An approach to the equation of state for the inner crust of neutron stars based on Skyrme-type forces is presented .
Working within the Wigner-Seitz picture , the energy is calculated by the TETF ( temperature-dependent extended Thomas-Fermi ) method , with proton shell corrections added self-consistently by the Strutinsky-integral method .
Using a Skyrme force that has been fitted to both neutron matter and to essentially all the nuclear mass data , we find strong proton shell effects : proton numbers Z = 50 , 40 and 20 are the only values possible in the inner crust , assuming that nuclear equilibrium is maintained in the cooling neutron star right down to the ambient temperature .

Convergence problems with the TETF expansion for the entropy , and our way of handling them , are discussed .
Full TETF expressions for the specific heat of inhomogeneous nuclear matter are presented .
Our treatment of the electron gas , including its specific heat , is essentially exact , and is described in detail .