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Rev. Lett."],"accepted":{"date-parts":[[2026,8,7]]},"abstract":"<jats:p>\n                    We solve the orientation recovery of a tumbling protein in the gas phase from single-event measurements of the spatial positions of its ions after an x-ray laser-induced explosion. We simulate diffracted x-ray signal and ion dynamics under experimental conditions and compare our method to conventional orientation recovery in single-particle imaging with x-ray free-electron lasers using only diffraction data. We reconstruct 3D diffraction intensities using orientations recovered from the ion signatures and retrieve the electron density with established phase-retrieval algorithms. We test our orientation recovery procedure on 56 proteins ranging from 14 to 52\u00a0kDa (1800 to 6500 atoms), achieving an angular error of around\n                    <a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\">\n                      <a:mrow>\n                        <a:mn>5<\/a:mn>\n                        <a:mi>\u00b0<\/a:mi>\n                      <\/a:mrow>\n                    <\/a:math>\n                    . The resulting 3D electron-density reconstructions are compared to ground-truth volumes simulated at the same nominal resolution and achieve the resolution at the edge of the detector in conditions similar to current single-particle imaging setups. We investigate the reconstruction quality and demonstrate that ion data can be used for reliable orientation recovery of particles in single-particle imaging, achieving orientation on par with or better than currently used recovery techniques. 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