In this master's thesis, the system ErTa$_7$O$_{19}$ is studied. ETO consists of a strongly frustrated 2D triangular lattice of magnetic Er$^{3+}$ ions (effective spin 1/2) with nearly Ising-like couplings, and it is believed that it possibly realizes a quantum liquid ground state. Experiments on ETO using elastic neutron scattering and muon spectroscopy yield conflicting results: neutrons indicate magnetic order at low temperatures, while muons detect no ordering. One leading explanation is that the muon, as a charged particle, locally distorts the crystal lattice. This explanation is tested using \textit{ab initio} simulations. Within density functional theory (DFT), muon stopping sites and their effects on the local atomic environment are determined. Four candidate sites are identified, of which only one matches the experimentally observed negative sign of the Knight shift. The formalism of Stevens operators is then used to compute changes in the crystal electric fields at the Er$^{3+}$ ions, first within the point-charge model (where all charge is assumed to be localized at ionic positions) and then within the distributed charge model (which accounts for the spatial distribution of electron density). A new procedure is introduced to compute only the corrections of the Stevens coefficients due to the muon from DFT results; these are added to experimental coefficients for the pristine crystal, yielding more accurate results than fully \textit{ab initio} estimates. Changes in the crystal-field levels and magnetic $g$ tensors are also calculated. Large differences between the two models and experimentally determined quantities are observed, suggesting that the muon sufficiently distorts the crystal to explain the discrepancy between muon and neutron experiments. Using DFT, the nuclear contribution to the low-temperature specific heat of ETO is additionally computed, enabling extraction of the purely magnetic contribution and a proper interpretation of the ground state of this quantum magnet.
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