Tunnelling, through barrier

From the Gamov formula, calculate the probabilities for an electron and for a proton to tunnel through barriers of 1 and 10 A thickness with a height of 1 eV. 

Why is it that one regards the proton as different from all other ions There are three reasons, all connected with its tiny size and small mass (1) The tiny size means that such an ion can go anywhere (e.g., diffusing in Pd). (2) Its small mass turns out to give rise to a mechanism of motion in solution quite different from that of any other ion (except its isotope, the deuteron). (3) In quantum mechanical tunneling (see also Chapter 9), low mass is a vital factor. The electron, the mass of which is nearly 2000 times less than that of a proton, can easily tunnel through barriers more than 2000 pm in thickness. The ability of the proton to tunnel is much less than that of the electron. 

The obvious defect of classical trajectories is that they do not describe quantum effects. The best known of these effects is tunnelling through barriers, but there are others, such as effects due to quantization of the reagents and products and there are a variety of interference effects as well. To circumvent this deficiency, one can sometimes use semiclassical approximations such as WKB theory. WKB theory is specifically for motion of a particle in one dimension, but the generalizations of this theory to motion in three dimensions are known and will be mentioned at the end of this section. More complete descriptions of WKB theory can be found in many standard texts [1, 2, 3, 4 and 5, 18]. 

The transmission factor k has received little discussion in enzyme reactions. A restricted k is theoretically expected where changes in electron spin occur, as in the oxygenation of haemoglobin (24)- Where tunneling through barriers is important, e.g., in electron-transfer reactions (26), this factor is of importance. 

For those interested in tunnelling through barriers, please look at p. 153 and subsequent material. 

FIGURE 2. Electron tunneling through barrier formed by field and image potential at the metal/vacuum interface/ ... 

In (a), a molecule alights onto a positive electrode surface, its electrons being attracted to the surface and its nuclei repelled. In (b), an electron has tunneled through a barrier onto the electrode, leaving a positive ion that is repelled by and moves away from the positive electrode. 

The time, t, taken to tunnel through the barrier (i.e. the time for the molecule to invert from one pyramidal form to the other) is given by... 

Torsional barriers are referred to as n-fold barriers, where the torsional potential function repeats every 2n/n radians. As in the case of inversion vibrations (Section 6.2.5.4a) quantum mechanical tunnelling through an n-fold torsional barrier may occur, splitting a vibrational level into n components. The splitting into two components near the top of a twofold barrier is shown in Figure 6.45. When the barrier is surmounted free internal rotation takes place, the energy levels then resembling those for rotation rather than vibration. 

High Field Emission - In this case, the electric field is high enough to narrow the work-function barrier and allow electrons to escape by tunneling through the barrier. 

We have seen that 10" M s is about the fastest second-order rate constant that we might expect to measure this corresponds to a lifetime of about 10 " s at unit reactant concentration. Yet there is evidence, discussed by Grunwald, that certain proton transfers have lifetimes of the order 10 s. These ultrafast reactions are believed to take place via quantum mechanical tunneling through the energy barrier. This phenomenon will only be significant for very small particles, such as protons and electrons. 

The transmission coefficient T in equation (2.58) is the relative probability that a particle impinging on the potential barrier tunnels through the barrier. The reflection coefficient R in equation (2.59) is the relative probability that the particle bounces off the barrier and moves in the negative v-direction. Since the particle must do one or the other of these two possibilities, the sum of T and R should equal unity... 

If the wavelengths of the reacting nuclei become comparable to barrier widths, that is, the distance nuclei must move to go from reactant well to product well, then there is some probability that the nuclear wave functions extend to the other side of the barrier. Thus, the quantum nature of the nuclei allows the possibility that molecules tunnel through, rather than pass over, a barrier. 

Thus, these results indicated the involvement of heavy atom tunneling in the localized biradicals. The rates of decay for 19,20, and 9 could be fitted with Bell s simple model of tunneling through a parabolic barrier. Assuming log A (s ) = 8.0, and... 

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