Barrier width

As this first example indicates, experiments involving superlattices have an additional complication compared to the ADQW treated previously. In the ADQW, essentially all quantities of interest can be calculated analytically. The well widths and barriers widths are designed such that the electronic levels move into tunneling resonance at a value of the DC field sufficient to immobilize the holes. This is not guaranteed to be the case in superlattices, where the electronic levels display a complicated pattern of repulsion and anticrossing as the DC field is varied. In addition, as shown... 

The important criterion thus becomes the ability of the enzyme to distort and thereby reduce barrier width, and not stabilisation of the transition state with concomitant reduction in barrier height (activation energy). We now describe theoretical approaches to enzymatic catalysis that have led to the development of dynamic barrier (width) tunneUing theories for hydrogen transfer. Indeed, enzymatic hydrogen tunnelling can be treated conceptually in a similar way to the well-established quantum theories for electron transfer in proteins. 

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. 

Equations 10.11 and 10.12 confirm our qualitative predictions. The degree of tunneling depends inversely both on the square root of the mass and on the barrier width. Moreover, it turns out that Q increases as temperature is lowered. The... 

Since the rate for the tunneling of a proton is strongly dependent on barrier width, it is necessary that the molecular systems to be studied constrain the distance of proton transfer. Also, since the various theoretical models make predictions as to how the rate of proton transfer should vary with a change in free energy for reaction as well as how the rate constant should vary with solvent, it is desirable to study molecular systems where both the driving force for the reaction and the solvent can be varied widely. 

Fig. 6.2 (a) Bell (parabolic) and Eckart barriers, both widely used in approximate TST calculations of quantum mechanical tunneling, (b) Transmission probability (Bell tunneling) as a function of energy for two values of the reduced barrier width, a... 

Piner, R., and Reifenberger, R. (1989). Computer control of the tunnel barrier width for the scanning tunneling microscope. Rev. Sci. Instrum. 60, 3123-3127. 

The reason that a compound ion can be field dissociated can be easily understood from a potential energy diagram as shown in Fig. 2.23. When r is in the same direction as F, the potential energy curve with respect to the center of mass, V(rn) is reduced by the field. Thus the potential barrier width is now finite, and the vibrating particles can dissociate from one another by quantum mechanical tunneling effect. Rigorously speaking, it... 

The diffusion voltage Vo and the barrier width W depend on the influence of diffusion and recombination (i. e. the Debye length Z-d = ]/D rrei, D = j = diffusion coefficient) and are therefore given by... 

One interesting analogy should be noted here. It is well known that the exponential factor that determines the rate of tunneling contains the product d. JmE, where d is the barrier width, E is its height, and m is the mass of tunneling particle. In chemical cases in ours and American works, d 10 8-5cm, m 30, E 0, 1 eV. In the spontaneous fission of nuclei, d 10 12 cm, m 100, E 106 eV. Thus the spontaneous fission of nuclei and molecular tunneling in chemical reactions can be treated to some extent as quite similar phenomena. 

An empirical correlation between the rate constants for internal conversion (or intersystem crossing) and the barrier width was found. From earlier work, Ross and co-workers10 were able to infer diatomic-like spectra for certain polyacenes and therefore treat their electronic spectra as those of diatomic molecules. Thus the potential curves of the different states could be represented by functions of one parameter R, as shown in Figure 2. R is the magnitude of... 

Although there is an apparent success in Ross treatment, the barrier width cannot be the only determining parameter. It is the authors belief (all other things being equal) that the energy gap between the two levels (and therefore v", the vibrational quantum number of the ground state) is the controlling factor. 

At these temperatures the distribution of occupied levels in the conduction bands ( the Fermi distributions ) in the two metal electrodes ( Fig.l ) are quite sharp, with a boundary between filled and empty states ( the Fermi level ) of characteristic width k T ( k =0.08617 meV/K=0.69503 cm Vk ). An applied bias voltage V between the two electrodes separates the Fermi levels by an energy eV. If the barrier oxide is sufficiently thin electrons can tunnel from one electrode to the other. This process is called tunneling since the electrons go through a potential barrier, rather than being excited over it. The barrier must be thin for an appreciable barrier to flow. For a typical 2 eV barrier the junction resistance is proportional to, where s is the barrier width in Angstroms (17). The... 

In this expression d is the barrier width corresponding to the zero-point energy in the initial state, and the factor a is of the order unity depending on the barrier shape. It equals 1/2, 2/tt, and 3/4 for rectangular, parabolic, and triangular barriers, respectively, and is unity for a barrier constructed from two shifted parabolas. For the parabolic barrier (1.5), Eq. (1.6) assumes the form... 

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