Searches for magnetic fields in white dwarfs have clarified both the frequency of occurrence and the global structure of the fields found down to field strengths of the order of 500 kG. Below this level, the situation is still very unclear. We are studying the weakest fields found in white dwarfs to determine the frequency of such fields and their structure. We describe a very sensitive new method of measuring such fields in DA (H-rich) white dwarfs, and search for a field in the brightest such star, 40 Eri B. Our new method makes use of the strongly enhanced polarisation signal in the sharp core of Halpha. We find that with one-hour integrations with the high-resolution spectropolarimeter ESPaDOnS on the 3.6-m CFHT, we can reach a standard error fo the longitudinal field of about 85 G, the smallest error ever achieved for any white dwarf. Nevertheless, we do not detect a magnetic field in this star. Observations with ISIS at the WHT, and the Main Stellar Spectrograph at the SAO, support the absence of a field at somewhat lower precision. The new method is very efficient; it is shown that for suitable DA stars the integration time, with ESPaDOnS on a 3.6-m telescope, to reach a 500 G standard error on a white dwarf of V = 12.5, is about half an hour, about the same as the time required on an ESO 8-m telescope with FORS using conventional low-resolution spectropolarimetry.
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