Correction terms numerical results

The fine structure was calculated by Pirenne [110] and independently by Berestetski [4J. Minor errors are corrected, and numerical results are given by Ferrell [45]. The approach used by these authors is to write down the Dirac equations for the two particles, and the interaction terms as they are expressed in quantum field theory. The equations can be transformed so that the particle spins appear explicitly. The interaction terms are found to comprise the Coulomb energy, the Breit interaction, and a term analogous to the Fermi expression for... 

While the data presented in Section 4.4 can be numerically integrated, they were generated by Lee and Kesler according to their equation of state. This equation can be analytically integrated to find values for enthalpy departure that can be used in Equation (5.50). The results are shown in Appendix E. Plots of the simple fiuid and correction terms that result are presented in Eigures 5.5 and 5.6, respectively Tables of their values are presented in Appendix C (Tables C.3 and C.4). 

Eqs. (40H41) are obtained from the analytical solution using the first two terms in the 0-series expansion of the concentration profile. As a result, they are accurate only for small values of meridional angle, 8. To correct for large values of 6, Newman [45] used Lighthill s transformation and Eq. (15) for the meridional velocity gradient to calculate the local mass transfer rate as Sc - oo. His numerical result is plotted in Fig. 5 in the form of Shloc/Re1/2 Sc1/3 vs. 8 as the thin solid line. The dashed line is... 

Recoil contributions in (10.6), and (10.7) are symmetric with respect to masses of the light and heavy particles. As in the case of the leading recoil correction, they were obtained without expansion in the mass ratio, and hence an exact dependence on the mass ratio is known (not just the first term in the expansion over m/M). Let us mention that while for the nonrecoil nonlogarithmic contributions of order Za), both to HFS and the Lamb shift, only numerical results were obtained, the respective recoil contributions are known anal3dically in both cases (compare discussion of the Lamb shift contributions in Subsect. 4.2.3). 

The term measurand, which might be new to some readers, is the quantity intended to be measured, so it is correct to say of a numerical result that it is the value of the measurand. Do not confuse measurand with analyte. A test material is composed of the analyte and the matrix, and so the measurand is physically embodied in the analyte. For example, if the measurand is the... 

The error in the up-dated value of Xi (thus x ) can be estimated using equation (b). Keeping in mind that m n and assuming that Vf(x ) is a well-conditioned matrix, leads to the conclusion that the error in the correction term x - Xi° is approximately F II II 6x [1. Numerical experiments have verified this result. 11 II... 

This PWE was used in [18] to obtain the numerical results. For the numerical implementation the B-spline approximation [21] was chosen that represents actually the refined version of the space discretization approach. In Table 1 the convergence of the PWE approach with the multicommutator expansion is presented for the lowest-order SE correction for the ground state of hydrogenlike ions with Z = 10. The minimal set of parameters for the numerical spline calcuations was chosen to be the number of grid points N = 20, the number of splines k = 9. This minimal set allowed to keep a controlled inaccuracy below 10%. What is most important for the further generalization of the PWE approach to the second-order SESE calculation is that with Zmax = 3 the inaccuracy is already below 10% (see Table 1). The same picture holds with even higher accuracy for larger Z values. The direct renormalization approach is not necessarily connected with the PWE. In [19] this approach in the form of the multicommutator expansion (Eq. (16)) was employed in combination with the Taylor expansion in powers of (Ea — En>)r 12 The numerical procedure with the use of B-splines and 3 terms of Taylor series yielded an accuracy comparable with the PWE-expansion with Zmax = 3. 

When mass-transfer rates are moderate to high, an additional correction term is needed in equations (6-101) and (6-102) to correct for distortion of the composition profiles. This correction, which can have a serious effect on the results, is discussed in detail by Taylor and Krishna (1993). An alternative approach would be to numerically solve the Maxwell-Stefan equations, as illustrated in Examples 1.17 and 1.18. The calculation of the low mass-transfer fluxes according to equations (6-94) to (6-104) is illustrated in the following example. 

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