We report a systematic first principles study on positron annihilation parameters in the $beta$-Ga$_2$O$_3$ lattice and Ga mono-vacancy defects complemented with orientation-dependent experiments of the Doppler broadening of the positron-electron annihilation. We find that both the $beta$-Ga$_2$O$_3$ lattice and the considered defects exhibit unusually strong anisotropy in their Doppler broadening signals. This anisotropy is associated with low symmetry of the $beta$-Ga$_2$O$_3$ crystal structure that leads to unusual kind of one-dimensional confinement of positrons even in the delocalized state in the lattice. In particular, the split Ga vacancies recently observed by scanning transmission electron microscopy produce unusually anisotropic positron annihilation signals. We show that in experiments, the positron annihilation signals in $beta$-Ga$_2$O$_3$ samples seem to be often dominated by split Ga vacancies.
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