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Cross Section
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Two-Body Collision Center of mass R Neglect external force. Motion is in a plane Reduced mass applies Compare to CM m2m2 r1r1 F 2 int r2r2 R m1m1 F 1 int r = r 1 – r 2
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Frame of Reference Two body interactions generally define a natural axis. One body at rest, so the moving body moves along xOne body at rest, so the moving body moves along x Objects as point masses, so use the axis connecting the points.Objects as point masses, so use the axis connecting the points. m1m1 m2m2 v x m1m1 m2m2 v1v1 x v2v2
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Scattering Angles Look only at one mass. Mass 2 at restMass 2 at rest Compare to CM frame. Set up velocity triangle in the plane of the collisionSet up velocity triangle in the plane of the collision x m1m1 The general solution is transcendental. Often special cases For m 2 >> m 1 (= ) Or m 2 = m 1 (= 2 )
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Elastic Scattering Scattering angle measured in CM. Convert to lab frame.Convert to lab frame. Depends on mass ratio.Depends on mass ratio. x
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Impact Parameter Two body results are kinematic, not dynamic. Predict angle from dynamic variables. Angular momentum JAngular momentum J Kinetic energy TKinetic energy T Define impact parameter b. For a given T, J(b)For a given T, J(b) recoil m1m1 m2m2 v x b scatter
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Differential Cross Section Predict angles from a initial distribution. Force decreases with distanceForce decreases with distance Trajectory must be asymptotic.Trajectory must be asymptotic. Start with incident flux I. Particles/time / area normal to beam.Particles/time / area normal to beam. Number of interactions N per solid angle that scatter at angle .
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Axial Symmetry Central forces are symmetric. Axis through force centerAxis through force center Integrate over azimuthal angleIntegrate over azimuthal angle Particles scatter in an angular range. b
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Lab Scattering The cross section can be determined in the CM or lab frame. Convert from CM to lab anglesConvert from CM to lab angles next
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