^2 \delta_s =4 \pi a_s^2 </math>

==See also==
*[Fermi pseudopotential](/source/Pseudopotential)
*[Neutron scattering length](/source/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 }}

Category:Quantum mechanics
Category:Scattering theory

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Adapted from the Wikipedia article [Scattering length](https://en.wikipedia.org/wiki/Scattering_length) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Scattering_length?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.