1. Introduction & Collision Kinematics

Background

Atomic and photonic collisions (atomic interactions) is a very wide topic, going all the way from thermal atom transfer collisions (chemical reactions) to relativistic collisions with the emission of subatomic particles. Here we will limit the discussion to processes that are of importance in important application of heavy particle collisions to materials processing.

The field of atomic interactions has a very venerable history, since it was central to the development of quantum mechanics, and nuclear physics. Atomic interactions are the basic means for probing the atomic structure of matter. The same can be said to subatomic collisions for probing subatomic structure. Notable landmarks started with the discovery of X-rays, radioactivity, the electron, proton, and alpha particles about one hundred years ago. There is an impressive list of Noble prize winners (Physics and Chemistry) who did research in the area. It includes M. Curie, J.J. Thompson, E. Rutherford, A. Einstein, N. Bohr, J. Franck, G. Hertz, A.H. Compton, V, Hess. C. Anderson, F. Aston, C. Davisson, G. Thomson, P. Debye, E. Fermi, P. Blackett, J. Cockcroft, E. Walton, F. Bloch, M. Born, W. Lamb, L. Landau, E. Wigner, C. Townes, N. Basov, A. Prokhorov, H. Bethe, A. Bohr, N. Mott, K. Siegbahn, P. Kapitsa, J. Polanyi, and A. Zewail.

The development of the field was propelled by an intrinsic interest in the basic science, and by applications. Important applications are to other fields of science, like the role in the establishment of quantum mechanics, nuclear physics, quantum chemistry, aeronomy, plasma physics, and astrophysics. Atomic collisions and laser methods are also the fundamental processes behind very important and ubiquitous methods of characterizations of gases, materials, and plasmas. Technological applications developed; roughly in historical order, they include in gaseous discharges, particle accelerators, isotope separators, nuclear fusion reactors, sputter deposition of thin films, ion implantation, surface analysis, plasma processing, laser ablation and film deposition, and nm ion beams.

The four basic division of the field are: electronic, atomic photonic, and rearrangement collisions, the latter being a term denoting chemical reactions. Due to the competing demands of breadth and depth, we will limit ourselves to interactions of energetic atoms (ions) and lasers with matter (mostly solids), with some important technical applications.

In the following, we consider basic equations that can be obtained from conservation laws, and we will restrict the treatment to non-relativistic collisions.

Fundamental Concepts in Atomic Interactions 

Collision Kinematics

The simplest atomic collision is the elastic scattering between two particles, under the action of a central force f(r) that is a function of interparticle separation r, but not of angle. Related to the force is the interaction energy U(r). In elastic scattering the total kinetic energy of the particles before and after the collision is the same, where in an inelastic collision, there is a transfer between kinetic energy and internal energy of the system (electronic, vibrational, rotational energies). Both f(r) and U(r) decay fairly rapidly for r exceeding atomic dimensions, making it possible in most cases to speak of the time before and after the collision, where the interactions are negligibly small.

Laboratory and CM coordinates - asymptotic aspects of non-relativistic collisions

There are two natural reference frames for describing a collision. The laboratory frame, where, initially, one particle (projectile) moves initially with velocity v0 and other is stationary, and the center of mass frame, which is at rest before and after the collision and where both particles move. Sometimes one also uses a projectile frame, which moves with the projectile. In this case, the We denote by M and m the masses of the projectile and target atom. The impact parameter b is the closest distance between the projectile and the target in the absence of an interaction.

Asymptotic diagram of the elastic collision in the lab and CM systems. After the collision m and M travel with velocities v, V and lab angles of , respectively in the lab system, and with angle Q in the CM system.

Velocities and kinetic energy

Since the total momentum needs to be the same in both systems, the magnitude of the center of mass velocity is given by

or

where the reduced mass is defined by:

Note the special cases: Mr is the mass of the lighter particle if there is a large difference in mass. Mr = 0.5 m = 0.5 M if m = M.

Since kinetic energy and momentum is conserved in an elastic collision,

Note that the interatomic forces / potential do not enter in this expression. Solving for the final velocities, one obtains:

An important quantity is the elastic energy transfer from the projectile to the target:

where E0 is the initial kinetic energy of the projectile and

Note the special cases:

 

 The first two conditions say that the energy transfer is very small when the particles differ greatly in mass.  The third equation gives the maximum energy transfer, which occurs when the masses are equal.

Relationships between lab and CM scattering angles are:

and

Again, it is interesting to consider special cases:

a)

That is, when the projectile is much lighter than the target, the lab and CM systems are nearly identical.

b) m = M

In this case of identical masses, there is no backscattering in the lab system. Head-on scattering (Q = p) gives J = p/2, q = 0)

c) m > M

There is a maximum scattering angle. As J increases to

Q increases from 0 to

b)

Inelastic collisions

Homework Problem

1) Express the final energy of the projectile and of the target, in the laboratory system, when there is an inelastic energy transfer in the collision Q (>0 exothermic, <0 endothermic). Analyze limiting cases (extremes of scattering angles, ranges of m/M).

2) For endothermic collisions, there will be a collision energy below which inelasticity is not possible. Analyze collisions for projectile energies close to the threshold.

References

E.W. McDaniel, Collision Phenomena in Ionized Gases (Wiley, NY 1964)

E.W. McDaniel, J.B.A. Mitchell, and M.E. Rudd, Atomic Collisions. Heavy Particle Projectiles (Wiley, NY, 1993)

 

Copyright 2003: Raśl A. Baragiola