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The first results of a new three-dimensional, finite temperatureSkyrme-Hartree-Fock+BCS study of the properties of inhomogeneous nuclear matterat densities and temperatures leading to the transition to uniform nuclearmatter are presented. Calculations are carried out in a cubic box representinga unit cell of the locally periodic structure of the matter. A constraint isplaced on the two independent components of the quadrupole moment of theneutron density in order to investigate the dependence of the totalenergy-density of matter on the geometry of the nuclear structure in the unitcell. This approach allows self-consistent modeling of effects such as (i)neutron drip, resulting in a neutron gas external to the nuclear structure,(ii) shell effects of bound and unbound nucleons, (iii) the variety of exoticnuclear shapes that emerge, collectively termed `nuclear pasta' and (iv) thedissolution of these structures into uniform nuclear matter as density and/ortemperature increase. In part I of this work the calculation of the propertiesof inhomogeneous nuclear matter in the core collapse of massive stars isreported. Calculations are performed at baryon number densities of $n_{\rm b}$= 0.04 - 0.12 fm$^{\rm -3}$, a proton fraction of $y_{\rm p}=0.3$ andtemperatures in the range 0 - 7.5 MeV. A wide variety of nuclear shapes areshown to emerge. It is suggested that thermodynamical properties changesmoothly in the pasta regime up to the transition to uniform matter; at thattransition, thermodynamic properties of the matter vary discontinuously.

Modeling Nuclear Pasta and the Transition to Uniform Nuclear Matter with  the 3D Skyrme-Hartree-Fock Method at Finite Temperature I: Core-Collapse  Supernovae