Context: The detection or absence of a polarisation signal in the diffuse interstellar band (DIB) profile can possibly give important clues on the identity of its carrier. For molecular gas-phase carriers the effect of polarisation on its electronic transitions depends, although uncertain how exactly, on its physical properties (e.g. molecular structure). On the other hand, if the carriers are grains, impurities or defects in grain mantles are also expected to show up in the polarisation of DIBs. Aims: Our aim is to detect a polarisation signal in a DIB profile or to derive stricter upper limits on the polarisation efficiency of DIB carriers. Methods: In order to detect and measure the linear and circular polarisation of the DIBs we observe reddened lines of sight showing continuum polarisation; for this study we select the stars HD 21219 , HD 198478 , HD 197770 , HD 183143 and HD 163472 . We use spectropolarimetry in the wavelength range 4480 to 6620 A with the MuSiCoS echelle spectrograph mounted at the Telescope Bernard Lyot. Results: Linear polarisation spectra are constructed in order to search for a polarisation signal in the DIB profiles. No significant change in the linear polarisation degree is found. We obtain a 2sigma polarisation detection limit (per DIB FWHM) of 0.01-0.04% for HD 21219 , 0.04-0.14% for HD 197770 , 0.01-0.14% for HD 183143 and 0.01-0.14% for HD 198478 , for the six investigated DIBs. We derive upper limits for the polarisation efficiency factor f for six strong narrow DIBs; f_max of 0.31, 0.44, 0.45, 0.18, 0.47 and 0.68 for the lambdalambda 5780, 5797, 6196, 6284, 6379 and 6613 DIBs, respectively. The derived detection limits are similar to those derived by Adamson & Whittet (1995), although for more lines of sight and more DIBs. Circular polarisation (Stokes V) spectra of high signal-to-noise have been obtained for the first time for the 5780, 5797, 6196, 6203, 6284, 6376, 6379 and 6613 A DIB profiles. No circular polarisation signal is detected. The 2sigmaV (per 0.1 A) noise level limits are 1.0-2.5% for the DIBs in the line of sight towards HD 197770 and 0.06-0.10% for the DIBs towards HD 163472 . Conclusions: . The lack of polarisation of the DIB profiles is consistent with DIB carriers that are not directly related to solid features, like impurities in grain mantles. However, large (carbonaceous) gas phase molecules remain viable DIB carrier candidates. Based on observation at the Observatoir du Pic du Midi, France with the MuSiCoS spectropolarimeter.
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