Aims. The HIFI instrument onboard Herschel has allowed high spectral resolution and sensitive observations of ground-state transi- tions of three molecular ions: the methylidyne cation CH+, its isotopologue 13CH+, and sulfanylium SH+. Because of their unique chemical properties, a comparative analysis of these cations provides essential clues to the link between the chemistry and dynamics of the diffuse interstellar medium. Methods. The CH+, 13CH+, and SH+ lines are observed in absorption towards the distant high-mass star-forming regions (SFRs) DR21(OH), G34.3+0.1, W31C, W33A, W49N, and W51, and towards two sources close to the Galactic centre, SgrB2(N) and SgrA*+50. All sight lines sample the diffuse interstellar matter along pathlengths of several kiloparsecs across the Galactic Plane. In order to compare the velocity structure of each species, the observed line profiles were deconvolved from the hyperfine structure of the SH+ transition and the CH+, 13CH+, and SH+ spectra were independently decomposed into Gaussian velocity components. To analyse the chemical composition of the foreground gas, all spectra were divided, in a second step, into velocity intervals over which the CH+, 13CH+, and SH+ column densities and abundances were derived. Results. SH+ is detected along all observed lines of sight, with a velocity structure close to that of CH+ and 13CH+. The linewidth distributions of the CH+, SH+, and 13CH+ Gaussian components are found to be similar. These distributions have the same mean (<deltau{psion}> ~ 4.2 km s-1) and standard deviation (sigma(deltau{psion}) ~ 1.5 km s-1). This mean value is also close to that of the linewidth distribution of the CH+ visible transitions detected in the solar neighbourhood. We show that the lack of absorption components narrower than 2 km s-1 is not an artefact caused by noise: the CH+, 13CH+, and SH+ line profiles are therefore statistically broader than those of most species detected in absorption in diffuse interstellar gas (e. g. HCO+, CH, or CN). The SH+/CH+ column density ratio observed in the components located away from the Galactic centre spans two orders of magnitude and correlates with the CH+ abundance. Conversely, the ratio observed in the components close to the Galactic centre varies over less than one order of magnitude with no apparent correlation with the CH+ abundance. The observed dynamical and chemical properties of SH+ and CH+ are proposed to trace the ubiquitous process of turbulent dissipation, in shocks or shears, in the diffuse ISM and the specific environment of the Galactic centre regions.
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