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The ionization fraction plays a key role in the chemistry and dynamics ofmolecular clouds. We study the H13CO+, DCO+ and HOC+ line emission towards theHorsehead, from the shielded core to the UV irradiated cloud edge, i.e., thePhotodissociation Region (PDR), as a template to investigate the ionizationfraction gradient in molecular clouds. We analyze a PdBI map of the H13CO+J=1-0 line, complemented with IRAM-30m H13CO+ and DCO+ higher-J line maps andnew HOC+ and CO+ observations. We compare self-consistently the observedspatial distribution and line intensities with detailed depth-dependentpredictions of a PDR model coupled with a nonlocal radiative transfercalculation. The chemical network includes deuterated species, 13Cfractionation reactions and HCO+/HOC+ isomerization reactions. The role ofneutral and charged PAHs in the cloud chemistry and ionization balance isinvestigated. The detection of HOC+ reactive ion towards the Horsehead PDRproves the high ionization fraction of the outer UV irradiated regions, wherewe derive a low [HCO+]/[HOC+]~75-200 abundance ratio. In the absence of PAHs,we reproduce the observations with gas-phase metal abundances, [Fe+Mg+...],lower than 4x10(-9) (with respect to H) and a cosmic-rays ionization rate ofzeta=(5+/-3)x10(-17) s(-1). The inclusion of PAHs modifies the ionizationfraction gradient and increases the required metal abundance. The ionizationfraction in the Horsehead edge follows a steep gradient, with a scale length of~0.05 pc (or ~25''), from [e-]~10(-4) (or n_e ~ 1-5 cm(-3)) in the PDR to a fewtimes ~10(-9) in the core. PAH^- anions play a role in the charge balance ofthe cold and neutral gas if substantial amounts of free PAHs are present ([PAH]>10(-8)).

The ionization fraction gradient across the Horsehead edge: An archetype for molecular clouds