Fractional parameters

The summative-fractioruition method was extended to apply to narrow-distribution polymers with polydispersity (Mw/ Mn) less than 1.12. A fractionation parameter H, previously defined and calculated for theoretical molecular weight distributions for normal polymers, was computed for narrow-distribution polymers. The calculations were made both with and without correction for fractionation errors, using the Flory-Huggins treatment. The method was applied to a well-characterized anionic polystyrene with Mw = 97,000, for which the polydispersity was estimated by this technique to be 1.02 (in the range 1.014-1.027, 95% confidence limits). 

Table I. Summative-Fractionation Parameter H for the Poisson Distribution...

Figure 1. The summative-fractionation parameter H (ideal-fractionation assumption) as a function of polydispersity K w/Kln for the Poisson, Schulz-Zimm exponential, Lansing-Kraemer logarithmic-normal, and rectangular...

Figure 2. The summative-fractionation parameter H as a function of polydispersity for the Poisson distribution. H is calculated for the ideal-fractionation assumption while H0.001 and H0 01 we calculated including the Flory-Huggins correction for imperfect fractionation, with the volume ratio R equal to 0.001 and 0.01, respectively.

The isotopic abundance of deuterium in the three hydrogen atoms of the hydrogen ion differs from that in the water. The fractionation parameter involved is designated by the symbol l, i.e. 

Again it is possible to replace the fractionation parameter for the transition state by the rate coefficient ratio in isotopically pure solvents, so that (Gold, 1960)... 

The predictions of various theoretical equations are superimposed on the experimental data in Fig. 3. Curve A is a plot of equation (29), in logarithmic form, using the. experimentally determined (NMR) value 0-96 for the fractionation parameter 1A for acetic acid (Gold and Lowe, 1968) so that... 

Evidently this does not give an entirely satisfactory description of the experimental results, even though the maximum discrepancy between theory and experiment amounts to only ca. 0-05 pK unit. These experiments represent the only existing study where the dependence of a protolytic equilibrium constant on n has been related only to independently measured fractionation parameters. ... 

The evaluation of these basic fractionation parameters provides a basis for the study of dioxan-H20-D20 mixtures with a constant proportion of dioxan but varying relative amounts of the isotopic waters. No significant measurements in such systems or on systems with a non-aqueous component other than dioxan appear as yet to have been reported. Hine and Haworth (1958) have successfully studied the... 

Fractional parameter giving proportion of Lorentzian lineshape... 

These protein solubilization conditions have a key impact on the protein fractionation that can be carried out afterwards, in the sense that they will restrict the choice to techniques with which they are compatible. As an example, ionic detergents are not compatible with any technique using protein charge or pi as the fractionation parameter. 

Three dimensional presentations are cumbersome and it is more convenient to transform the Hansen parameters into fractional parameters as defined by [14]... 

The advantages of this kind of formulation stand out not only in terms of elegance and beauty (the moment method, the Lanczos method, and the recursion method are relevant but particular cases of the memory function equations), but also in the possibility of providing insight into a number of problems, such as the asymptotic behavior of continued fraction parameters and their relationship with moments, the possible inclusion of nonlinear effects, the introduction of the concept of random forces, and so on. 

The memory function formalism leads to several advantages, both from a formal point of view and from a practical point of view. It makes transparent the relationship between the recursion method, the moment method, and the Lanczos metfiod on the one hand and the projective methods of nonequiUbrium statistical mechanics on the other. Also the ad hoc use of Padd iqiproximants of type [n/n +1], often adopted in the literature without true justification, now appears natural, since the approximants of the J-frac-tion (3.48) encountered in continued fraction expansions of autocorrelation functions are just of the type [n/n +1]. The mathematical apparatus of continued fractions can be profitably used to investigate properties of Green s functions and to embody in the formalism the physical information pertinent to specific models. Last but not least, the memory function formaUsm provides a new and simple PD algorithm to relate moments to continued fraction parameters. 

A number of moments (3.65) are reported for convenience in the first row of Table I. Using the memory function PD algorithm (see Chapter III and ref. 25), summarized by Eq. (3.54), we obtain the moments for the first few memory functions (also rq>orted in Table I), and hence the continued fraction parameters (3.64) are recovered. Notice also that the last row of Table I is constituted by powers of the same number, as foreseen by Eq. (3.54) in the case of exact truncation. 

The presence of the gap produces undamped oscillations in the continued fraction parameters. In particular, a oscillates between (E + 4) G and b between G), while the multiplicity of the period of oscillations de-... 

In spite of these useful investigations, further work must still be done to assess the accuracy in realistic situations such as those encountered in band structure calculations on real crystals. Notice, furthermore, that any kind of extrapolation of continued fraction parameters must be consistent with the theory of error bounds as provided by the first few exact available moments. 

It may seem that an extended impurity problem could require an impossibly large cluster size. This is not true, however, basically because of the physical implications of translational symmetry away from the impurity region. In fact, the asymptotic limits of the continued fraction parameters and the cut in the real energy axis are determined by the perfect crystal only this allows a guideline for appropriate extrapolation of the recursion codffidents. 

In this section we shall apply the algorithm developed in the second part of Section III.B. This algorithm makes it possible to obtain directly the continued fraction parameters. As in Section III.B, we shall choose for F the expansion basis provided by Eqs. (3.37) and (3.38), which in this case results in the following coefficients ... 

Via the same procedure we can calculate the continued fraction parameters and then obtain the ESR lineshape. In the following we show the effect on the lineshape in the presence of the orienting potential of varying the r parameter defined in Section III.A. 

Merlivat and Jouzel (1979) showed that the initial isotope concentration in an air parcel above the ocean, 8 q, may, under certain simplifying assumptions, be expressed as a function of the fractionation coefficient a at sea surface temperature, T, of relative humidity, h, of a kinetic fractionation parameter k depending on wind speed (Merlivat, 1978), and of the isotopic composition of the sea surface, Socean. as... 

Figure 35. Real and imaginary parts of the complex susceptibility x(rlco) versus normalized frequency r C0 for y — 10 and — 3 and various values of the fractional parameter a.

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