We present detailed calculations of the electric field gradient (EFG) using a point charge approximation in Ba$_2$NaOsO$_6$, a Mott insulator with strong spin-orbit interaction. Recent $^{23}$Na nuclear magnetic resonance (NMR) measurements found that the onset of local point symmetry breaking, likely caused by the formation of quadrupolar order, precedes the formation of long range magnetic order in this compound. An extension of the static $^{23}$Na NMR measurements as a function of the orientation of a 15 T applied magnetic field at 8 K in the magnetically ordered phase is reported. Broken local cubic symmetry induces a non-spherical electronic charge distribution around the Na site and thus finite EFG, affecting the NMR spectral shape. We combine the spectral analysis as a function of the orientation of the magnetic field with calculations of the EFG to determine the exact microscopic nature of the lattice distortions present in low temperature phases of this material. We establish that orthorhombic distortions, constrained along the cubic axes of the perovskite reference unit cell, of oxygen octahedra surrounding Na nuclei are present in the magnetic phase. Other common types of distortions often observed in oxide structures are considered as well.
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