  He accretion onto carbon-oxygen white dwarfs (CO WDs) plays a fundamental
role when studying the formation of type Ia supernovae (SNe Ia). Employing the
MESA stellar evolution code, we calculated the long-term evolution of
He-accreting CO WDs. Previous studies usually supposed that a WD can grow in
mass to the Chandrasekhar limit in the stable He burning region and finally
produce a SN Ia. However, in this study we find that off-centre carbon ignition
occurs in the stable He burning region if the accretion rate is above a
critical value (~2.05*10^{-6}{M$_\odot$}{yr}^-1), resulting in
accretion-induced collapse rather than a SN Ia. If the accretion rate is below
the critical value, explosive carbon ignition will eventually happen in the
centre producing a SN Ia. Taking into account the possibility of off-centre
carbon ignition, we have re-determined the initial parameter space that
produces SNe Ia in the He star donor channel, one of the promising channels to
produce SNe Ia in young populations. Since this parameter space is smaller than
was found in the previous study of Wang et al. (2009), the SN Ia rates are also
correspondingly smaller. We also determined the chemical abundance profile of
the He-accreting WDs at the moment of explosive carbon ignition, which can be
used as initial input for SN Ia explosion models.
