We found that a heavy target such as Pb is most suitable for determining the proton distribution radii of unstable nuclei through charge-changing cross-section (${\sigma }_{\mathrm{cc}}$) measurements. Experimentally, low-$Z$ targets are routinely used to determine nucleon distribution radii of unstable isotopes. This approach has recently been extended to study proton distribution radii from ${\sigma }_{\mathrm{cc}}$ measurements. However, empirical scaling factors have to be introduced to apply the Glauber models. In the present Letter, we systematically investigated the scaling factor using 39 new ${\sigma }_{\mathrm{cc}}$ data of $18p$-shell nuclei on hydrogen, carbon, silver, and lead targets at around $240\mathrm{MeV}/\mathrm{nucleon}$. Together with the existing data, we reveal a universal dependence of the scaling factor on both the masses of target nuclei and the separation energies of projectile nuclei. The scaling factors decrease with increasing target-nucleus mass and converge to 1 for the highest-$Z$ target, making the scaling unnecessary. We conclude that instead of a low-$Z$ target, employing a heavy target such as Pb in ${\sigma }_{\mathrm{cc}}$ measurements is the best option to determine the proton distribution radii of unstable nuclei.

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• Nuclear reactions
• 20 ≤ A ≤ 38
• 39 ≤ A ≤ 58
• 6 ≤ A ≤ 19
• Radioactive beams