Abstract
The phenomenon of shape coexistence and rapid shape phase transitions expected in deformed odd-$Z$ proton emitters from $Z=51$ to 75 are investigated using the triaxially deformed Nilsson potential and Strutinsky's prescription for the evaluation of shell correction. Calculations suggest prolate and triaxial shapes with large deformations except for a few oblate deformations, along with the occurrence of some coexisting configurations with different intrinsic shapes. Shape coexistence is observed in ${}^{113}$Cs with two energy minima for different shapes at similar energies. Location of the first proton unbound nuclei for odd-$Z$ rare-earth region is predicted and compared with the available experimental data.
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