We report a systematic study of the anomalous rapid oscillation ͑RO͒ phenomena in the quasi-onedimensional organic metal ͑TMTSF͒ 2 ClO 4 in pulsed magnetic fields up to 51 T. We argue that the temperature and magnetic-field dependence of the RO amplitudes in the high-field state result from the reconstructed, nested Fermi surface topology at low temperatures in high magnetic fields. In this topology, the RO amplitudes depend on competing magnetic breakdown and Bragg reflection probabilities, along with Lifshitz-Kosevich reduction factors. ͓S0163-1829͑96͒01422-1͔Organic conductors based on the TMTSF ͓tetramethyltet-raselenafulvalene͔ molecule are model systems in which to investigate the fundamental physics of low-dimensional electronic systems. 1 When slowly cooled through its anion ordering transition at 24 K, ͑TMTSF͒ 2 ClO 4 exhibits a number of magnetic-field-dependent phases, as shown in the inset of Fig. 1. Here the magnetic field is parallel to the c* axis, the least conducting direction. At ambient pressure ͑TMTSF͒ 2 ClO 4 is a superconductor below 1.3 K and up to 30 mT. A cascade of phase transitions into field-inducedspin-density-wave ͑FISDW͒ phases occurs at fields above 4 T. Near 15 T the second-order FISDW transition temperature reaches a maximum of Ϸ5.5 K. Another phase boundary is present at B RE ϳ28 T, and the high-field state ͑HFS͒ above B RE is also known as the reentrant phase. 2 Recently extensive studies of transport and magnetization have indicated that the B RE line is not connected to the second-order FISDW line, and hence the HFS is not a reentrant metallic state. 3 Aside from the complication of the B RE phase boundary, the HFS strongly represents the final Nϭ0 nested state predicted by the ''standard model. '' 1 As a function of field quantum oscillations, reminiscent of de Haas-van Alphen and Shubnikov-de Haas ͑SdH͒ oscillations seen in conventional metals, are observed in transport, 4-6 thermodynamic, 4,7,8 far-infrared reflectivity, 9 and acoustic 10 properties. In Figs. 1͑a͒ and 1͑b͒ these oscillations are clearly seen, superimposed on the background magnetoresistance, for transport in both the a-axis and c*-axis directions in the title material. However, since the zero-field Fermi surface is open, their origin must arise from some explanation other than quantum oscillations seen in standard closed orbit quasi-two-dimensional metals. Although a number of theories have been proposed for the rapid oscillations 11-15 none are completely consistent, even qualitatively, with all the effects that are observed.In this paper we explore the rapid oscillation ͑also known as ''RO''͒ behavior in the high-field state in ͑TMTSF͒ 2 ClO 4 . Previous pulsed-field studies by Agosta et al. 16 ͑see also Osada et al. 17 ͒ have shown that the more well ordered the sample is ͑the ClO 4 anions order at about 24 K͒, the greater is the ratio of the first-to-second harmonic amplitude, as seen in the Fourier transform ͑FT͒ spectrum. Although the temperature dependence of the FT amplitude of both the first ͑F...