Double tunneling

Figure 33 Double-tunnel-injection electroluminescence (metal contacts). (From Ref. 270.)...

SET Single-electron transistor. Double tunnel junctions with a central island serving as a gate electrode. 

Kornyshev, A.A., Kumetsov, A.M., and Ulstrup, J. (2005) Double-tunnel nanoscale switch with a redox mediator operational principles and tunneling spectroscopy. ChemPhysChem, 6,... 

Abbreviations P, perovskite T, tetragonal H, hexagonal I, intergrowth Is, single row of tunnels in parent WO3 matrix Id, double tunnels PC, pentagonal column bronzes related to tetragonal TB type. 

FIGURE 4.15. Inteigrowth tungsten bronze structure of the pyrochlore type (double tunnel) [4.30]. 

The Forth Bridge, designed by Sir Benjamin Baker and opened in 1890, is perhaps the most impressive in existence. For centuries the traveller wishing to cross the Forth at its junction with the sea had been dependent on ferry-boats, and many a one had been overtaken by a gale which prevented his ever reaching the other side. In 1805 it was proposed to construct a double tunnel, some 15 feet wide, under the bed of the Forth, one tunnel for the comers and one for the goers as was quaintly explained but nothing came of it. 

In the current understanding of PCET reactions, both electron and proton are treated quantum-mechanically, and therefore the tunnelling probability must be accounted for both particles. In fact, concerted processes can be described as double tunnelling (proton and electron), with a single transition state. " For a description of the reaction coordinate, four adiabatic states (reactants, products and intermediates) described by paraboloids, are usually considered. The expression for the semi-classical rate constant in this case incorporates elements derived from electron and proton transfer theories... 

Fig. 2. The fluctuating difference between the proton potential at the product side relative to that at the reactant side (the difference between the two wells in a double-well proton potential). Whenever this difference is close to zero, tunneling conditions are favourable.

Reaction times can be as short as 10 minutes in a continuous flow reactor (1). In a typical batch cycle, the slurry is heated to the reaction temperature and held for up to 24 hours, although hold times can be less than an hour for many processes. After reaction is complete, the material is cooled, either by batch cooling or by pumping the product slurry through a double-pipe heat exchanger. Once the temperature is reduced below approximately 100°C, the slurry can be released through a pressure letdown system to ambient pressure. The product is then recovered by filtration (qv). A series of wash steps may be required to remove any salts that are formed as by-products. The clean filter cake is then dried in a tray or tunnel dryer or reslurried with water and spray dried. 

In 1979, a viable theory to explain the mechanism of chromium electroplating from chromic acid baths was developed (176). An initial layer of polychromates, mainly HCr3 0 Q, is formed contiguous to the outer boundary of the cathode s Helmholtz double layer. Electrons move across the Helmholtz layer by quantum mechanical tunneling to the end groups of the polychromate oriented in the direction of the double layer. Cr(VI) is reduced to Cr(III) in one-electron steps and a colloidal film of chromic dichromate is produced. Chromous dichromate is formed in the film by the same tunneling mechanism, and the Cr(II) forms a complex with sulfate. Bright chromium deposits are obtained from this complex. 

Aside from merely calculational difficulties, the existence of a low-temperature rate-constant limit poses a conceptual problem. In fact, one may question the actual meaning of the rate constant at r = 0, when the TST conditions listed above are not fulfilled. If the potential has a double-well shape, then quantum mechanics predicts coherent oscillations of probability between the wells, rather than the exponential decay towards equilibrium. These oscillations are associated with tunneling splitting measured spectroscopically, not with a chemical conversion. Therefore, a simple one-dimensional system has no rate constant at T = 0, unless it is a metastable potential without a bound final state. In practice, however, there are exchange chemical reactions, characterized by symmetric, or nearly symmetric double-well potentials, in which the rate constant is measured. To account for this, one has to admit the existence of some external mechanism whose role is to destroy the phase coherence. It is here that the need to introduce a heat bath arises. 

Coleman s method can be applied to finding the ground state tunneling splitting in a symmetric double well [Vainshtein et al. 1982], for some... 

As seen from this table, the WKB approximation is reasonably accurate even for very shallow potentials. At 7 = 0 the hindered rotation is a coherent tunneling process like that studied in section 2.3 for the double well. If, for instance, the system is initially prepared in one of the wells, say, with cp = 0, then the probability to find it in one of the other wells is P( jn, t) = 5sin (2Ar), while the survival probability equals 1 — sin ( Ar). The transition amplitude A t), defined as P( + t) = A t), is connected with the tunneling frequency by... 

Isotope effect between the HH, HD, DH, and DD isotopomers was used as an important tool to determine the mechanism of the double-proton transfer. For concerted degenerate double-proton transfers in the absence of tunneling, the rule of the geometrical mean (RGM) should hold in good approximation, which states that /chh/ hd = /cdh/ dd-Tunneling may lead to a breakdown of this rule but the relation /chh > hd = dh > dd should remain valid. In the absence of secondary isotope effects the relation /chh HD = DH = 2 /cdd sliould liold for a stepwise pathway, even if tunneling is involved. 

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