Bethe stopping power theory for heavy-element targets and relativistic projectiles

An approach is described by which the Bethe theory for the stopping power of heavy-element targets is extended to treat the projectile in a fully relativistic way. The target electrons are treated semirelativistically by use of a modified Foldy-Wouthuysen transformation, also known as the Douglas-Kroll transformation. This approach allows results to be obtained that are valid to all orders in v²/c², with v being the initial speed of the projectile. The contributions from both the longitudinal and transverse parts of the differential cross section are included, yielding a full expression for the stopping power of these systems. Our results show that the relativistic nature of the electrons in heavy-element targets leads to significant corrections which can be greater than 3% of Bethe's original results.