We present a detailed spectroscopic analysis of 115 helium-line (DB) and 28 cool, He-rich hydrogen-line (DA) white dwarfs based on atmosphere fits to optical spectroscopy and photometry. We find that 63% of our DB population show hydrogen lines, making them DBA stars. We also demonstrate the persistence of pure DB white dwarfs with no detectable hydrogen feature at low effective temperatures. Using state-of-the art envelope models, we next compute the total quantity of hydrogen, M H, that is contained in the outer convection zone as a function of effective temperature and atmospheric H/He ratio. We find that some (T eff, M H) pairs cannot physically exist as a homogeneously mixed structure; such a combination can only occur as stratified objects of the DA spectral type. On that basis, we show that the values of M H inferred for the bulk of the DBA stars are too large and incompatible with the convective dilution scenario. We also present evidence that the hydrogen abundances measured in DBA and cool, helium-rich white dwarfs cannot be globally accounted for by any kind of accretion mechanism onto a pure DB star. We suggest that cool, He-rich DA white dwarfs are most likely created by the convective mixing of a DA star with a thin hydrogen envelope; they are not cooled down DBAs. We finally explore several scenarios that could account for the presence of hydrogen in DBA stars.
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