Opacity Project (OF) and OPAL opacities were used to calculate non-linear hydrodynamic models of 0.7-M. radially pulsating extreme helium stars, having mean (averaged over a pulsation cycle) effective temperatures and luminosities in the ranges 10(4) K less than or equal to [T-eff] less than or equal to 5 x 10(4) K and 1282 less than or equal to [L]/L. less than or equal to 8091 respectively. Separated helium and metal ionizing layers cause local maxima (or bumps) in the depth-dependent Rosseland mean opacity, which were found to be responsible for two distinct regions of pulsation instability; they will be referred to as the helium instability region (HeIR) and Z-bump instability region (ZBIR). At [L] < 3000 L. the cooler HeIR and hotter ZBIR are separated by a region of stability where radial pulsations are either unexcited or characterized by a very small radial displacement amplitude (Delta R/R similar to 10(-3)). At [L] = 1282 L. for example, the stability region is bounded by [T-eff](red) approximate to 12 000 K and [T-eff](blue) approximate to 17 500 K; it becomes narrower with increasing luminosity, so that both instability regions merge at [L] approximate to 3000 L. where [T-eff](red) approximate to [T-eff](blue) approximate to 15 000 K. All ZBIR models represent fundamental mode radial pulsators; for these cases an approximate formula was derived to express the pulsation constant (Q) in terms of [L] and the mean mass-radius ratio. BD + 13 degrees 3224 (V652 Her) remains the only hot extreme helium star known to be a radial pulsator; published photometric and spectroscopic data were used to test non-linear radial pulsation models and, indirectly, new opacities upon which they were based. The best, and remarkably good, agreement between observed and theoretical (radial velocity and luminosity) curves was obtained with M = 0.72 M., [T-eff] = 23 500 K and [L] = 1062 L. when OP opacities (with mixture X = 0.0015, Y = 0.98287 and Z = 0.01563) were adopted. As a consequence, BD + 13 degrees 3224 was identified as a ZBIR fundamental mode radial pulsator having a mean surface gravity of log g = 3.7.
