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Schottky diode X-/ γ-ray detectors based on semi-insulating Cl-doped CdTe crystals have been developed and investigated. Both the Schottky and Ohmic contacts were formed by vacuum deposition of Ni electrodes on the opposite faces of (111) oriented CdTe crystals pretreated by Ar ion bombardment with different conditions. Record-low leakage current in the fabricated Ni/CdTe/Ni structure at high voltages (~5 nA at 300 K for the area of 10 mm2 at bias voltage of 1500 V) was achieved. The charge transport mechanisms in the detectors are interpreted on the basis of known theoretical models. The developed detectors have shown the record-high energy resolution in the measured spectra of 137Cs (0.42% FWHM). From a comparison of the spectra taken with the detector irradiated from the Schottky contact side and from the opposite side with an Ohmic contact, the concentration of uncompensated impurities (defects) in the CdTe crystals has been determined. The obtained value has been found to be close to the optimal one determined from the calculation results.