From superconductors to atomic nuclei, strongly-interacting many-body systems are ubiquitous in nature. Measuring the microscopic structure of such systems is a formidable challenge, often met by particle knockout scattering experiments. While such measurements are fundamental for mapping the structure of atomic nuclei, their interpretation is often challenged by quantum mechanical initial- and final-state interactions (ISI/FSI) of the incoming and scattered particles. Here we overcome this fundamental limitation by measuring the quasi-free scattering of 48 GeV/c 12C ions from hydrogen. The distribution of single protons is studied by detecting two protons at large angles in coincidence with an intact 11B nucleus. The 11B detection is shown to select the transparent part of the reaction and exclude the otherwise large ISI/FSI that would break the 11B apart. By further detecting residual 10B and 10Be nuclei, we also identified short-range correlated (SRC) nucleon-nucleon pairs, and provide direct experimental evidence for the separation of the pair wave-function from that of the residual many-body nuclear system. All measured reactions are well described by theoretical calculations that do not contain ISI/FSI distortions. Our results thus showcase a new ability to study the short-distance structure of short-lived radioactive atomic nuclei at the forthcoming FAIR and FRIB facilities. These studies will be pivotal for developing a ground-breaking microscopic understanding of the structure and properties of nuclei far from stability and the formation of visible matter in the universe.
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