  We have investigated the Aharonov-Bohm effect in mesoscopic semiconductor
GaAs/GaAlAs rings in low magnetic fields. The oscillatory magnetoconductance of
these systems is systematically studied as a function of electron density. We
observe phase-shifts of $\pi$ in the magnetoconductance oscillations, and
halving of the fundamental $h/e$ period, as the density is varied.
Theoretically we find agreement with the experiment, by introducing an
asymmetry between the two arms of the ring.
