4 In this chapter it is convenient to define dΩ = sin θ d θ d φ rather than d 2 Ω = sin θ d θ d φ as in earlier chapters.
240 Chapter 12: Scattering Theory 12.3 Cross-sections and scattering experiments Children sometimes test their skill by taking turns to throw pebbles at a dis- tant target, perhaps a rock. If a pebble hits, it will bounce off in a different direction, whereas a pebble that misses will simply continue undisturbed. Each throw will not be repeated exactly, and after a long time we might imagine that the children have thrown pebbles randomly, such that the dis- tribution of throws per unit area is uniform over a region surrounding the target. If so, we can estimate the area of the target that the children see by simply counting the number of pebbles that hit it – if N in pebbles are thrown in per unit area, and N sc of them hit the rock, the rock has cross-sectional area A ≃ N sc /N in . (12 . 47) With more care, we can measure the angle through which throws are de- flected. Pebbles that strike nearby points of a smooth rock will bounce off in roughly the same direction, whereas a jagged rock may deflect pebbles that hit closely spaced points very differently. Hence, counting the number of pebbles that end up going in a given direction gives us information about the rock’s shape. We define the differential cross-section δσ to be the area of the target that deflects pebbles into a small solid angle δ Ω. If there are N ( θ,φ ) δ Ω such pebbles, then δσ ≡ N ( θ,φ ) δ Ω N in or δσ δ Ω = N ( θ,φ ) N in , (12 . 48) and the total cross-section is σ tot ≡ integraldisplay d σ = integraldisplay dΩ d σ dΩ = integraldisplay dΩ N ( θ,φ ) N in = N sc N in (12 . 49) as above. This may seem a rather baroque manner in which to investigate rocks, but when you go out on a dark night with a torch, you probe objects in a very similar way by throwing photons at them. A more complete analogy can be drawn between pebble-throwing children and physicists with particle accel- erators: a beam containing a large number N b of particles is fired towards a target, and detectors measure the number of particles that scatter off into each element of solid angle δ Ω. Long before the collision, a typical particle in the beam looks like a free state | φ ) , so the probability density of each particle is |( x | φ )| 2 and the number of particles per unit area perpendicular to the beam direction is n in ( x ⊥ ) = N b integraldisplay d x bardbl |( x | φ )| 2 , (12 . 50) where the integral is along the beam direction. When | φ ) is expanded in terms of momentum eigenstates, equation (12.50) becomes n in ( x ⊥ ) = N b (2 π ¯ h ) 3 integraldisplay d x bardbl d 3 p d 3 p ′ e i( p − p ′ ) · x / ¯ h φ ( p ) φ ∗ ( p ′ ) = N b (2 π ¯ h ) 2 integraldisplay d 3 p d 3 p ′ δ ( p bardbl − p ′ bardbl )e i( p ⊥ − p ′ ⊥ ) · x ⊥ / ¯ h φ ( p ) φ ∗ ( p ′ ) , (12 . 51) where the integral over x bardbl produced the delta function of momentum along the beam direction. Experimental beams are highly collimated, so φ ( p ) vanishes rapidly unless the momentum is near some average value ¯ p . In