Abstract We have performed combined photoionization and photodissociation region (PDR) modelling of a Crab nebula filament subjected to the synchrotron radiation from the central pulsar wind nebula, and to a high flux of charged particles; a greatly enhanced cosmic-ray ionization rate over the standard interstellar value, ζ0, is required to account for the lack of detected C i emission in published Herschel SPIRE FTS observations of the Crab nebula. The observed line surface brightness ratios of the OH+ and ArH+ transitions seen in the SPIRE FTS frequency range can only be explained with both a high cosmic-ray ionization rate and a reduced ArH+ dissociative recombination rate compared to that used by previous authors, although consistent with experimental upper limits. We find that the ArH+/OH+ line strengths and the observed H2 vibration–rotation emission can be reproduced by model filaments with nH = 2 × 104 cm−3, ζ = 107ζ0 and visual extinctions within the range found for dusty globules in the Crab nebula, although far-infrared emission from O i and C ii is higher than the observational constraints. Models with nH = 1900 cm−3 underpredict the H2 surface brightness, but agree with the ArH+ and OH+ surface brightnesses and predict O i and C ii line ratios consistent with observations. These models predict HeH+ rotational emission above detection thresholds, but consideration of the formation time-scale suggests that the abundance of this molecule in the Crab nebula should be lower than the equilibrium values obtained in our analysis.