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A  -rigid solution of the Bohr Hamiltonian is derived for 0   utilizing the Davidson potential in the  variable. This solution is going to be called X(3)-D. The energy eigenvalues and wave functions are obtained by using an analytic method which has been developed by Nikiforov and Uvarov. BE(2) transition rates are calculated. A variational procedure is applied to energy ratios to determine whether or not the X(3) model is located at the critical point between spherical and deformed nuclei.

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In this study, the neutron and proton density distributions of 208 Pb nucleus have been analytically calculated by using a new, simple and physically meaningful analytical expression obtained for the Fermi integral which is situated in the nucleon density distributions. The results obtained for 208 Pb have been compared with those of the other theoretical and experimental data and found in good agreement.

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