The contribution to the Fokker-Planck eqttation. for the distribution function for gases, due to particle-particle interactions in which the fundamental two-body force obeys an inverse square law, is investigated. The coefficients in the equation, (~) (the average change in velocity in a short time) and f# ~), are obtained using the collision cross sections for such forces. These coefficients are obtained in terms of two fundamental integrals which are dependent on the distribution'function itself. The transformation of the equation to polar c.oordtnates in a case of axial symmetry is carried out. By expanding the distribution function in Legendre functions of the angle the equation is cast into the form of an infinite set of one dimensional coupled nonlinear integra-differential equations. If the distribution function is approximated by a finite series, the reaultant Fokker-~lanck equation may be treated numerically using a compUting machine. Keeping only one or two terms in the series corresponds to the approximations of Chandrasekhar, and Cohen, Spitzer and McRoutly, respectively .

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A two-year study of recirculating induction heavy ion accelerators as low-cost driver for inertial-fusion energy applications was recently completed. The projected cost of a 4 MJ accelerator was estimated to be about $500 M (million) and the efficiency was estimated to be 35%. The principal technology issues include energy recovery of the ramped dipole magnets, which is achieved through use of ringing inductive/capacitive circuits, and high repetition rates of the induction cell pulsers, which is accomplished through arrays of field effect transistor (FET) switches. Principal physics issues identified include minimization of particle loss from interactions with the background gas, and more demanding emittance growth and centroid control requirements associated with the propagation of space-charge-dominated beams around bends and over large path lengths. In addition, instabilities such as the longitudinal resistive instability, beam-breakup instability and betatron-orbit instability were found to be controllable with careful design.