Scattering kinematics : Transformation of differential cross sections between two moving frames

The primary datum of a scattering experiment is the differential cross section. It is defined as the number of particles (events) per second per solid angle observed by the detector divided by the number of particles per unit area per second incident per target particle (scattering center). 1 In many scattering experiments the detector is most conveniently described using the fixed reference frame of the laboratory, while the scattering events are most conveniently analyzed in a reference frame moving with uniform velocity with respect to the laboratory frame. In such cases it is necessary to relate the differential cross section measured in one frame to the differential cross section that would be measured in the other. This kinematic problem can be vexing, causes much confusion, and is the subject of this article.