Barrier shape

K (bottom) and 480 K (top). The curvature of the isothenns is interpreted as a temperature-dependent barrier shape [89],... 

Figure 4 demonstrates that in order to variationally describe a realistic barrier shape (Eckart potential) by an effective parabolic one, the frequency of the latter, should drop with decreasing temperature. At high temperatures, T > T, transitions near the barrier top dominate, and the parabolic approximation with roeff = is accurate. 

Another difficulty with the infrared method is that of determining the band center with sufficient accuracy in the presence of the fine structure or band envelopes due to the overall rotation. Even when high resolution equipment is used so that the separate rotation lines are resolved, it is by no means always a simple problem to identify these lines with certainty so that the band center can be unambiguously determined. The final difficulty is one common to almost all methods and that is the effect of the shape of the potential barrier. The infrared method has the advantage that it is applicable to many molecules for which some of the other methods are not suitable. However, in some of these cases especially, barrier shapes are likely to be more complicated than the simple cosine form usually assumed, and, when this complication occurs, there is a corresponding uncertainty in the height of the potential barrier as determined from the infrared torsional frequencies. In especially favorable cases, it may be possible to observe so-called hot bands i.e., v = 1 to v = 2, 2 to 3, etc. This would add information about the shape of the barrier. 

Permeation When a fluid contacts one side of an elastomer membrane, it can permeate right through the membrane, escaping on the far side. The process again combines adsorption and diffusion as above, but with the additional process eventually of evaporation—treated mathematically as negative adsorption. (Permeation could also be viewed as combining one-way absorption and evaporation.) Wherever these conditions for permeation exist the phenomenon occurs, whatever the shape of the elastomer barrier— but the associated mathematics becomes complex for irregular barrier shapes. 

In reality, as the barrier becomes narrower, it deviates from the square shape. One often used model is the parabolic barrier (dashed line in Fig. 1). When the barrier is composed of molecules, not only is the barrier shape difficult to predict, but the effective mass of the electron can deviate significantly from the free-electron mass. In order to take these differences into account, a more sophisticated treatment of the tunneling problem, based on the WKB method, can be used [21, 29-31]. Even if the metals are the same, differences in deposition methods, surface crystallographic orientation, and interaction with the active layer generally result in slightly different work functions on either side of the barrier. 

Equation (1) suggests that tunnel junctions should be ohmic. This is true only for very small bias. A much better description of the tunneling current results when the effects of barrier shape, changes in barrier with applied potential, and effective mass of the electron are all included. An example of such an improved relationship is given by (2), where / is the current density, a is a unitless parameter used to account empirically for non-rectangular barrier shape and deviations in the effective electron mass, and barrier height given by B = (L + work function of the left-hand metal ... 

The essential point that emerges from this first discussion of P is that only a fraction of the potential difference across the double layer, not the whole potential difference, is operative on the reaction. That there is a fraction P becomes clear what the fraction is remains a problem as long as the barrier shape is not known. This point of view must only be considered as the first murmuring of a theory of p, the symmetry factor. 

Using the asymptotic form of the solutions of the Schrodinger equation to the left and to the right of the barrier, it is possible to show that, regardless of barrier shape, the ratio... 

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... 

In order to calculate the height Fmax of the potential barrier from the measured tunneling frequencies [see Eq. (6), (7) ref. 3-14>] it is necessary to know the geometry of the molecule (especially the height of the nitrogen pyramid), which may also be obtained from analysis of the microwave spectra. Fitting then a barrier shape function 13> to the observed spectral data leads to the value of Fmax. Thus an approximate shape of the potential curve is obtained. 

A second type of model, derives the potential barrier from vibrational force constants and molecular geometrical parameters alone, assuming a certain barrier shape. Such a procedure is much less accurate (especially for high barriers) than the previous one, but it may be used for estimating barrier heights in systems for which no level splittings have been observed. It may thus be of appreciable practical usefulness. 

The main significance of the works [8] was in revealing the existence, irrespective of the barrier shape, of the finite low-temperature limit of the rate constant K(0). Even for Eckart barrier V x)= V /ch (2x/d), having an infinite width at = 0, the tunneling probability remains finite due to the existence of zero-point vibrations. 

J.L. Kurz, The Relationship of Barrier Shape to Linear Free Energy Slopes and Curvatures, Chem. Phys. Lett., 1978, 57, 243 J.C. Harris J.L. Kurz, A Direct Approach to the Prediction of Substituent Effects in Transition-state Structures, J. Am. Chem. Soc., 1970, 92, 349 R.P. Bell, Proc. Roy. Soc. London, 1936, 154A, 414. 

The coefficients of the truncated Fourier series of Eq. (2) or Eq. (3) are the molecular parameters for the hindering potential. The use of a limited number of those potential parameters implicitly means an assumption of the barrier shape. Usually, the first term (V3 or ) is the only term that is determined from the experiment. Using only one term of Eq. (2) means the assumption of a pure sinusoidal shape of the potential with the height of V3. If also higher terms V3n are included, the shape and height change. 

AH -r -r represents the minimum energy for tunneling to occur as described above and is assumed to be isotope independent. Note that a similar effect on the Arrhenius curves may be obtained by using more complex barrier shapes [10]. 

Basran, J., Patel, S., Sutcliffe, M. J., Scrutton, N. S. (2001) Importance of barrier shape in enzyme-catalyzed reactions - vibrationally assisted tunneling in tryptophan tryptophyl-... 

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