The transition state is fundamental to modern theories of reaction dynamics: essentially, the transition state is a structure in phase space that all reactive trajectories must cross. While transition-state theory (TST) has been used mainly in chemical physics, it is possible to apply the theory to considerable advantage in any collision problem that involves some form of reaction. Of special interest are systems in which chaotic scattering or half-scattering occurs such as the ionization of Rydberg atoms in external fields. In this paper the ionization dynamics of a hydrogen atom in crossed electric and magnetic fields are shown to possess a transition state: We compute the periodic orbit dividing surface (PODS) which is found not to be a dividing surface when projected into configuration space. Although the possibility of a PODS occurring in phase space rather than configuration space has been recognized before, to our knowledge this is the first actual example: its origin is traced directly to the presence of velocity-dependent terms in the Hamiltonian. Our findings establish TST as the method of choice for understanding ionization of Rydberg atoms in the presence of velocity-dependent forces. To demonstrate this TST is used to (i) uncover a multiple-scattering mechanism for ionization and (ii) compute ionization rates. In the process we also develop a method of computing surfaces of section that uses periodic orbits to define the surface, and examine the fractal nature of the dynamics
