Microscopic Scattering Cross Section
A flux of JP projectiles/cm2-sec impinge on a dilute gas containing Nt targets (dilute means no interaction or shielding between targets). The intensity Is of projectiles scattered over polar angle J and azimuthal angle j in an element of solid angle dW is given by
where dq is called the differential scattering cross section (dimensions are cm2/target particle) which can be interpreted as the area presented by each target particle for scattering into dW.
The total cross section q is given by:
Commonly, there are no azimuthal variations (no j dependence). Thus,
Atomic potentials are long range, they decrease slowly to
large distances. Since small
potentials mean small scattering angles, the classical differential cross
section diverges for small J.
For instance, in Coulomb scattering,
and the total cross section diverges logarithmically:
Relationship of cross sections in the LAB and CM systems
The relation between cross sections is just given by the transformation of angles since the integrated fluxes must be conserved. From the relation between scattering angles in the LAB and CM systems, we obtain
The relationship between the impact parameter b and the scattering angles are given in the figure below.
Mean Free Path
For
a projectile beam of intensity I0 going through a dilute gas
of density N of thickness dx, the current of projectiles scattered
out of the beam is dI = Inqdx. If the target is "thin" so that the particle is not
scattered back into the incident beam direction,
The mean free path (MFP) is the mean distance
between collisions:
Reaction rates and particle flux
Consider a beam of particles with density n (projectiles/cm3 ) with a defined velocity v passing through a gas with a density N of stationary target particles. Each projectile makes, on average, v/l collisions per second. The total number of collisions per cm3 per second, the reaction rate, is given by
The projectile flux is given by
(projectiles/cm2-s). Thus, the reaction rate is qJ/cm3-sec.
Quantum Effects
The classical picture of deflections is valid if the observation doesn't radically perturb the collision.
If scattering occurs in a finite volume of dimension a, with potential of the order U, then locating a particle within a produces an uncertainty Dp in its momentum p, given by
The classical picture is valid if
The deflection, J is approximately Dp/p, while the energy change in the interaction, DE » 2pDp/m » U. Thus, 2vDp » U, or Dp » U/2v. Then the classical picture is valid if:
Note that a/v is the interaction time Dt. For a classical description, the energy uncertainty
must be much smaller than the potential U, which again implies that
In a scattering experiment, the conditions given above imply a minimum scattering angle below which quantum mechanics must be taken into account:
Since smaller angles cannot be observed, the quantum mechanical effect limits the divergence of the differential cross section.