On the low order approximation of radiation fields generated by some hollow-cylindrical ion beams accelerated to produce exoergic nuclear reactions

Increasing interest is being shown in obtaining accurate predictions concerning radiation fields produced by hollow-cylindrical ion beams impinging on homogeneous plane targets, the net effect of this process being exoergic nuclear reactions. Previous theoretical studies by the authors have focused on radiation fields generated by homogeneous plane ring-shaped sources, based on a unified treatment of the radiation field distribution developed by Hubbell and co-workers. In the case of an equivalent homogeneous source anisotropically emitting in nondispersive media, the Legendre polynomial series expansion method for emissivity function can be successfully applied when conditions for the convergence of the approximating series are satisfied. We have developed an analytical expression for the radiation field distribution around a homogeneous plane target bombarded by hollow-cylindrical ion beams whose elementary areas anisotropically emit in non-dispersive media. The expression includes summation of four terms in a cos-type approximation, yielding a low order approximation of the angular distribution of source emissivity. The resulting expression is a linear combination of common and elliptic functions. Particular interest focuses upon the evolution of the shape of the curves close to the source and to the discontinuity at the source boundary. Results of this investigation can be extended to experimental situations in which the assumption of an omni-directional distribution of nuclear reaction emissivity over the accelerator target surface or other kinds of axi-symmetric plane sources of radiation, is no longer valid