The shell correction and self-consistent deformation energies

The Hartree-Fock deformation energy of the nucleus is represented as the sum of two terms one of which (E^σ) is due to the re-distribution of the nuclear density and depends on the microscopically non-self-consistent parameters σ of the nuclear shape. The other component (E^π) is related to the coherent distortion of the quasiparticle wave functions in the occupied states and is the same as the deformation energy considered in theories of microscopic vibrations for a fixed quasiparticle distribution. Quantities averaged over the particle-hole distribution are introduced which satisfy the condition of statistical self-consistency. It is shown that the shell correction energy represents the averaged effect of the re-distribution of the single-particle states. Finally, corrections are formulated for the shell-model potential which does not satisfy the condition of statistical self-consistency.