Scattering of particles - topic 1 - june 2007 Particle Scattering: –Differential cross section –Trajectories and currents –Mean free path Quantal Scattering: –Currents –Differential cross section –Integral Equation Classical Scattering: –Trajectories; impact parameter –Differential cross section –Total cross section –Example: Hard sphere scattering

Cross section - mean free path - macroscopic cross section

Number of scattered particles into : Differential Cross Section: Total Cross Section: The Scattering Cross Section N out N

Number of particles Number of particles : seen as ”particles” in a current, or probability density current Number of particles : seen as ”particles” in a number of possible trajectories (impact parameters In both cases: DIFFERENTIAL CROSS SECTION In both cases: probabilistic formulations

Example - Classical scattering:     Hard Sphere scattering: Independent of angles! = Geometrical Cross sectional area of sphere!

Quantal Scattering - No Trajectory! (A plane wave hits some object and a spherical wave emerges ) Solve the time independent Schrödinger equation Approximate the solution to one which is valid far away from the scattering center Write the solution as a sum of an incoming plane wave and an outgoing spherical wave. Must find a relation between the wavefunction and the current densities that defines the cross section. Procedure:

Current Density: Incomming current density: Outgoing spherical current density:

The Schrödinger equation - scattering form: Now we must define the current densities from the wave function…

The final expression:

Summary Then we have: …. Now we can start to work Write the Schrödinger equation as: Asymptotics:

Integral equation With the rewritten Schrödinger equation we can introducea Greens function, which (almost) solves the problem for a delta-function potential: Then a solution of: can be written: where we require: because….

This term is 0 This equals Integration over the delta function gives result : Formal solution : Transforms formally differential equation to integral equation

Green’s function ”Proof”: The result is well known (function of scalar distance only!): Easier to solve for r’ = 0 ( | r | instead of | r - r’ | )

One obtains: The Born series (first Born approximation usually used: Schrödinger equation as: Asymptotics: SUMMARY

The potential is assumed to have short range, i.e. Active only for small r’ : 1) Asymptotics - Detector is at near infinite r 2) Asymptotic excact result: Still formal expression

The Born approximation: The scattering amplitude is then: Suitable for simple evaluations: Fourier transform of the potential for the value of the momentum change or ”momentum transfer” Use incomming wave instead of Under integration sign:

Spherically Symmetric potentials - typical evaluation: Total Cross Section: momentum transfer

A feature - 1’st. Born Approximation: Because scattering angle is related to MOMENTUM TRANSFER INTEGRAL OVER MOMENTUM TRANSFER

Example - Hard sphere Classical Hard Sphere scattering Differenial cross section constant, no angular dependence!

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