$\overline{\text{P}}$ANDA is a new experiment in hadron physics at the upcoming FAIR facility in Darmstadt, Germany. The combination of $\overline{\text{P}}$ANDA and the antiproton beam, provided by the antiproton storage ring HESR, yields high production rates of strange hyperon-antihyperon pairs. This enables multiple experiments in strangeness nuclear physics which allow to study the interaction of hyperons and antihyperons within nuclear matter. This is essential to understand the composition of neutron star matter and solve the ``hyperon puzzle''. The modularity of $\overline{\text{P}}$ANDA allows to design and integrate a dedicated setup for the high resolution X-ray and $\gamma$ spectroscopy of heavy $\Xi^-$ hyperatoms and double $\Lambda$ hypernuclei. The germanium detector array PANGEA (PANda GErmanium Array) is mandatory for these experiments. Its optimization and integration into the $\overline{\text{P}}$ANDA target spectrometer is discussed in this thesis. During the experiments at $\overline{\text{P}}$ANDA, the HPGe (High Purity Germanium) crystals of PANGEA will suffer from inevitable hadronic background. %during the experiments. Especially fast neutrons will damage the lattice structure of the crystal and deteriorate its resolution. This effect has been experimentally studied in irradiation tests at the COSY accelerator in J\"ulich, Germany, with up to \SI{5.6e9}{neutrons\per\centi\meter\squared}. A large fraction of the performance loss of the detector could be corrected by analyzing the pulse shape of the detector response. The initial crystal performance could be restored by annealing of the crystal in the laboratory after the irradiation. The effects of the irradiation had to be kept in mind when the feasibility of the hyperatom experiment was studied. $\overline{\text{P}}$ANDA is unique in its ability to study the $\Xi^-$ nucleon interaction in the neutron-rich periphery of $\Xi^-$-\ce{^{208}Pb} hyperatoms. Full simulations of the experiment show that multiple experimental observables will allow to measure the real and imaginary part of the $\Xi^-$ optical potential with a precision of \SI{\pm 1}{\mega\eV}.

The PANDA experiment (antiProton ANnihilation at DArmstadt) is one of the four key experiments to be operated at FAIR (Facility for Antiproton and Ion Research), which is currently under construction near Darmstadt/Germany. This fixed target experiment will address a wide range of open questions in the field of hadron physics. The detector consists of at target as well as a forward spectrometer to fully exploit the forward boosted collisions of antiprotons with dense hydrogen or nuclear targets. Phase-space cooled antiprotons with momenta in the range of 1.5 GeV/c to 15 GeV/c provided by the High Energy Storage Ring (HESR) allow for high precision line-shape scans. The ability to perform exclusive reconstruction of arbitrary final states enables a physics program including topics such as spectroscopy in the charmonium and open-charm region, proton structure, and hyperon and hypernuclear physics. The talk will give an overview of the PANDA experiment and highlight the most important aspects of the physics program.