Microscopic Bumper Cars

SpectrumLogoInvestigating the Particle Physics Behind Perfectly Elastic Atomic and Subatomic Collisions.

Raza Syed

SPECTRUM Writer

April 20, 2016

 

An elastic collision is a collision in which both momentum and kinetic energy are conserved in the collision. This implies that there is no dissipative force acting during the collision and that all of the kinetic energy of the objects before the collision are still in the form of kinetic energy afterward. Momentum is a physics term; it refers to the quantity of motion possessed by an object and can be defined as “mass in motion”.  The amount of momentum possessed by an object is dependent upon the variables of mass and velocity. Therefore, linear momentum is equal to the product of mass and velocity. Kinetic energy is the energy possessed by a body because of its motion. There are many different forms of kinetic energy- vibrational, rotational, and translational kinetic energy. In terms of simplicity, the focus of this investigation is geared towards translational kinetic energy which is the energy due to motion from one location to another. Perfectly elastic collisions are rare at the macroscopic level. Though the collisions between billiard balls represent fairly elastic collisions, they are not perfectly elastic and thus kinetic energy is not completely conserved. However, collisions between particles, atoms and ions are considered 100% elastic. At the atomic level, elastic “collisions” occur where the particles do not come into direct contact with one another, but instead, a repulsive Coulomb force keeps the particles out of direct contact with each other.

An example of this at the atomic level is Rutherford scattering. Rutherford’s famous gold foil experiment led to the discovery of the atomic nucleus, but also demonstrated particle scattering through a central repulsive Coulomb force. In the experiment, alpha particles from a radioactive source were allowed to strike a thin gold foil. Alpha particles produce a tiny, but visible flash of light when they strike a fluorescent screen. To Rutherford’s amazement, some alpha particles were found deflected at extreme angles to their linear displacement from the radioactive source to the gold foil, while some were even found to be completely back scattered. Thus, the experiment manifested the idea of the nuclear atom.

The scattering of alpha particles from nuclei can be modeled from the Coulomb force and treated as an orbit. The scattering process can be treated statistically in terms of the cross-section for interaction with a nucleus which is considered to be a point charge Ze. When a detector is placed at a specific angle with respect to the incident beam, the number of particles per unit area striking the detector is given by Rutherford’s formula (right):

The Coulomb force is responsible for electromagnetism at the atomic level; the Coulomb force between two or more charged bodies is the force between them due to Coulomb’s law. If the particles are both positively and negatively charged, the force is repulsive; if they are of opposite charge, the force is attractive.

With respect to studying atomic collisions in one dimension only, a head-on elastic collision model must be assumed. A head on-elastic collision is one in which the colliding bodies move along the same axis of motion before and after the collision. For a head-on collision between a projectile and a stationary object of equal mass, the projectile will come to a rest and the target will move off with equal velocity similar to a head-on shot with the cue ball on a pool table; the velocities will always be exchanged. In a head-on elastic collision where the projectile is much more massive than the target, the velocity of the target particle after the collision will be about twice that of the projectile and the projectile will be essentially unaffected. In a head-on elastic collision between a small projectile and a much more massive target, the projectile will bounce back with approximately the same speed and the massive target will be delivered a very small velocity. An example of this is a ball bouncing back from the Earth when it is thrown vertically down.

The mass of a hydrogen atom is  kg whereas the mass of a helium atom is  kg. Therefore, when a helium atom in motion hits a hydrogen atom at rest, the massive projectile scenario of a head-on elastic collision can be considered. After the collision, the velocity of the hydrogen atom will approximately be twice that of the helium atom and the velocity of the helium atom will be relatively the same after the collision compared to before the collision. It is important to label the quantity of motion as velocity instead of speed so that both direction and speed of the objects in question can be interpreted from an analysis of the situation. In terms of direction, both objects are moving forward in the positive direction following the collision. The miniscule amount of momentum lost by the massive helium atom is gained by the hydrogen atom at rest according to the Law of Conservation of momentum. Since the hydrogen atom has a very small mass compared to the helium atom, the small loss in momentum of the helium atom upon the collision is equal to the momentum gained by the hydrogen atom providing it with a large velocity due to the miniscule mass of the hydrogen atom. Therefore, the velocity of the hydrogen atom will be approximately double in magnitude than that of the helium atom.

The mass of a neutron is 1.6749 x 10-27 kg whereas the mass of a hydrogen atom is kg. Since the masses of the two entities are approximately the same in value, when the neutron in motion hits the hydrogen atom at rest, the neutron will come to a rest while the hydrogen atom will move off with equal velocity. This observation can be based on the generalization of head-on elastic collisions where two colliding objects have relatively the same mass which states that the velocities of the objects in question will always exchange. In this scenario, the neutron will have approximately zero velocity after the collision. After the collision, the hydrogen atom will have the same velocity as the neutron before the collision in the forward direction, where the forward velocity vector is positive. Conclusively, the hydrogen atom will have a slightly higher velocity of the two entities following the elastic collision.

 

 

References

http://hyperphysics.phy-astr.gsu.edu/hbase/elacol.html

http://hyperphysics.phy-astr.gsu.edu/hbase/rutsca.html

http://hypernews.slac.stanford.edu/slacsite/aux/HiPPP/scattering/