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