The crust of a neutron star accounts for less than 1% of its mass, however any information of the core gets filtered by the crust. Additionally, a good understanding of the crust is important in order to understand the phenomenon of glitches, to construct models of neutron stars, and to predict order of magnitude of gravitational waves emitted by neutron stars. The crust is a Coulomb crystal formed by neutron rich ions arranged in a lattice of embedded in a relativistic background of electrons. Using the Wigner-Seitz approximation to model the structure of the crystal, I apply relativistic mean field formalism to calculate the composition and the equation of state of the crust of a cold, isolated neutron star. I find the average composition in the crust as a function of density and use the obtained Equation of State to model neutron stars.

We also study fusion reactions in the crust of accreting neutron stars. These reactions are an important source of heat, and the depth at which these reactions occur is important to determine the temperature profile of the star. Fusion reactions depend strongly on the atomic number Z; nuclei with Z < 6 can fuse at low densities in the liquid ocean, however, nuclei with Z = 8 or higher may not burn until higher densities where the crust is solid and electron capture reactions result in neutron-rich nuclei. We calculate the astrophysical S factor for fusion reactions of neutron rich nuclei, including ²⁴O+ ²⁴O and ²⁸Ne+²⁸Ne, using a simple barrier penetration model. We calculate the rate of thermonuclear fusion for ²⁴O+²⁴O and find that ²⁴O should burn at densities near 10¹¹g/cm³. The energy released from this and similar reactions may be important for the temperature profile of the star.