  Neutron captures and delayed decays of reaction products are common sources
of backgrounds in ultra-rare event searches. In this work, we studied
$^{13}$C($\alpha,n)^{16}$O reactions induced by $\alpha$-particles emitted
within the calibration sources of the \textsc{Majorana Demonstrator}. These
sources are thorium-based calibration standards enclosed in carbon-rich
materials. The reaction rate was estimated by using the 6129-keV $\gamma$-rays
emitted from the excited $^{16}$O states that are populated when the incoming
$\alpha$-particles exceed the reaction Q-value. Thanks to the excellent energy
performance of the \textsc{Demonstrator}'s germanium detectors, these
characteristic photons can be clearly observed in the calibration data.
Facilitated by \textsc{Geant4} simulations, a comparison between the observed
6129-keV photon rates and predictions by a TALYS-based software was performed.
The measurements and predictions were found to be consistent, albeit with large
statistical uncertainties. This agreement provides support for background
projections from ($\alpha,n$)-reactions in future double-beta decay search
efforts.
