APEX observations of several transitions of N₂H⁺ and N₂D⁺ towards the dust peak of the pre-stellar core I16293E (black histograms) and results of the LOC non-LTE radiative transfer model (red curve). The physical model of I16293E computed from Herschel SPIRE and PACS data is shown in the upper right panel. The black curve (left y-axis) shows the total volume density $\left({\text{n}}_{H_2}\right)$, the red curve (right y-axis) shows the dust temperature, and the vertical dashed line indicates the 3000 au (21″) inner radius of the infalling envelope (see Sect. 3).