  We report on precision measurements of spin-dependent interaction-strengths
in the 87Rb spin-1 and spin-2 hyperfine ground states. Our method is based on
the recent observation of coherence in the collisionally driven spin-dynamics
of ultracold atom pairs trapped in optical lattices. Analysis of the Rabi-type
oscillations between two spin states of an atom pair allows a direct
determination of the coupling parameters in the interaction hamiltonian. We
deduce differences in scattering lengths from our data that can directly be
compared to theoretical predictions in order to test interatomic potentials.
Our measurements agree with the predictions within 20%. The knowledge of these
coupling parameters allows one to determine the nature of the magnetic ground
state. Our data imply a ferromagnetic ground state for 87Rb in the f=1
manifold, in agreement with earlier experiments performed without the optical
lattice. For 87Rb in the f=2 manifold the data points towards an
antiferromagnetic ground state, however our error bars do not exclude a
possible cyclic phase.
