Based on two constants of motion, H and Pθ, where H is the total energy of a particle and Pθ is its canonical angular momentum, particle confinement criteria are derived which impose constraints on H and Pθ. With no electric field at the ends of field-reversed magnetic configurations, confinement criteria for closed-field and absolute confinements are obtained explicitly, including both lower and upper bounds of Pθ/q, where q is the charge of the species considered, for a class of Hill's vortex field-reversed magnetic configurations. The commonly used criterion for the Hamiltonian, H < −ω0Pθ, where ω0 ≡ qB0/mc, is deduced from a more general form as a special case. In this special case, it is found necessary to impose a new criterion, , where Rw is the wall radius and B0 is the vacuum field, which reduces the confinement region in (H, Pθ) space. With the presence of electric fields at the ends of field-reversed magnetic configurations, confinement criteria are obtained for two interesting cases. In addition to lower and upper bounds of H, both lower and upper bounds of Pθ/q are found. For axially confined particles, the lower bound of Pθ/q reduces the confinement region in (H, Pθ) space and represents a new criterion. These results can be applied to calculations for field-reversed mirrors and field-reversed theta pinches.
Theoretical limitations to the Hall acceleration mechanism in the polytron machine