Zero-range potential

Various choices have been made for the binding potential Vi (and the corresponding wave functions) and the interaction Vi2, which will be discussed below. If the former is described by a regularized zero-range potential so that V] (r) three-body contact potential V(ri,r2) <5(ri — r2)S(r2), then the amplitude (4.1) can be reduced... 

Demkov, Yu.N. and OstrovskA ,. N. (1988) Zero Range Potentials and Their Applications in Atomic Physics, Plenum New YorA ... 

The non-steady-state optical analysis introduced by Ding et al. also featured deviations from the Butler-Volmer behavior under identical conditions [43]. In this case, the large potential range accessible with these techniques allows measurements of the rate constant in the vicinity of the potential of zero charge (k j). The potential dependence of the ET rate constant normalized by as obtained from the optical analysis of the TCNQ reduction by ferrocyanide is displayed in Fig. 10(a) [43]. This dependence was analyzed in terms of the preencounter equilibrium model associated with a mixed-solvent layer type of interfacial structure [see Eqs. (14) and (16)]. The experimental results were compared to the theoretical curve obtained from Eq. (14) assuming that the potential drop between the reaction planes (A 0) is zero. The potential drop in the aqueous side was estimated by the Gouy-Chapman model. The theoretical curve underestimates the experimental trend, and the difference can be associated with the third term in Eq. (14). 

The influence of the single crystal face of the Pb electrode on zero charge potential has been analyzed. The difference between these potentials is within the range of 60 mV [8] and the potential value increases with lowering atomic density [11]. The increase of the inner layer capacitance at zero charge in the sequence Pb(lOO) < Pb(llO) < Pb(112) < polycrystalline Pb < Pb(lll) has been explained by increasing hydrophilicity of the surface [6,11]. 

Experimental data on the deuterium-hydrogen isotope shift (see Table 12.4 below) have an accuracy of about 0.1 kHz and, hence, a more accurate theoretical result for the polarizability contribution is required. In order to obtain such a result it is necessary to go beyond the zero range approximation, and take the deuteron structure into account in more detail. Fortunately, there exist a number of phenomenological potentials which describe the properties of the deuteron in all details. Some calculations with realistic proton-neutron potentials were performed [40, 41, 42, 43]. The most precise results were obtained in [43]... 

The contributions from the short-range repulsive potential and the long-range attractive potential are shown explicitly in Fig. 12.6. Also shown are the full Stockmayer potential for three different orientations of the dipole moments. The curve listed as no dipole is for orientation angles ft = Gj = rjr — n/2. In this case the x term in Eq. 12.9 is zero. The potential has a minimum at r, - = 21/6a,y, with an attractive well depth of e,y. The curve listed as attractive dipole has orientation angles ft = 6j = 0. Thus x = 2, and this orientation has the maximum (attractive) contribution from the dipole-dipole term. The well depth in this case is almost a factor of 6 deeper due to the dipole interaction. The repulsive... 

As a result of its broad and symmetric, but still mainly non-hydrogen-bonding, cr-profile, the solvent nitromethane has an almost zero cr-potential in the non-hydrogen-bonding cr-range. Here it behaves closest to the perfect conductor limit. Because it has no... 

For systems with larger values of aR, rainbow scattering may be completely quenched [2, 15,41,42]. This occurs, for example, in K. + Br2 and K + CBr4. The long-range potential parameters must then be calculated in a different manner [15], based on the supposition that the reaction has a zero threshold energy, so that elastic scattering occurs only from the portion... 

For a short-range potential there is a point ro beyond which the potential can be considered to be zero. A bound particle has a negative energy eigenvalue and we define the absolute value of the imaginary wave number by... 

The current density presented in Figure 5.4(b) cem be described as being a linear function of potential over a narrow range of potential near the zero-current potential. Taylor series expansions of the exponential terms in equation (5.14) yield... 

Chen and Liu (1977) utilized the spontaneous oxidation of NADH by potassium ferricyanide for the construction of a potentiometric LDH electrode. The coupled reduction of ferricyanide ions to ferrocyanide ions results in a measurable electrochemical zero-current potential. The potential was found to be Nemstian in nature and directly proportional to the logarithm values of lactate concentration over the range 0.02 to 50 mmol/1. The response time was as high as 10 min. 

The pll dependence of (-potential, surface speciation and colloid stability ratios is presented in Figure 4. In panel (a) of Figure 4, the solid line represents the surface potential from the model, and the data points correspond to ( potential from electrokinetic measurements. The dashed line represents a reduced potential calculated at a distance of 4.0 nm from the surface. In panel (b). the surface species distribution is calculated by the SCF/DLM model. Under these conditions, the observed zero iiiterfacial potential is in the range pH 7.2 to 7.5, which suggests... 

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