Scattering from a hard ellipsoid

Present · Nov 8, 2014

Homework Statement

Diagram of an ellipse in the x-z plane with semi-axes a and b, an incoming beam parallel to the x-axis hitting its boundary, a dashed normal, an angle psi, and an impact parameter s; above it the printed cross-section formula sigma(theta) = (1/4) b^2 a^2 b^2 over [a^2 sin^2(theta/2) + b^2 cos^2(theta/2)]^2.

a,b is constant and s is impact parameter, θ is scattering angle.
i know that ψ in the picture is <ψ=(π-θ)/2>

Homework Equations

differential scattering crossection dσ/dΩ = (s/sinΘ) I ds/dθ I

and σ(θ)=∫(dσ/dΩ)dΩ

The Attempt at a Solution

i guessed, first step is that finding a relation of impact parameter s and scattering angle θ.
but i couldn't. This picture is my attempt at a solution.

Handwritten sketch of a hard ellipsoid scattering geometry with x and z axes, labelled a and b, a ray hitting the surface at an angle psi, and the ellipse equation x = a(1 - s^2/b^2)^(1/2) and x^2/a^2 + z^2/b^2 = 1 written below.

i think for getting a relation of <s and θ> to use tan(θ/2) but i do not find.
How can i get impact parameter s(θ)?

haruspex · Nov 8, 2014

calculate, or look up, the equation for the tangent to an ellipse.

Scattering from a hard ellipsoid

Homework Statement

Homework Equations

The Attempt at a Solution

Discussion

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