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Quasi-elastic scattering measurements have been performed using $^{16}$O and$^{24}$Mg projectiles off $^{90}$Zr at energies around the Coulomb barrier.Experimental data have been analyzed in the framework of coupled channels (CC)calculations using the code CCFULL. The quasi-elastic scattering excitationfunction and derived barrier distribution for $^{16}$O + $^{90}$Zr reaction arewell reproduced by the CC calculations using the vibrational coupling strengthsfor $^{90}$Zr reported in the literature. Using these vibrational couplingstrengths, a Bayesian analysis is carried out for $^{24}$Mg + $^{90}$Zrreaction. The $\beta_{2}$ and $\beta_{4}$ values for $^{24}$Mg are determinedto be $+0.43 \pm 0.02$ and $ - 0.11 \pm 0.02$, respectively. The $\beta_{2}$parameter determined in the present work is in good agreement with resultsobtained using inelastic scattering probes. The hexadecapole deformation of$^{24}$Mg has been measured very precisely for the first time. Present resultsestablish that quasi-elastic scattering could provide a useful probe todetermine the ground state deformation of atomic nuclei.

Determination of hexadecapole ($\beta_{4}$) deformation of the light-mass nucleus $^{24}$Mg using quasi-elastic measurement