Translationally invariant treatment of the charge density in nuclei

An expression is derived for the translationally invariant (transition-) charge density in the center of mass rest frame of the nucleus in between arbitrary wave functions. An explicit formulation is given for the charge density of spherically symmetric Hartree-Fock type ground states. The method is applied to the nuclei⁴He,¹⁶O,⁴⁰Ca and⁹⁰Zr using Woods-Saxon single nucleon orbits for the construction of the ground state wave functions. For the two lighter nuclei in addition realistic Hartree-Fock wave functions are investigated. The results are compared to those obtained with various approximate treatments of the center of mass motion. It turns out that at least in the lighter nuclei the usual center of mass correction, which is based on the assumption of pure non-spurious oscillator configurations and commonly used in the analysis of, e.g., electron scattering data, does not produce reliable (or even predictable) results. Thus here indeed the full translationally invariant treatment seems to be required. For the heavy nucleus⁹⁰Zr on the other hand, as expected, the center of mass effects become negligibly small.