Quasi-elastic processes of the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mrow /><mml:mn>48</mml:mn></mml:msup></mml:math>Ca + <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mrow /><mml:mn>120</mml:mn></mml:msup></mml:math>Sn system and the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msup><mml:mrow /><mml:mn>48</mml:mn></mml:msup></mml:math>Ca nuclear matter density

We present the results of a high-precision quasi-elastic excitation function measurement for the ${}^{48}$Ca + ${}^{120}$Sn system at ${\ensuremath{\theta}}_{\mathrm{LAB}}$ = 160\ifmmode^\circ\else\textdegree\fi{} at near-barrier energies in steps of 1.0 MeV. The corresponding quasi-elastic barrier distribution is derived. A large-scale coupled-channel calculation was performed to investigate the role of several reaction channels in the reaction mechanism. An excellent agreement between theory and data was obtained for the barrier distribution. The first quadrupole vibrations of the ${}^{48}$Ca and ${}^{120}$Sn, the 2$n$, and the ${}^{4}$He transfers have a strong influence on the reaction mechanism and are responsible for the good agreement achieved. The 1$n$ transfer has a minor importance in the result when compared with the 2$n$ transfer, which suggests that the pairing correlation might play an important role in the 2$n$-neutron transfer process. However, if the octupole vibration of the projectile is included in the coupling scheme, the agreement with the data gets worse. The comparison of the coupled-channel calculations with experimental data leads to the conclusion that the nuclear matter diffuseness of the ${}^{48}$Ca nucleus is 0.56 fm in agreement with most of the double-magic nuclei.