HOME

TheInfoList



OR:

The scattering length in
quantum mechanics 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 chemistr ...
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 Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
: : \lim_ k\cot\delta(k) =- \frac\;, where a is the scattering length, k is the
wave number 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 ...
, and \delta(k) is the
phase shift In physics and mathematics, the phase of a 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 denoted \phi(t) and expressed in such a scale that it ...
of the outgoing spherical wave. The elastic
cross section Cross section may refer to: * Cross section (geometry) ** Cross-sectional views in architecture & engineering 3D *Cross section (geology) * Cross section (electronics) * Radar cross section, measure of detectability * Cross section (physics) **Abs ...
, \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 Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wave ...
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 in
classical electrodynamics Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model; It is, therefore, a classical fi ...
), where one expands in the
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
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 atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in an ...
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, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering va ...
. 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 linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
(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 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 ...
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 of a 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 denoted \phi(t) and expressed in such a scale that it ...
(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 Cross section may refer to: * Cross section (geometry) ** Cross-sectional views in architecture & engineering 3D *Cross section (geology) * Cross section (electronics) * Radar cross section, measure of detectability * Cross section (physics) **Abs ...
\sigma. In 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 along the z-axis): :\psi(r,\theta)=e^+f(\theta) \frac where f is the
scattering amplitude In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process.differential cross section In physics, the cross section is a measure of the probability that a specific process will take place when some kind of radiant excitation (e.g. a particle beam, sound wave, light, or an X-ray) intersects a localized phenomenon (e.g. a particle o ...
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 In physics, the cross section is a measure of the probability that a specific process will take place when some kind of radiant excitation (e.g. a particle beam, sound wave, light, or an X-ray) intersects a localized phenomenon (e.g. a particle o ...
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 physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applica ...
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 A neutron may pass by a nucleus with a probability determined by the nuclear interaction distance, or be absorbed, or undergo scattering that may be either coherent or incoherent. The interference effects in coherent scattering can be computed via t ...


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