PHY6200 W07

Chapter 14: Collision Theory


In Taylor, read sections 14.1 - 14.2 for today, and 14.2 to 14.4 for Wednesday.

Two of the most important tools for studying phenomenon are oscillations and scattering. We studied simple harmonic, damped, and forced oscillations in chapter 5, and coupled oscillators in chapter 11. Now we will study scattering, learning concepts like impact parameter and cross section, and lookint at the details of a seminal scattering experiment carried out by Ernest Rutherford and his team.

Many phenomena can be viewed as scattering problems: a comet passing near the sun; a spaceship attempting to enter earth's atmosphere; the collision of billiard balls. The prototypical scattering experiment involves objects at the atomic scale or below, i.e. nuclei or subatomic particles. In such setups, the orbit of the projectile around the target cannot be known in detail, both because we can't know the position of projectile and target with sufficient accuracy, and because of quantum mechanical limitations. Instead, we imagine many projectiles randomly distributed with respect to the target, and ask with what probability the projectile will scatter in a particular direction. This approach takes some getting used to.

The Scattering Angle and Impact Parameter

The prototypical model of a collision experiment (whether with subatomic particles or billiard balls) is a projectile approaching a stationary target from a great distance where its initial motion is unaffected by the target. More precisely, initially the projectile-target system is assumed to have zero potential energy, so the initial energy of the system is just the kinetic energy of the projectile. The projectile approaches the target is is deflected by their interaction. The projectile (and sometimes the target as well) is scattered, and moves away, where again the potential energy of interaction approaches zero, so the final energy of the system is again purely kinetic (though it could be the kinetic energy of both projectile and target).

The angle between the final and initial velocities of the projectile is called the scattering angle θ. The angle θ = 0 means there was no scattering, the projectile moved along a straight line. The maximum scattering angle is θ = π, meaning that the projectile bounced straight back from the target.

The inpact parameter, b, is the closest distance the projectile would approach the target if it moved without deflection. In subatomic scattering experiments, it is impossible to know the impact parameter a priori, so this is the quantity that we average over when determing the probability to scatter in a particular direction. The minimum impact parameter is b = 0, corresponding to a head-on collision. The larger the impact parameter, the more the projectile misses the target, normally yielding smaller scattering angles.

A basic assumption is one-to-one correspondence between impact parameter and scattering angle -- that a particle scattered in a particular direction had a particular impact parameter. So, first we need to find the relation between impact parameter and scattering angle, and then average over the possible impact parameters to find the probability for scattering in a particular direction.

Experimentally, it is easy to measure the scattering angle, while basically impossible to measure the impact parameter directly.

The Collision Cross Section

Since we can't measure the trajectories of projectile and target well enough to predict if a collision will occur in subatomic processes, we generate a probability. The apparent size of the target is called the cross section.

© 2007 Robert Harr