Neutron-irradiated quartz is a promising model system to learn more about t
he low-energy excitations (tunneling states) in vitreous silica and in simi
lar glasses. We present the first systematic study of the elastic propertie
s of neutron-irradiated quartz at low frequencies and very low temperatures
. Using the vibrating reed technique at frequencies of several kHz we have
measured the sound velocity and internal friction of six quartz crystals ir
radiated with different neutron doses over a wide range of temperatures (7
mK-300 K). The results are analyzed using the tunneling model and several e
xtensions of this theory. Comparisons are made with recent low-frequency me
asurements on normal and compacted vitreous silica and with ultrasound expe
riments on neutron-irradiated quartz.