CLASSICAL ION MOTION IN ELECTROSTATIC DIPOLE FIELDS

The classical unbound motion of an ion in an electrostatic dipole field V=kcosθr2 has been solved for analytically and displays many very interesting features. In addition to total energy E and the azimuthal angular momentum pφ, α=pθ2+pθ2sin2θ+2mkecosθ is also conserved. For all potentials of the form F(θ, φ)r2, r2 is a symmetric quadratic function of time. For the fixed point dipole problem, r2(τ)=2Eτ2m+α2Em; both E, α>0 for scattering orbits. For meridan plane motion (pφ=0), the polar angle θ(τ) is expressed in terms of Jacobian elliptic functions. Special attention is paid to turning points and to the specification of asymptotic orbits via energy, impact parameter b, and initial asymptotic angle θi=θ(-). Representative meridian plane orbits are shown and scattering angle discussed as a function of Eb2. For general nonplanar scattering (pφ 0), limitations on the range of θ arise, as in the related problems of the spherical pendulum and symmetric heavy top. Azimuthal angle φ(τ) is expressed as an incomplete elliptic integral of the third kind: π(x, β)=0xdu(1-β2sn2u), which can usefully be expressed in terms of theta functions. Thus the simplest case of classical ion-dipole scattering is already quite complex as regards solution by analytical means. © 1969 The American Physical Society.