Pressurized drift-tube chambers are efficient detectors for high-precision tracking over large areas. The Monitored Drift-Tube (MDT) chambers of the muon spectrometer of the ATLAS detector at the Large Hadron Collider (LHC) reach a spatial resolution of 35 micons and almost 100% tracking efficiency with 6 layers of 30 mm diameter drift tubes operated with Ar:CO2 (93:7) gas mixture at 3 bar and a gas gain of 20000. The ATLAS MDT chambers are designed to cope with background counting rates due to neutrons and gamma-rays of up to about 300 kHz per tube which will be exceeded for LHC luminosities larger than the design value of 10-34 per square cm and second. Decreasing the drift-tube diameter to 15 mm while keeping the other parameters, including the gas gain, unchanged reduces the maximum drift time from about 700 ns to 200 ns and the drift-tube occupancy by a factor of 7. New drift-tube chambers for the endcap regions of the ATLAS muon spectrometer have been designed. A prototype chamber consisting of 12 times 8 layers of 15 mm diameter drift tubes of 1 m length has been constructed with a sense wire positioning accuracy of 20 microns. The 15 mm diameter drift-tubes have been tested with cosmic rays in the Gamma Irradiation Facility at CERN at counting rates of up to 1.85 MHz.
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