An improved nuclear diagnostic method identifies a contained target material by measuring on-axis, mono-energetic uncollided particle radiation transmitted through a target material for two penetrating radiation beam energies, and applying specially developed algorithms to estimate a ratio of macroscopic neutron cross-sections for the uncollided particle radiation at the two energies, where the penetrating radiation is a neutron beam, or a ratio of linear attenuation coefficients for the uncollided particle radiation at the two energies, where the penetrating radiation is a gamma-ray beam. Alternatively, the measurements are used to derive a minimization formula based on the macroscopic neutron cross-sections for the uncollided particle radiation at the two neutron beam energies, or the linear attenuation coefficients for the uncollided particle radiation at the two gamma-ray beam energies. A candidate target material database, including known macroscopic neutron cross-sections or linear attenuation coefficients for target materials at the selected neutron or gamma-ray beam energies, is used to approximate the estimated ratio or to solve the minimization formula, such that the identity of the contained target material is discovered.