8—Multivariable Calculus 235Cross Section, Scattering
There are many types of cross sections besides absorption, and the next simplest is the scattering cross section,
especially the differential scattering cross section.
θ bb+db
θ∆Ω
The same flux of particles that you throw at an object may not be absorbed, but may scatter instead. You
detect the scattering by using a detector. (You were expecting a catcher’s mitt?) The detector will have an
area∆Afacing the particles and be at a distancer from the center of scattering. The detection rate will be
proportional the the area of the detector, but if I doublerfor the same∆A, the detection rate will go down by a
factor of four. The detection rate is proportional to∆A/r^2 , but this is just the solid angle of the detector from
the center:
∆Ω = ∆A/r^2 (28)
The detection rate is proportional to the incoming flux and to the solid angle of the detector. The proportionality
is an effective scattering area,∆σ.
∆R=f∆σ, sodσ
dΩ=
dR
fdΩThis is the differential scattering cross section.
You can compute this if you know something about the interactions involved. The one thing that you need
is the relationship between where the particle comes in and the direction in which it leaves. That is, the incoming
particle is aimed to hit at a distanceb(called the impact parameter) from the center and it scatters at an angle
θfrom its original direction. Particles that come in at distance betweenbandb+dbwill scatter into directions
betweenθandθ+dθ.