This thesis describes SR observations of the internal magnetic field distribution n(B) as a function of temperature in LuNi2B2C ( , ), under a magnetic field of applied parallel to the crystal axis. The SR polarisation signal is fitted to a nonlocal London model, assuming a square vortex lattice. By incorporating first order nonlocal corrections, this model achieves significantly better fits than the local London model. The fitted penetration depth temperature dependence follows the form expected for a BCS s-wave superconductor, although the dependence is also consistent with a slight linear increase in the penetration depth with rising temperature. The rate of any such linear growth, however, is smaller than would be expected for an energy gap with line nodes. The fitted core radius temperature dependence reveals a Kramer-Pesch effect, or linear contraction of the vortex core radius upon cooling at low temperatures , that is weaker than predicted. The Kramer-Pesch effect found for this nearly three-dimensional superconductor is almost identical to that seen in quasi two-dimensional NbSe2, implying that quasiparticles behave similarly in LuNi2B2C and NbSe2 despite their different dimensionalities, and that longitudinal disorder of vortices has negligible impact on SR determinations of the vortex core radius . The surprisingly small magnitude of the Kramer-Pesch effect suggests that future theoretical work on the temperature dependence of vortex structure should consider zero point motion of vortices and intervortex interactions.