  We prepare and study a two-component Mott insulator of bosonic atoms with two
particles per site. The mapping of this system to a magnetic spin model, and
the subsequent study of its quantum phases, require a detailed knowledge of the
interaction strengths of the two components. In this work, we use radio
frequency (RF) transitions and an on-site interaction blockade for precise,
empirical determination of the interaction strengths of different combinations
of hyperfine states on a single lattice site. We create a map of the
interactions of the lowest two hyperfine states of $^7$Li as a function of
magnetic field, including measurements of several Feshbach resonances with
unprecedented sensitivity, and we identify promising regions for the
realization of magnetic spin models.
