  We study matter wave scattering from an ultracold, many body atomic system
trapped in an optical lattice. We determine the angular cross section that a
matter wave probe sees and show that it is strongly affected by the many body
phase, superfluid or Mott insulator, of the target lattice. We determine these
cross sections analytically in the first Born approximation, and we examine the
variation at intermediate points in the phase transition by numerically
diagonalizing the Bose Hubbard Hamiltonian for a small lattice. We show that
matter wave scattering offers a convenient method for non-destructively probing
the quantum many body phase transition of atoms in an optical lattice.
