Can There be Quark Matter Core in a Strongly Magnetized Neutron Star?

The effect of strong quantizing magnetic field on the nucleation of quark matter droplets and on the chemical evolution of nascent quark phase at the core of a neutron star are investigated. The surface energy of quark phase diverges logarithmically. As a consequence there can not be a first order transition to quark phase. However, a metal-insulator type of second order transition is possible unless the field strength exceeds ${10}^{20}$G. The study of chemical evolution of newborn quark phase shows that in $\beta$-equilibrium the system becomes energetically unstable.