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The KATRIN experiment is designed for a direct and model-independentdetermination of the effective electron anti-neutrino mass via a high-precisionmeasurement of the tritium $\beta$-decay endpoint region with a sensitivity on$m_\nu$ of 0.2$\,$eV/c$^2$ (90% CL). For this purpose, the $\beta$-electronsfrom a high-luminosity windowless gaseous tritium source traversing anelectrostatic retarding spectrometer are counted to obtain an integral spectrumaround the endpoint energy of 18.6$\,$keV. A dominant systematic effect of theresponse of the experimental setup is the energy loss of $\beta$-electrons fromelastic and inelastic scattering off tritium molecules within the source. Wedetermined the \linebreak energy-loss function in-situ with a pulsedangular-selective and monoenergetic photoelectron source at varioustritium-source densities. The data was recorded in integral and differentialmodes; the latter was achieved by using a novel time-of-flight technique.  We developed a semi-empirical parametrization for the energy-loss functionfor the scattering of 18.6-keV electrons from hydrogen isotopologs. This modelwas fit to measurement data with a 95% T$_2$ gas mixture at 30$\,$K, as used inthe first KATRIN neutrino mass analyses, as well as a D$_2$ gas mixture of 96%purity used in KATRIN commissioning runs. The achieved precision on theenergy-loss function has abated the corresponding uncertainty of$\sigma(m_\nu^2)<10^{-2}\,\mathrm{eV}^2$ [arXiv:2101.05253] in the KATRINneutrino-mass measurement to a subdominant level.

Precision measurement of the electron energy-loss function in tritium and deuterium gas for the KATRIN experiment