XMASS-I is a single-phase liquid xenon detector whose purpose is direct detection of dark matter. To achieve the low background requirements necessary in the detector, a new model of photomultiplier tubes (PMTs), R10789, with a hexagonal window was developed based on the R8778 PMT used in the XMASS prototype detector. We screened the numerous component materials for their radioactivity. During development, the largest contributions to the reduction of radioactivity came from the stem and the dynode support. The glass stem was exchanged to the Kovar alloy one and the ceramic support were changed to the quartz one. R10789 is the first model of Hamamatsu Photonics K. K. that adopted these materials for low background purposes and provided a groundbreaking step for further reductions of radioactivity in PMTs. Measurements with germanium detectors showed 1.2$pm$0.3 mBq/PMT of $^{226}$Ra, less than 0.78 mBq/PMT of $^{228}$Ra, 9.1$pm$2.2 mBq/PMT of $^{40}$K, and 2.8$pm$0.2 mBq/PMT of $^{60}$Co. In this paper, the radioactive details of the developed R10789 are described together with our screening methods and the components of the PMT.
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