Isotope effect in hydrogen diffusion in metals

Using a quantum barrier crossing model based upon inelastic phonon scattering, we have derived a formula for the diffusion coefficient, D(i)(T), for hydrogen atoms in metals, where T is the absolute temperature, and i = 1, 2, 3 runs over the isotropic mass number of the H-atom. Parameters in the theory include theta(D), the Debye temperature of the metal lattice; theta(i), the H-atom local mode vibrational temperature; B, the electronic energy binding the H-atom to an interstitial site in the lattice; and D(i)', a constant related to the high temperature limiting value, D(i)(infinity) of D(i)(T). By selection of the values for B, theta(i), and D(i)', consistent with their predicted variation (in the case of B, lack of variation) with H-atom isotopic mass, we are able to fit the measured isotopic mass dependence of D(i)(T) in the case of Fe, V, Nb, Hf, Ni, Cu, and Pd, at all temperatures and in the case of Ta at high temperatures. We draw the following conclusions: (1) When theta(i) > theta(D), plots of log D(i)(T) vs. 1/T made over sufficiently wide ranges of 1/T can curve upward with the curvature increasing with decreasing isotopic mass. (2) Defining a local Arrhenius activation energy, E(i), we find at sufficiently low temperatures a "normal" isotope effect where E(1) < E(2) < E(3), as in the case of Fe, V, Nb, and Ta, and find at sufficiently high temperatures an "inverse" isotope effect where E(3) < E(2) < E(1), as in the case of Ni, Cu, and Pd. (3) At intermediate temperatures, there are two cross-over regions, where E(3) < E(1) < E(2) and E(1) < E(3) < E(2). (4) By comparing values of B and theta(i), we find, for Fe, V, Nb, and Ta, that the localization of the H-atom in the lattice is due to a stiff. low energy bond, while for Hf, Ni, Cu, and Pd, it is due to a flexible, high energy bond. (5) For all the metals considered, D(i)(infinity) decreases with increasing isotopic mass.