Energy loss of slow protons in solids: Deviation from the proportionality with projectile velocity

Recent experimental determinations [J. E. Valdes, G. Martinez-Tamayo, G. H. Lantschner, J. C. Eckardt, and N. R. Arista, Nucl. Instrum. Methods B 73, 313 (1993)] of the energy loss of low-energy protons in Ag and Au show a departure from the velocity proportionality predicted by theory, while data for other metallic elements, such as Al, Sb, and Bi, are in agreement with this dependence. In this work we present measurements in Cu foils which show a similar behavior to that in Ag and Au. We give an interpretation of the differences between these cases due to the existence of small binding energies for the d electrons in Cu, Ag, and Au. We present a model based on transport cross section and density-functional theory, which introduces threshold effects in the calculation of the transport cross section of nonfree electrons, and explains the deviations from the velocity-proportional dependence