We perform atomic relativistic many-body calculations of the field isotope shifts and calculations of corresponding nuclear parameters for all stable even-even isotopes of Yb$^+$ ion. We demonstrate that if we take nuclear parameters of the Yb isotopes from a range of the state-the-art nuclear models which all predict strong quadrupole nuclear deformation, then calculated non-linearity of the King plot, caused by the difference in the deformation in different isotopes, is consistent with the non-linearity observed in the experiment (Ian Counts {em et al}, Phys. Rev. Lett. {bf 125}, 123002 (2020)). The changes of nuclear RMS radius between isotopes extracted from experiment are consistent with those obtained in the nuclear calculations.
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