This paper presents the solution of a special scattering problem which may be important
in the theory of slowing-down of atomic particles in crystals . A projectile moves along the center
axis of a regular ring of n equal atoms which are free and do not interact with each other. The
interaction between the projectile and each ring atom is described by a Born-Mayer potential,
and the scattering is assumed to be elastic and governed by the classical equations of motion.
Because of symmetry, the problem can be reduced to plane motion of a particle in a potential
of elliptic symmetry. The elliptic force field is approximated by a spherical one, which is dependent
on the initial conditions of the individual scattering problem . For the spherical symmetrical
potential, scattering angles and related quantities have been tabulated, but simple analytical
approximations can be used too. As a result, one obtains the asymptotic velocities of the ring
atoms as well as the energy loss of the projectile. Furthermore, it can be decided whether the
projectile is reflected by the ring. Both the feasibility of assumptions specifying the problem
and the validity of different approximations made in the transformation from the elliptic to
the spherical potential are investigated. Special attention is paid to proper definitions of collision
time and collision length which are important in collisions in crystals. Limitations to classical
scattering arising from the uncertainty principle prove to be more serious than assumed previously.
Inelastic contributions to the energy loss can easily be included . The oscillator forces binding
lattice atoms turn out to influence the scattering process only at very small energies. The validity
of the so-called momentum approximation and a related perturbation method are also investigated.
|Title of host publication||Matematisk-Fysiske Meddelelser udgivet af Det Kongelige Danske Videnskabernes Selskab|
|Place of Publication||København|
|Publisher||Det Kongelige Danske Videnskabernes Selskab|
|Publication status||Published - 1966|
|Series||Matematisk-fysiske meddelelser udgivet af Det Kongelige Danske Videnskabernes Selskab|