An X-ray detector with thin Si(Li) crystals has been fabricated and employed to detect the characteristic X-rays by positron impact. Use of thin detector crystals is essential for the measurements of the characteristic X-rays induced by positron impact. Otherwise the background produced in the crystals by the annihilation gamma-rays is too large to isolate the X-ray peaks. The data has been analyzed to obtain the inner-shell ionization cross sections by positron impact.