Structural Studies of Metastable and Ground State Vortex Lattice Domains in MgB$_2$*

Small-angle neutron scattering (SANS) studies of the vortex lattice (VL) in the type-II superconductor MgB$_2$ have revealed an unprecedented degree of metastability that is demonstrably not due to vortex pinning, [C. Rastovski $et$ $al.$, Phys. Rev. Lett. {\bf 111}, 107002 (2013)]. Application of an AC magnetic field to drive the VL to the ground state revealed a stretched exponential behavior in the metastable volume fraction as a function of the number of applied AC cycles. Here, we report on detailed structural studies of both metastable and ground state VL domains. These include measurements of VL correlation lengths as well as spatially resolved SANS measurements showing the VL domain distribution within the MgB$_2$ single crystal. We discuss these results and how they may help to resolve the mechanism responsible for stabilizing the metastable VL phases.