A quantitative description of the change in ground-state neutron occupancies between $^{136}$Xe and $^{136}$Ba, the initial and final state in the neutrinoless double-$beta$ decay of $^{136}$Xe, has been extracted from precision measurements of the cross sections of single-neutron adding and -removing reactions. Comparisons are made to recent theoretical calculations of the same properties using various nuclear-structure models. These are the same calculations used to determine the magnitude of the nuclear matrix elements for the process, which at present disagree with each other by factors of 2 or 3. The experimental neutron occupancies show some disagreement with the theoretical calculations.
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