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Up to now, all charge radius measurements of the proton and deuteron assumeduniform spheroidal charge distribution. We investigate the nuclear deformationeffects on these charge radius measurements by assuming a uniform prolatecharge distribution for the proton and deuteron. We solve the energy levels ofthe corresponding muonic and electric atoms with such deformed nucleus andpresent how the purely quadruple deformation of proton and deuteron affectstheir Lamb shifts. The numerical results suggest that the deformation of protonand deuteron leads to that the charge radius extracted from the electronicmeasurement should be smaller than the corresponding one in the muonicmeasurement which assumed uniform spheroidal charge distribution. If thecentral values of newest measurements for the proton are adopted, the protonwould have a prolate structure with the 0.91 $\mathrm{fm}$ long axis and 0.73$\mathrm{fm}$ short axis. Further improved precise charge radius measurementsof the proton and deuteron will help us to pin down their shape deformation.

Nuclear deformation effects on charge radius measurements of the proton and deuteron