A systematic survey is made of free‐boundary, large‐orbit ion ring equilibria for the case where the distribution function is narrowly peaked at a given value of the single‐particle energy (H) and the canonical angular momentum (P&thgr;). Suitably normalized, such equilibria depend on only two free parameters. The survey covers essentially all of the parameter space where field‐reversed equilibria exist. Results are presented as contour plots in the (P¯,H¯) plane of various physical quantities. The results may be reinterpreted so as to apply to relativistic electron rings on which there are experimental observations.