Protons, quantum jumps

If one is going to consider protons actually jumping from one quasi-stationary water molecule to the next, the classical view would be to ask what fraction of them would be sufficiently activated to get over the top of the corresponding energy barrier. Another possibility was discussed not long after the introduction of quantum mechanics in 1928 by Bernal and Fowler. In a famous paper of 1933, they applied quantum mechanics to the possibility of tunneling through a barrier. 

The two-site jump model appears to work well for this polymer. However, it should be noted that the density of transitions is large here due to the larger spin quantum number of deuterium (/= 1), the fact that there are three of them in the isotopically substituted polymeric radical, and that the coupling constant for each deuterium is smaller by a factor of 6.4 compared to the protonated radical. Coupling these facts to the visual fitting process, these fits may not be unique. In fact, when the same model is applied to the temperature dependence of the protonated PMMA spectra (Fig. 14.2), reasonable visual fits could not be obtained with this model. Deuteration of the... 

A pair of ion states H3O+ and OH is formed, at least from a formal point of view, when a proton jumps from the normal end of a bond to a position near the other end as shown in fig. 7.1 a, thus violating rule (ii). According to the semi-empirical calculations of Lippin-cott Schroeder (1955) plotted in fig. 2.6 there is a subsidiary potential minimum for a proton at this position on the bond, though the simplified quantum-mechanical treatment of Weissmann ... 

It is, of course, not possible to include a statistical weighting function exp — WJkT) directly in the proton wave function, since this must actually be inserted in the density matrix. The total jump rate, however, amounts essentially to a summation of quantum contributions like (9.68) from energy levels below the barrier top together with a classical contribution like (9.59) frorh higher levels. 

Tunnelling spectroscopy is unique to observing quantum nonlinear dynamics in crystals. Evidence for proton transfer along hydrogen bonds is another outstanding contribution of INS. It is another manifestation of the decoupling of proton dynamics from the crystal lattice. The quantum nature of proton transfer dynamics even at room temperature is quite unforeseen and contrasts with mechanisms based on semiclassical diffusion jumps. 

Similar considerations may be applied to free transport of protons (cf. Fig.5.11). For dilute solutions of protons in an oxide essentially all nearest neighbour oxygen ions are available, and thus in this case Nd is unity. However, the specification of Z, s and co is not straightforward in this case. The dynamics of free proton diffusion in oxides are complicated by 1) the multistep process (jump+rotation), 2) the dependency on the dynamics of the oxygen ion sublattice, and 3) the quantum mechanical behaviour of a light particle such as the proton. 

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