Potential barriers, penetration

R. Atkinson and F. Houtermans apply Gamow s theory of potential barrier penetration by quantum tunnelling to suggest how stars can release nuclear energy by synthesis of hydrogen into helium by an (unspecified) cyclic process. 

In essence, the goal of this review is to describe the evolution of notions about tunneling, from the Gamov one-dimension model potential barrier penetration to multidimensional quantum motion with account for an active role of the medium. Such evolution goes on in various branches of modern chemical physics and the physics of solids. Therefore, the analysis of the common features of developing theories seems timely and important. 

Figure 19 Diagram for single-potential barrier penetration and reflection.

Electrons can penetrate the potential barrier between a sample and a probe tip, producing an electron tunneling current that varies exponentially with the distance. 

Tethering may be a reversible or an irreversible process. Irreversible grafting is typically accomplished by chemical bonding. The number of grafted chains is controlled by the number of grafting sites and their functionality, and then ultimately by the extent of the chemical reaction. The reaction kinetics may reflect the potential barrier confronting reactive chains which try to penetrate the tethered layer. Reversible grafting is accomplished via the self-assembly of polymeric surfactants and end-functionalized polymers [59]. In this case, the surface density and all other characteristic dimensions of the structure are controlled by thermodynamic equilibrium, albeit with possible kinetic effects. In this instance, the equilibrium condition involves the penalties due to the deformation of tethered chains. 

Suppose a particle of mass m and energy E coming from the left approaches the potential barrier. According to classical mechanics, if E is less than the barrier height Vq, the particle will be reflected by the barrier it cannot pass through the barrier and appear in region 111. In quantum theory, as we shall see, the particle can penetrate the barrier and appear on the other side. This effect is called tunneling. 

The basis of the scanning tunnelling microscope, illustrated schematically in Figure 3.5, lies in the ability of electronic wavefunctions to penetrate a potential barrier which classically would be forbidden. Instead of ending abruptly at a... 

According to the Sommerfeld model electrons in a metal electrode are free to move through the bulk of the metal at a constant potential, but not to escape at the edge. Within the metal electrons have to penetrate the potential barriers that exist between atoms, as shown schematically below. 

The potential curve for the electrons near the tip surface is shown in Fig. 1.38. The relevant dimensions are much smaller than the radius of the tip end. Therefore, a one-dimensional model is adequate. In the metal, the energy level of the electrons is lower than the vacuum level by the value of the work function c ). From the point of view of classical mechanics, the electrons cannot escape from the metal even with a very high external field, that is, the potential barrier is impenetrable. From the point of view of quantum mechanics, there is always a finite probability that the electrons can penetrate the potential barrier. In the semiclassical (WKB) approximation, the transmission coefficient for a general potential barrier is (Landau and Lifshitz, 1977) ... 

Fig. 2.28 Particle penetration probability in field dissociation of 4HeRh + and 3HeRh2+ from a vibrational state 300 K and 500 K above the bottom of the potential energy curve. At the same field, the particle barrier penetration probability for 3HeRh2+ is three to four orders of magnitude smaller than that for 4HeRh2+, in good agreement with the experiment.

Further consideration, however, reveals that even in the case of a metal, there is what might be termed a dipole layer. The situation is roughly as follows (Fig. 6.41) The physical surface of the metal tries to confine the free electrons inside the metal. It is as if the electrons are in potential wells." But electrons have the characteristic of being able to penetrate potential barriers. If they succeed, a positive charge is left behind for every electron jumping out of the metal. This is tantamount to a charge separation and a dipole layer. Thus, there is a surface potential for metals, too. and therefore a %M (see Section 6.6.8). 

Another important quantum mechanical problem of interest to nuclear chemists is the penetration of a one-dimensional potential barrier by a beam of particles. The results of solving this problem (and more complicated variations of the problem) will be used in our study of nuclear a decay and nuclear reactions. The situation is shown in Figure E.5. A beam of particles originating at — oo is incident on a barrier of thickness L and height V0 that extends from x = 0 to x = L. Each particle has a total energy E. (Classically, we would expect if E < V0, the particles would bounce off the barrier, whereas if E > V0, the particles would pass by the barrier... 

Usually tunneling through a potential barrier is considered on the basis of the stationary Schroedinger equation with the use of matching conditions. A different approach has been developed by Bardeen (34). Bardeen s method enables one to describe tunneling as a quantum transition and to use the Golden Rule in order to evaluate the probability of penetration through the barrier. A similar method has been used in Section III to describe vibrational predissociation. This section contains a short description of Bardeen s method (see refs. 39,82-84). 

Another conceivable therapeutic approach for Alzheimer s disease could be based on the administration of neurotrophic factors. Nerve growth factor (NGF) is a 118 amino acid polypeptide with no blood-brain barrier penetrance. Other substances with neurotrophic activity such as epidermal growth factor, brain-derived neurotrophic factor, gangliosides, and the (11-28 peptide of the b-amyloid protein might also have a therapeutic potential. Intracerebroventricular (ICV) administration of NGF has been shown to partially reverse lesion-induced deficits of cortical AChE and choline acetyltransferase (CHAT) activities to promote survival of septal cholinergic neurons after fimbrial transection in adult rats and to reverse behavioral deterioration in rats with such lesions. 

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