A model of tunnelling

The model in Fig. 6.2 has been widely used for two-level tunnelling systems in disordered glass structures, although the details of the tunnelling units in glass are unknown. For H in metals, the particle mass m and the distance d between the wells are known, so that the various dynamic effects can be checked for quantitative consistency much better than in glasses where crude averages have to be taken over most parameters. 

If the barrier Vq is fitted in Fig. 6.2, we can calculate the tunnel splittings of the oscillation states as approximately  

The asymmetry A between the potentials in the two wells will partly localize the H to the left (L) or right (/ ) well in symmetrized ground eigenstates 

These states are split in energy by A o = [ AEj-ff + and the minimum splitting AEfo has in some cases been observed directly with inelastic neutron scattering or specific heat measurements. The excited eigenstates and are similarly partly localized and split by 

The asymmetry A in (6.4) is caused by interactions with neighbouring hydrogens and with other impurities and defects. The values of A therefore have some distribution over the well pairs in a given sample, and the observed effects must be an average over A in the sample. The magnitude 

---

Quantitative agreement between the results of kinetic experiments and predictions on the basis of a model of tunnel scavenging of efr by acceptors... 

In order to model glass properties and to discuss experimental observations, we need a model of tunneling systems. A schema of a TLS, and its potential along the generalized coordinate is presented in Fig. 1. Independently of the nature of this coordinate, which determines the mass m of the tuimeling particle, the TLS is characterized by the barrier width d (taken as the distance between minima in each well), the barrier height V, and the asymmetry A or energy difference between the wells. 

A calculation of tunneling splitting in formic acid dimer has been undertaken by Makri and Miller [1989] for a model two-dimensional polynomial potential with antisymmetric coupling. The semiclassical approximation exploiting a version of the sudden approximation has given A = 0.9cm" while the numerically exact result is 1.8cm" Since this comparison was the main goal pursued by this model calculation, the asymmetry caused by the crystalline environment has not been taken into account. 

They point out that at the heart of technical simulation there must be unreality otherwise, there would not be need for simulation. The essence of the subject linder study may be represented by a model of it that serves a certain purpose, e.g., the use of a wind tunnel to simulate conditions to which an aircraft may be subjected. One uses the Monte Carlo method to study an artificial stochastic model of a physical or mathematical process, e.g., evaluating a definite integral by probability methods (using random numbers) using the graph of the function as an aid. 

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... 

Figure 3.44. Dissociation of 02 adsorbed on Pt(lll) by inelastic tunneling of electrons from a STM tip. (a) Schematic ID PES for chemisorbed Of dissociation and illustrating different types of excitations that can lead to dissociation, (b) Schematic picture of inelastic electron tunneling to an adsorbate-induced resonance with density of states pa inducing vibrational excitation (1) competing with non-adiabatic vibrational de-excitation that creates e-h pairs in the substrate (2). (c) Dissociation rate Rd for 0 as a function of tunneling current I at the three tip bias voltages labeled in the figure. Solid lines are fits of Rd a IN to the experiments with N = 0.8, 1.8, and 3.2 for tip biases of 0.4, 0.3, and 0.2 V, respectively and correspond to the three excitation conditions in (a). Dashed lines are results of a theoretical model incorporating the physics in (a) and (b) and a single fit parameter. From Ref. [153].

It is convenient to consider a model of an anisotropic recombination region the reflecting recombination sphere (white sphere) with black reaction spots on its surface [77, 78], The measure of the reaction anisotropy here is the geometrical steric factor Q which is a ratio of a black spot square to a total surface square. Such a model could be actual for reactions of complex biologically active molecules and tunnelling recombination when the donor electron has an asymmetric (e.g., p-like) wavefunction. Note the non-trivial result that at small Q, due to the partial averaging of the reaction anisotropy by rotational motion arising due to numerous repeated contacts of reactants before the reaction, the reaction rate is K() oc J 1/2 rather than the intuitive estimate Kq oc Q. 

Following the method of Huddleston and Miller [91], let us consider a model of the tunnel capture of efr by such a bifunctional molecule Bj-Bj containing two acceptor fragments Bj and B2 located at a fixed distance l from each other (Fig. 19). Let V1(t) and V2(t) be the reaction volumes of the Bj and B2 acceptors relative to the capture of e, and Rtl(t) and i t2(f) be the tunneling distances (radii) corresponding to these volumes. It will be recal-... 

Wind tunnel test methods were developed to determine wind induced stresses in cooling towers using aeroelastic models as part of a detailed model of a power station site. The turbulence and shear in the atmospheric wind are simulated. Tests on a model of Ferrybridge C Power Station show that resonant stresses are significant at the design wind speed. These increase as the fourth power of wind speed and can be greatly enhanced by turbulent wakes of upstream structures. 6 refs, cited. 

Conceiving a corrrelated model of tunnelling protons we will first discuss the smallest case N = 4. It follows immediately that the number of pairings is... 

The barrier height is about 15 kcal/mol by analogy to the similar gas-phase reaction H2C + CH4- 2CH3, which was calculated by Bausch-licher et al. [1976], The C-C distance calculated in the same work is found to be equal to 2.65 A. Like reaction (6.41) any attempt to reconcile the experimental dependence k(T) with a model of one-dimensional tunneling in the barrier of indicated height leads to a tunneling distance d that is far shorter than could be reasonably rationalized based on this C-C distance. 

Fig. 10.6. Percolation cluster model of tunnel current in composite film containing M/SC nanoparticles (a) two-sphere model of spherical M/SC nanoparticle of radius Rq surrounded by outer sphere (radius Rd) that is defined by a degree of electron delocalization extending the nanoparticle and characterizes electron tunneling (see text) (b) the distribution of conductivity G(r) over the two-sphere particle (c) two-dimensional pattern of cluster from overlapping two-sphere particles (overlapping areas of outer spheres are shown).

Janssens M, Huczek J, Sauceda A. Development of a model of the ASTM E84 Steiner tunnel test. Ninth International IAFSS Symposium. Interscience Communications Ltd. Greenwich, London, 2008. 

The temperature conductivity data of iodine doped PMQ3 (1.14 I per repeat from element analysis and weight gain.)(Figure 14) can be fitted not only with Equation 4, but also to a model of fluctuation-induced carrier tunneling (Equation 5). Either of them can be expected only when the metallic domain concentration increases to above the percolation threshold from broken paths. ... 

Recently, a controversial debate has arisen about whether the optimization of enzyme catalysis may entail the evolutionary implementation of chemical strategies that increase the probability of tunneling and thereby accelerate reaction rates [7]. Kinetic isotope effect experiments have indicated that hydrogen tunneling plays an important role in many proton and hydride transfer reactions in enzymes [8, 9]. Enzyme catalysis of horse liver alcohol dehydrogenase may be understood by a model of vibrationally enhanced proton transfer tunneling [10]. Furthermore, the double proton transfer reaction in DNA base pairs has been studied in detail and even been hypothesized as a possible source of spontaneous mutation [11-13]. 

In order to distinguish between the different contributions a comparison with sophisticated theoretical calculations is mandatory. However, it is necessary to test these models on small prototype systems. High resolution gas phase spectroscopy can now provide exact experimental data on proton transfer tunneling and thereby allow a test of tunneling criteria which could be used for more complex systems. 

---