Ion Implantation and Synthesis of Materials - Studium

3.5 Motion under a Central Force 31coordinate system. Since potential-energy involves only the radial distance, theproblem has spherical symmetry, indicating any rotation about a fixed axis canhave no effect on the solution; i.e., if either particle is located at the origin, theforce on the other is given by a central force, F(r), which only depends on theseparation distance, r.In the problem examined in this section, we will assume that, in the laboratorysystem, one of the particles is practically at rest at the origin, O, while the otherone moves with velocity v – a good approximation when the stationary particle ismuch heavier than the moving particle.3.5.1 Energy Conservation in a Central ForceFor conservative central forces and a defined interaction potential, V(r), we canwrite a statement for the total mechanical energy for a particle of mass M, adistance r away from a central force F asM 2 2E = ⎡ +θ⎤+( ),2 ⎣vrv ⎦ V r(3.20)where v r and v θ are the radial and transverse velocities, respectively. The first termon the right-hand side of (3.20) represents the kinetic energy in polar coordinates.In addition to the total energy equation given above, we also have the conditionof conservation of angular momentum,l=Mrvθ(3.21)The quantities E and l are the constants of motion, while V(r) is the potentialenergyfor a particle of mass M in a central field. We rewrite (3.20) in the form2 2Mv= ( ) = r lE E r + + V( r).22 2Mr(3.22)All terms in (3.22) are a function of r only. The first term is the kinetic energyfor the radial component; the term l2 /2Mr 2 is referred to as the centrifugal energy;and V(r) is the interatomic potential-energy. The centrifugal energy is the portionof the kinetic energy term that is due to the particle’s motion transverse to thedirection of the radius vector. It is because the centrifugal energy can be describedas a function of radial position r alone that we can treat the radial motion of aparticle as a one-dimensional problem in r. Equation (3.22) is now simply afunction of r only.