Fine structure features in very low-energy electron diffraction (VLEED) near beam thresholds are sensitive to the form of the surface barrier potential. The VLEED features give information about the barrier for an electron incident on the barrier from the crystal side almost parallel to the surface and with total energy above the barrier height. Theoretically, the barrier form is known to depend on the kinetic energy and angle of approach, that is the momentum, of the incoming electron, but the various theoretical methods of calculation presently provide information which is quantitatively different. The calculation of the fine structure features from models which do not take into account the energy dependence of the barrier height and saturation for the electrons involved in VLEED have not provided consistent and definitive information about the barrier form. It is shown that this information can be extracted from a large data set over an extended energy range 8-19 eVon Cu(0 0 1). To obtain the required information, it is necessary to have overlapping energy ranges which cover all of the vacuum emergence energies for at least one specific VLEED electron beam.