Hopping of ions in ice

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Abstract
The effects of intermolecular tunneling by protons in ice and other protonic semiconductors on thermodynamic and transport properties are estimated on the basis of an idealized model. The model involves a simple tight‐binding Hamiltonian on the infinite‐dimensional set of molecular configurations in the generally proton‐bonded but otherwise disordered structure. The cycle‐poor topology of the state set is approximated by that of a cycle‐free Bethe lattice, i.e., an infinite, homogeneous Cayley tree. For coordination q and hopping matrix element V the distribution of energy levels is given by the function g(u) = g(E/V) = q[4(q − 1) − u2]1/2/2π(q2 − u2), where q = 3 for the set of hopping options available to an ion in ice. The thermal average of the group velocity v = [4(q − 1) − u2]1/2 V d / ℏ on the Bethe lattice with lattice spacings d determines a finite coefficient of diffusion in real three‐dimensional space, where paths on the Bethe lattice are represented by random walks in 3 space with only a finite measure of correlation between the directions of successive steps. These results agree with recent computations by Minagawa and with the results of various parallel efforts in the theory of electron tunneling. Some questions of principle are resolved by an analysis of the corresponding eigenvalue problem for a symmetrically constructed finite Cayley tree, and an effective upper bound for the error incurred by disregarding cycles is obtained from a computation for a periodic graph in three dimensions. While the ionic mobilities in ice are not yet well known, even the greatest claimed values of about 0.075 cm2/V · sec are compatible with matrix elements somewhat smaller than 1 mV, which would entail tunneling corrections to the partition function for a hydrogen ion of less than 2% near the freezing point.

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