The BeEST (Beryllium Electron capture in Superconducting Tunnel junctions) experiment searches for physics beyond the standard model (BSM) in the neutrino sector through momentum conservation in electron capture decay (EC) of $^7$Be . For this purpose, $^7$Be$^+$ ions are directly implanted into high-resolution superconducting tunnel junction (STJ) quantum sensors , which allows the direct measurement of $^7$Li recoils after the $^7$Be decay. Despite the simple atomic structure of Li and Be, a precise model of the atomic relaxation following EC decay is essential to understand details of the spectrum and to distinguish BSM physics from known effects. In the first two phases of the experiment, some details on the spectra were found to disagree with simple models of the atomic structure. We have therefore started an effort to precisely model the atomic relaxation process, specifically the energy and the intensities of all possible transitions and of the shake-up and shake-off process after EC. To accomplish this, two complementary approaches have been applied, one using the Multiconfiguration Dirac-Fock and General Matrix Element (MCDFGME)  Code and another using the Flexible Atomic Code (FAC) . Here we present the steps towards the calculation of a full theoretical recoil spectrum that includes all details of the atomic relaxation process.
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