  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, $\zeta_0$, 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 H$_2$ vibration-rotation emission can be reproduced by model
filaments with $n_{\rm{H}} = 2 \times 10^4$ cm$^{-3}$, $\zeta = 10^7 \zeta_0$
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 $n_{\rm{H}} = 1900$ cm$^{-3}$
underpredict the H$_2$ 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 timescale suggests
that the abundance of this molecule in the Crab Nebula should be lower than the
equilibrium values obtained in our analysis.
