Prof. Reinisch, EEAS /511

4.1 Simple Collision Parameters (1) There are many different types of collisions taking place in a gas. They can be grouped into two classes, elastic and inelastic. Elastic Collisions: the particles conserve their masses, and the kinetic energy and momentum is conserved. Inelastic Collisions: kinetic energy can be transformed into rotational or vibrational energy, or excitation and ionization.

Prof. Reinisch, EEAS / Simple Collision Parameters (2) Collision Time and Frequency: Assume a molecule with radius r 0 moves with velocity v through a cloud of electrons (F. 4.1). In the time  t it sweeps out a cylindrical volume V =  v  t that was previously filled with nV electrons. Here  =  r 0 2 is the collisional cross section of the molecule. If n is the number of electrons per unit volume, i.e., the electron number density there will be nV = n  v  t collisions in the time  t. The mean time  per collision is then  =  t/ (n  v  t) =1/n  v. The inverse is called the collision frequency c (Greek symbol, not velocity v): c  1/  = n  v. Mean Free Path Length: mfp  v  n Generally, the electrons have thermal (random) velocities and the relative velocities must be considered.

Prof. Reinisch, EEAS / Binary Elastic Collisions (1) The collision process between particles of species s and t is controlled by their relative velocities and the inter-particle force. We want to find the differential cross section  st (g st,  ) required to calculate the Boltzmann collision integral (3.9). Here is the magnitude of the relative velocity, and  is the scattering angle. If the two colliding particles have comparable masses m s and m t, it is advantageous to perform the calculations in the center-of-mass system defined in equations (4.6) to (4.13). Using the laws of conservation of momentum and energy, it is easy to show that g st = g st ‘ (g st before, g st ‘after collision). The direction of the relative velocity vector g changes changes at the collision, see Fig.4.3. We illustrate the collision process for the simple case of a Coulomb collision between an ion and and electron. Since the ion mass is so much larger than the electron mass it barely changes its velocity in response to the collision, i.e., the center-of-mass (CM) system is essentially anchored in the ion.

Prof. Reinisch, EEAS / Binary Elastic Collisions (2) Coulomb Collision The geometry of the electron-ion collision is shown in Fig The ion is at rest in the ion frame of reference. ‘Far away’ (before the collision) the electron has the momentum m e v 0. A line through the center of the ion parallel to v 0 has the distance b 0 from the electron when the electron is still far away. This distance is called the impact parameter. The Coulomb force is a so- called central force, i.e., it acts along the line connecting the two charges. The form of the Coulomb law suggests the use of polar coordinates r,  in the plane through the two particles (Fig. 4.4).

Prof. Reinisch, EEAS / Binary Elastic Collisions (3) Coulomb Collision (kinetic + potential energy) after = (kinetic + potential energy) before

Prof. Reinisch, EEAS / Binary Elastic Collisions (4) Coulomb Collision

Prof. Reinisch, EEAS / Binary Elastic Collisions (5) Coulomb Collision

Prof. Reinisch, EEAS / Binary Elastic Collisions (6) Coulomb Collision

Prof. Reinisch, EEAS /511