Scattering and Collisions| Scattering and Collisions |
Scattering in General > s.a. diffraction [in field
theory]; huygens' principle; resonance.
* Geometric collisions:
For particles of diameter d, mean speed v and
number density n, the mean free path is l
= (√2π
d2n)−1,
and the collision frequency f = v/l
= √2π
d2vn;
The mean free time is of course t = 1/f ;
> s.a. brownian motion.
* Optical theorem: (a.k.a. optical
cross-section theorem) The total cross section for an elastic scattering and
absorption process (the potential can possibly be complex) is
σtot = (4π/k) Im f(k, k) ;
The result is often credited to N Bohr, R Peierls and G Placzek,
but it is actually due to E Feenberg and Lord Rayleigh.
@ Geometric collisions: Jakoby EJP(09)
[relaxation time, mean free path, and electronic conductivity];
Palk AJP(14)jun
[mean free path as statistical mean of the distribution of free path lengths].
@ Optical theorem: in Schiff 68;
Newton AJP(76)jul;
Bussey PLA(86) [and wave-function collapse];
Mansuripur AJP(12)apr [new perspective].
@ Techniques: Barlette et al AJP(01)sep [integral equations, partial waves];
Liu et al JHEP(14)-a1403 [without large-distance asymptotics].
Special Cases and Applications
* Important experiments:
Thomson scattering; Rutherford scattering; Deep inelastic scattering,
that showed the composite nature of baryons.
* Deep inelastic scattering:
High-energy scattering of electrons off nucleons; Provides direct evidence for
the existence of quarks inside the proton; Parameters are x = fraction of
nuclear momentum carried by q, Q2
= square of momentum transfer between nucleon and beam particle;
> s.a. critical phenomena.
@ Coulomb potential:
Yafaev JPA(97) [n-dimensional, quantum];
Ahmed qp/03 [quantum];
Mineev TMP(04) [1D, self-adjoint extension];
Glöckle et al PRC(09) [screening and renormalization factor];
Neilson & Senatore ed JPA(09)#21;
Abramovici & Avishai JPA(09) [1D];
Collas a2102 [Born approximation, pedagogical].
@ Gravitational, relativistic:
Barrabès & Hogan CQG(04);
Barbieri & Guadagnini NPB(05)gq [massless particles off rotating bodies];
Nikishov a0807 [classical and quantum];
Betti a1411-th [transplackian scattering];
> s.a. motion of gravitating bodies.
@ Bounded / point obstacle: Athanasiadis et al JMP(02) [acoustic and electromagnetic waves].
@ Off defects: Katanaev & Volovich AP(99);
Spinelly et al CQG(01) [conical, cosmic string].
> Specific processes: see Bhabha Scattering;
Drell-Yan Process; neutron;
Preons; Rutherford, Superradiant,
Thomas, Thomson Scattering.
> Specific theories:
see atomic physics; dirac fields;
graviton; molecular physics;
photon phenomenology; pilot-wave
quantum mechanics; wave phenomena.
Multiple Scattering > s.a. Debye Length;
light; Rayleigh Scattering.
@ General references:
Huang PLA(04) [perturbative];
Ramm mp/06,
PLA(07)mp/06,
JPA(08) [off many small bodies],
PLA(07),
PLA(08) [waves off many particles];
Ramm & Rona a0910;
Ramm RPMP(13) [transmission boundary conditions].
@ Random scatterers: Mathur & Yeh JMP(64) [finite size, electromagnetic waves];
Külske mp/01,
Dean et al JPA(04) [point scatterers];
Field 09;
Basile et al a1307 [diffusion limit];
Ramm JMP(14)-a1402 [electromagnetic wave scattering by small perfectly conducting particles].
In Quantum Theory
> s.a. quantum mechanical tunneling [delay time]; S-Matrix.
* Levinson's theorem: A fundamental
theorem in quantum scattering theory, which shows the relation between the number of
bound states and the phase shift at zero momentum for the Schrödinger equation.
* Scattering amplitudes in quantum field
theory: They can be expressed using a path integral over all possible classical
field configurations, or starting from first principles and using recursion relations.
@ General references:
Amrein qp/01 [large-time behavior];
Ignatovich qp/04 [problems?];
Cannata et al AP(07) [PT-symmetric quantum mechanics, 1D];
Hussein et al JPA(08)-a0807 [new formulation];
Norsen a0910;
Carron & Rosenfelder NJP(10)-a0912 [path-integral description];
Karlovets a1710-conf [beyond the plane-wave approximation];
Sakhnovich a1905
[scattering operator, scattering amplitude and ergodic property];
> s.a. quantum systems.
@ Levinson's theorem: Wellner AJP(64)oct;
Lin PRA(97)qp/98,
PRA(98)qp/98 [2D];
Rosu NCB(99)gq/97 [in quantum cosmology];
Dong & Ma IJTP(00) [1D Schrödinger equation];
Sheka et al PRA(03)qp/02;
Boya & Casahorrán IJTP(07)qp/05-conf [from spectral density];
Kellendonk & Richard qp/05,
JPA(08)-a0712 [topological version];
Ma JPA(06) [rev];
Jia et al a1007
[for potentials with critical decay 1/r 2];
Kellendonk & Richard a1009;
Childs & Strouse JMP(11)-a1103,
Childs & Gosset JMP(12) [for scattering on a graph];
Nicoleau et al JMP(17)-a1611 [extended version for systems with complex eigenvalues];
> s.a. topology in physics.
@ Semiclassical:
Ford & Wheeler AP(00);
Rothstein JMP(04) [1D and 2D];
Berera NPA(07)ht [scattering of large objects in quantum field theory and classical description];
Adhikari & Hussain AJP(08)dec [2D].
@ In quantum field theory:
Buchholz & Summers mp/05-en;
Biswas a0807 [alternative approach];
Toth CEJP(12)-a0904 [definition of scattering states];
Rubtsov et al PRD(12)-a1204 [Lorentz-violating theories];
Arkani-Hamed et al 16 [Grassmannian geometry];
Taylor PRP(17)-a1703 [gauge theories];
> s.a. bogoliubov transformations.
@ Recursion relations for scattering amplitudes:
Cheung et al PRL(16)
+ Kosower Phy(16) [effective field theories].
@ Electron scattering:
Tyutin pr(74)-a0801 [by solenoid];
Dybalski NPB(17)-a1706 [non-perturbative description of colliding electrons].
@ Yang-Mills theory scattering amplitudes:
Britto et al PRL(05);
Bjerrum-Bohr NPB(16)-a1605 [analytic expressions].
@ Other types of situations: Mostafazadeh PRA(96) [on curved surfaces];
de la Torre AJP(97)feb [wave packet in a central potential, distorsion];
Esposito JPA(98)ht [singular potentials];
Pérez Prieto et al JPA(03) [Gaussian wave function and square barrier];
Roux & Yafaev JMP(03),
Duch a1906 [long-range Vs];
Oeckl a2105 [evanescent massive Klein-Gordon particles].
References > s.a. Inverse Scattering;
Perturbation Methods; special potentials.
@ General: Reed & Simon 79;
Ramm in(80)mp/00 [scalar + vector waves, arbitrary shapes].
@ Relativistic: Aichelburg et al CQG(04)gq/03 [ultrarelativistic charges].
@ In curved spaces: Beig APA(82) [scalar fields, Schwarzschild spacetime];
Ito & Skibsted a1109
[on non-compact, connected, complete Riemannian manifolds].
@ In non-commutative theories: Alavi MPLA(05)ht/04,
Bellucci & Yeranyan PLB(05)ht/04 [quantum];
Kumar & Rajaraman PRD(06)ht/05.
@ Bohm-Gadella theory controversy: de la Madrid JPA(06)qp;
Gadella & Wickramasekara JPA(07);
de la Madrid JPA(07)-a0704;
Baumgärtel a0704;
de la Madrid a0705.
@ Related topics:
Fabbrichesi et al NPB(94) [Planck energies];
Visser & Wolf PLA(97) [with field discontinuities];
Laura IJTP(97)qp/99;
de Vries et al RMP(98) [waves, point scatterer];
Horan et al JMP(00) [weak convergence];
Albeverio & Gottschalk CMP(01)mp/05,
mp/05 [with indefinite metric];
Stetsko a0912 [in spaces with minimal length];
Sassoli de Bianchi CEJP(12)-a1010 [time delay, introduction];
> s.a. Lennard-Jones Potential [scattering length].
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