Neutron-induced background on natural tellurium relevant to $^{\mathrm{130}}$Te 0$\nu \beta \beta $ decay searches at CUORE and SNO$+$

Gamma-ray production cross-section data have been obtained for the reactions $^{\mathrm{126,128,130}}$Te(n,n'$\gamma )$ at five mean neutron energies between 3.5 and 10 MeV. We report data for the $\gamma $-ray energy region relevant to 0$\nu \beta \beta $ decay of$^{\mathrm{\thinspace 130}}$Te with Q$_{\mathrm{\beta \beta }}$-value of 2527.515 keV. For CUORE only the $\gamma $-ray transitions of excited states of $^{\mathrm{130}}$Te at 2527.06 keV and of $^{\mathrm{126}}$Te at 2533.85 keV are of interest. For SNO$+$ with its inferior energy resolution, additional excited state decays of $^{\mathrm{130}}$Te levels at 2575.2, 2581.15, and 2607.33 keV, of $^{\mathrm{128}}$Te levels at 2494.20, 2508.06, 2516.64, 2550.52, 2571.17, 2587.14, 2598.99, and 2630.14 keV, and of $^{\mathrm{126}}$Te levels at 2496.83, 2503.55, 2577.822 and 2585.462 keV are important. The highest cross-section values were found for cascade $\gamma $-ray transitions to the ground state, while direct transitions to the ground state are very weak and have been observed only for $^{\mathrm{130}}$Te at 2607.31 keV, for $^{\mathrm{128}}$Te at 2508.04 keV, and for $^{\mathrm{126}}$Te at 2503.32 keV. However, both the CUORE and SNO$+$ detectors may not be able to distinguish between cascade transitions to the ground state and direct transitions, making especially the neutron-induced excitation of the 2527.06 keV state of $^{\mathrm{130}}$Te a potential problem for 0$\nu \beta \beta $ decay searches of $^{\mathrm{130}}$Te.