Distortion potential

The wavefunctions in Eq. (2.34) are different from the wavefunctions of the free tip and free sample. The effect of the distortion potential (V = Us — Uso and V = Us - Uso), can be evaluated through time-independent perturbation. In the following, we present an approximate method based on the Green s function of the vacuum (see Appendix B). To first order, the distorted wavefunction i)i is related to the undistorted one, i]jo, by... 

Using the properties of the Green s function (see Appendix B), the evaluation of the effect of distortion to transmission matrix elements can be greatly simplified. First, because of the continuity of the wavefunction and its derivative across the separation surface, only the multiplier of the wavefunctions at the separation surface is relevant. Second, in the first-order approximation, the effect of the distortion potential is additive [see Eq. (2.39)]. Thus, to evaluate the multiplier, a simpler undistorted Hamiltonian might be used instead of the accurate one. For example, the Green s function and the wavefunction of the vacuum can be used to evaluate the distortion multiplier. 

The MBA provides another simple explanation of why the image potential is not observable. According to Eq. (2.42), as long as the integral of the distortion potential is a constant (that is, the shaded area in Fig. 2.8 remains constant while varying the tip-sample distance), the effect of distortion is a constant independent of the barrier thickness. Therefore, the effect of barrier lowering due to image force is not observable. 

Controlled-current chronoabsorptometry involves the simultaneous optical monitoring of the product or other redox component in the electrode mechanism during a chronopotentiometry experiment [14]. Although this technique has been demonstrated with Sn02 optically transparent electrodes, it has generally received little use, since the resistance effects in thin-film electrodes can give unequal current densities across the electrode face. This results in distorted potential-time and absorbance-time responses. Consequently, the more prevalent spectro-electrochemical methods utilize potential rather than current as the excitation signal. 

It would be convenient for solving the Lippmann—Schwinger equation (6.73) if we could make the potential matrix elements as small as possible. For example, we could hope to find a transformed equation whose iteration would converge much more quickly. This is achieved by a judicious choice of a local, central potential U, which is called the distorting potential since the problem is reformulated in terms of the distorted-wave eigenstates of U rather than the plane waves of (6.73). An important particular case of U is the Coulomb potential Vc in the case where the target is charged. The Hamiltonian (6.2) is repartitioned as follows... 

Up to this stage the distorting potential U has been arbitrary. We now derive an optimum form for it. According to (6.85) the exact explicit form for the T-matrix element in the case 1=5 = 0 is... 

Note that the potential (7.15) is uncharged, since there is one nuclear charge for each electron charge. For electron—ion scattering Vq is replaced by Vq — Uq, where the distorting potential Uq is charged so that... 

Thus, substituting distortion vector by u = Uq + grad (p, the respective Laplace equation for the distortion potential tp will be Atp(r) = 0. Consider now a grain of phase A, which has several border interfaces with other A grains as well as with B-matrix and the internal volume of this grain will be a positive direction (Fig.l). The picture on Fig.l is taken from the real structure of W-Cu infiltrated composite with a graded distribution of diamonds [4]. The equilibrium requirements stipulate that distortion potential on positive (inside) and negative (outside) sides of the interfaces must be coupled as follows, so far the material is not a subject of an application of external forces ... 

Walker and Wyatt (1972) have also performed a distorted-wave calculation for H + H2, based on the Porter-Karplus surface. They constructed reactant and product distortion potentials assuming adiabatic vibrational motion in each case, and obtained numerical solutions for the relative motions. Their results show that by choosing adequate potential parameters it is possible to reproduce the threshold behaviour, but that probabilities grow above unity soon after the threshold energy. 

Very recently there has been a distorted-wave calculation (Walker and Wyatt, 1973) for planar reactive H + H2 using two different distortion potentials. Results were used to approximate three-dimensional total and differential cross sections. 

Elementary substitution reactions of type I + R2R3 -> R1R2 + R3. with Rk a molecular group, have been described in the context of the coupled-channel method by Brodsky and Levich (1973). These authors introduced distortion potentials for reactants and products and a parametrized, isotropic potential coupling. In practice, transition amplitudes were calculated... 

Fig. 17. Potential energy curves for (a) Jahn-Teller distortion, (b) near-degeneracy, small distorting force, and (c) near-degeneracy, large distorting force. The restoring potential is shown as a thin line, the distorting potential as a thick line, the resulting potential as a dashed line.

As a brief aside let us show here that if an external distorting potential wcxl as in Eq. (10) is used, calculating the A matrices could in some cases be simplified due to the relation reminiscent to those used in Ref. 16 for wcx being the imaginary potential. Consider... 

Other metals - the presence of galvanizing, galvanic anodes or conduits in the concrete can distort potential measurements. 

ABSTRACT. A review is given of recent applications of the distorted wave (DW) method to the theory of chemical reactions. A brief account of the following topics is included the formal DW theory of reactions, static and adiabatic methods for choosing the distortion potentials, and the removal of the 3 Euler angles from the 6 dimensional DW integral. Applications of various DW theories to the H+F2 0( P)+H2. 0( P) -C(CH3) 4. 

When y is approximated by xy- as In the DWBA. it is clear that T a now depends on the choice of distortion potentials V and Vp. In fact, since there are an infinite number of ways of partitioning Vy into Vy + Vy, the term DWBA covers an infinity of possible approximations. 

The distortion potentials used in the DWBA are chosen so that they only give rise to elastic scattering in the incident and final channels. (They are discussed In more detail In Section 2,2). Thus and... 

Assumptions (a) and (b) of the DWBA can be partially removed by going to the CCDW approximation, in this method, the distortion potential Vy is chosen so that It allows inelastic as well as elastic non-reactive collisions. Formally we can write... 

In the DWBA. distortion potentials are introduced to describe the elastic scattering in the initial and final channels. There are many ways of choosing these distortion potentials. One way (the static approximation) is to assume that the reactant or product molecule is unperturbed by the incoming or receding atom respectively. Another way (the adiabatic... 

It Is evident that a different static distortion potential is obtained for each value of V and j. 

Now for some highly exoergic reactions such as T + F2. there are over 1000 product TF vibrational-rotational states open. To avoid having to calculate a different distortion potential for each j. the usual rigid-rotor approximation can be made. In this case, we can write... 

A different kind of static distortion potential has been used for the H+D2 HD4-D reaction CIO. 31). In this reaction, the centre of... 

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