Aims: Imaging and spectroscopy of X-ray extended sources require a proper characterisation of a spatially unresolved background signal. This background includes sky and instrumental components, each of which are characterised by its proper spatial and spectral behaviour. While the X-ray sky background has been extensively studied in previous work, here we analyse and model the instrumental background of the ACIS-I detector on board the Chandra X-ray observatory in very faint mode. Methods: Caused by interaction of highly energetic particles with the detector, the ACIS-I instrumental background is spectrally characterised by the superimposition of several fluorescence emission lines onto a continuum. To isolate its flux from any sky component, we fitted an analytical model of the continuum to observations performed in very faint mode with the detector in the stowed position shielded from the sky, and gathered over the eight-year period starting in 2001. The remaining emission lines were fitted to blank-sky observations of the same period. We found 11 emission lines. Analysing the spatial variation of the amplitude, energy and width of these lines has further allowed us to infer that three lines of these are presumably due to an energy correction artefact produced in the frame store. Results: We provide an analytical model that predicts the instrumental background with a precision of 2% in the continuum and 5% in the lines. We use this model to measure the flux of the unresolved cosmic X-ray background in the Chandra deep field south. We obtain a flux of 10.2 0.5-0.4 × 10-13 erg cm-2 deg-2 s-1 for the [ 1 - 2 ] keV band and (3.8 ± 0.2) × 10-12 erg cm-2 deg-2 s-1 for the [ 2 - 8 ] keV band.
