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Quasi-elastic (QEL) scattering measurements have been performed using$^{28}$Si projectile off a $^{90}$Zr target at energies around the Coulombbarrier. A Bayesian analysis within the framework of coupled channels (CC)calculations is performed in a large parameter space of quadrupole andhexadecapole deformations ($\beta_{2}$ and $\beta_{4}$) of $^{28}$Si. Ourresults clearly show that $^{28}$Si is an oblate shaped nucleus with$\beta_{2}$=-$0.38 \pm 0.01$ which is in excellent agreement withelectromagnetic probes. A precise value of hexadecapole deformation for$^{28}$Si, $\beta_{4}$=+$0.03 \pm 0.01$, along with a consistent value ofquadrupole deformation has now been determined for the first time using QELscattering. A remarkable agreement between the experimental $\beta_{4}$ valueof $^{28}$Si and Skyrme-Hartree-Fock based calculations is obtained. The QELresults obtained previously for $^{24}$Mg (prolate) and the present result for$^{28}$Si (oblate) hereby affirm the strong sensitivity of the quasi-elasticscattering to ground state deformations, thus reinforcing its suitability as apotential probe for rare exotic nuclei.

Precise determination of quadrupole and hexadecapole deformation parameters of the $sd$-shell nucleus, $^{28}$Si