Isotope effect in hydrogen diffusion in metals

Using a quantum barrier crossing model based upon inelastic phonon scatteri ng, we have derived a formula for the diffusion coefficient, D-i(T), for hy drogen 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 theo ry include theta(D), the Debye temperature of the metal lattice; theta(i), the H-atom local mode vibrational temperature; B, the electronic energy bin ding the H-atom to an interstitial site in the lattice; and D-i', a constan t 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 isot opic 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 sufficien tly 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 wh ere E-1 < E-2 < E-3, as in the case of Fe, V, Nb, and Ta, and find at suffi ciently 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, tha t the localization of the H-atom in the lattice is due to a stiff. low ener gy 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 incr easing isotopic mass.