A crystalline quark-hadron mixed phase in neutron stars

The mixed phase of a substance undergoing a first order phase transition has entirely different behavior according as the substance has more than one conserved charge or only one, as in the text book examples. In the latter case the pressure and nature of the phases are constants throughout the coexistence phase. For systems with more than one conserved charge (or independent component) we prove two theorems: (1) The pressure and the nature of the phases in equilibrium change continuously as the proportion of the phases varies from one pure phase to the other. (2) If one of the conserved charges is the Coulomb force, an intermediate-range order will be created by the competition between Coulomb and surface interface energy. Their sum is minimized when the coexistence phase assumes a Coulomb lattice of one phase immersed in the other. The geometry will vary continuously as the proportion of phases. We illustrate the theorems for a simple description of the hadron to quark phase transition in neutron stars and find a crystalline phase many kilometers thick. However the theorems are general and pertain to chemical mixtures, nuclear systems, either static as in stars or dynamic as in collisions, and have possible application to phase transitions in the early universe