Parabolic approximation

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. 

Xiu, G. H., Modeling breakthrough curves in a fixed bed of activated carbon fiber - exact solution and parabolic approximation, Chem. Eng. Sci., 1996, 51(16), 4039 4041. 

The JKR theory, much like the Hertz theory, assumes a parabolic approximation for the profile of sphere, which is valid for small ratios of contact radii to the sphere s radius. Maugis [34] has shown that for small particles on a soft substrate, this ratio could be so large that such parabolic approximation is no longer valid. Under such conditions, the use of exact expression for the sphere profile is necessary for the applicability of the JKR theory, which is expressed as... 

Within the BHF approach, it has been verified [15, 22] that a parabolic approximation for the binding energy of nuclear matter with arbitrary proton fraction x is well fulfilled,... 

From the known pH dependence of the activity of other enzymes, a domain of approximately 4 pH units should be sufficient to bracket the optimum, yet not be so wide as to seriously invalidate a parabolic approximation to the true behavior of the system. The chosen treatment combinations will therefore be pH = 8, 9, 10, 11, and 12. We will code them as —2, —1, 0, +1, and +2 (c, = 10, = 1 see Section 8.5). 

Parabolic approximation .) Subsequently, as x h, the second integral gets its main contribution from the neighborhood x" a and may be replaced with... 

In Eq. 7.16,0 can be approximated using Eq. 7.18 the integral s value is dominated by 5rji 0, and therefore its limits can be taken to be oo, even though the parabolic approximation is valid only near equilibrium,... 

The terms represent successive approximations to a function at small values of its argument by the value of the function at the origin, the straight-line approximation of the function at the origin, the parabolic approximation of the function at the origin, and so forth. 

Fk,. 2.—Validity of the parabolic approximation for the total interaction potential in the evaluation of the integral in eqn (12). Solid curves are exact values, while points represent quantities determined... 

Except for K, all the integrals are centered at extrema. This allows the use of the parabolic approximation for the interaction potential. One obtains... 

Substituting the parabolic approximation (14) into the integrand yields ... 

A simple procedure consists of using the parabolic approximation of the enthalpy in the vicinity of its minimum at... 

The effective masses of electrons and holes are estimated by parabolic approximation a large curvature corresponds to a small effective mass and a small curvature corresponds to a large mass. With this band concept, light absorption and luminescence are interpreted as follows Light is absorbed by the transition from valence band to conduction band. Therefore, the broadening of the absorption spectrum originates basically from the one dimensionality of the joint density of states, which is described by (E - g) . Excited electrons and holes relax to the bottom of the bands and then recombine radiatively. Therefore, the photoluminescence of the spectrum is very sharp. The energy difference between two peaks is called the Stokes shift. 

Figure 10.7 Comparison of the results of numerical calculations of the relaxation and of the parabolic models with experimental results. Solid lines experimental results by Seidel-Morgenstern [41]. Dot-dashed line profiles calculated with the relaxation model. Dotted lines profiles calculated with the parabolic approximation. The figure on each subfigure indicates the duration of the rectangular injection. Reproduced from F. James, M. Pastel, M. Sepulveda, Physica D, 138 (2000) 316.(Fig. 8).

James et al. [37] discussed the numerical solutions of fwo above mentioned models in defail and compared the numerical solution of fwo mefhods for the same value of e for various initial and boimdary values. The differences between the solution of fhe relaxation method and the solution obtained using the parabolic approximation increases with increasing the value of e, of the injection voltune and of the concentration of the solute injected into the system. They also showed that the parabolic approximation is valid in a relatively wide range of injected amounts and of relaxation parameters. The parabolic approximation has the big advantage of requiring a much shorter computation time than the calculation of a solution of the complete relaxation model. 

Figure 3.6. Frozen solvent reaction coordinate potentials for aqueous C1 +CH3C1 reaction near standard conditions versus y = x — xK As in Figure 3.5A, the instantaneous potential V(S, x) is the sum of the gas phase potential U x) and the cage potential v S x) U x) is that of Figure 3.2 and v S x) is estimated [23] from the molcular dynamics results of ref. 14a. The parabolic approximation to V S x) of Eq. (3.48) is also shown.

In the simple case in which the parabolic approximation holds, F(k) = h (k +kj + kl)/2m = h k /2m for a conduction band with a constant effective mass m of the electrons, we have... 

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