Discrimination exponent

Inherent to all mass bias models stemming from Eq. (5.7) is the built-in variable (discrimination exponent) that distinguishes the various mass discrimination phenomena. Hence we have linear, exponential, equilibrium, power, and other discrimination models. This variable, in turn, is often used to identify the presence of a particular discrimination model [32, 41], Consequently, considerable effort has been spent in extracting the numerical value of the mass bias discrimination 

To date, interrogation of the efficacy of mass bias correction models has largely resorted to attempts to determine the value of the discrimination exponent. In such experiments, the slope of the log-linear two-isotope ratio regression is used, which, in turn, leads to the discrimination exponent by solving the following expression [43-46]  

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Qearly, there are many ways according to which one can express the caUbration factor Ki/j as functions of time and mass. Equation (5.4), for example, is commonly known as the linear correction law, Eq. (5.5) as Russell s law, and Eq. (5.6) as the exponential law. To add flexibility to the fitting function in the mass domain, it has been proposed that a discrimination exponent, n, be introduced into the mass bias correction factor [19] ... 

Assuming=fk/i t), one can solve for n. Iterative solutions are required to obtain n from b (and nuclide masses, m) since the above expression carmot be solved analytically. The fallacy of this approach is that one implicitly assumes that the ratio of discrimination functions is unity, that is, fi/j=fk/i [17]. Without knowing this a priori, the discrimination exponent remains a dummy variable, the value of which cannot be independently evaluated by the currently proposed methods. To illustrate this, consider a system of Hg/ Hg and Hg/ Hg. The experimentally obtained slope b = 1.537(6) corresponds to = —3.2(4) if fi/fi = I OO w = —0.5(4) i fi/f2 = 1.02. Therefore, even the slightest variations in the ratio of the discrimination functions greatly affect the perceived numerical value of n. Hence the catch 22 situation to estimate n one has to know/ but for that, one has to know n. 

Figure 5.3 Ratio of nuclide mass functions jiij = m - mf) as a function of the discrimination exponent (n) for selected isotopes of mercury. Adapted from [17],...

Although the true functional relation between K and Kj is unknown, it can be approximated using Taylor s theorem, which asserts that any sufficiently smooth function can locally be approximated by polynomials. This can be achieved by expanding InK as a function of m , where n is an arbitrary discrimination exponent [19] ... 

This is the model equation for the calibration of isotope amount ratios based on the log-linear temporal isotope amount ratio regression. Note that a and b are perfectly correlated (p = +1) if Rtp < 1 (inRup < 0) and perfectly anti-correlated (p = —1) if Rk/i > 1 (InRfe/ > 0). It is important to stress that this calibration method is fundamentally different from the conventional mass bias correction la vs. Since the regression model does not invoke the principle of time-mass separation, it does not need either the discrimination exponent or the equality of the discrimination functions [17]. 

Practidly the same value for i/ (cs — 2.86) has been observed also in the case of Cu(l 10). We have yet no explanation for this imusual exponent, but it raises at least some questions concerning the analysis of He scattering data measured without energy discrimination. 

The exponent A is set at 2 by GREGPLUS to emphasize discrimination initially as Hill, Hunter and Wichern did. 

It should also be observed that, in the case of an infinitely ramified fractal structure fed from all the perimeter sites, an exponent 0 fairly close to 0 = 1/2 (i.e. the regular case) is obtained, 0.40 < 0 < 0.50. The small difference between the exponent 0 obtained for infinitely ramified fractals and in the regular Euclidean case makes it difficult to discriminate between the two scaling behaviors, especially if one makes use of numerical simulations on small lattices. 

Discriminating branched and star polymers from linear ones can always be achieved by measuring the properties in dilute solution. In fact, molecules having the same molar mass but different macromolecular architectures exhibit different transport and light scattering properties. More specifically, a branched macromolecule is more compact than a linear molecule having the same molar mass, and therefore it will display less friction and will diffuse more easily in the solvent. Viscometry can be used to detect branched structures, since the Mark-Houwink-Sakurada exponent (Eq. 2.23) for branched and star-shaped polymers is lower tiian that for the corresponding linear chain. Unfortunately, in order to measure the difference, one must have a sample made exclusively... 

Schafer and Witten" have applied the RG to excluded volume, and established scaling laws , for example for the osmotic pressure. One of the objects of the RG method is to establish such scaling laws, and to demonstrate scale invariance . Then experimentally observable correlation functions can be shown to obey particular scaling behaviour, and the critical exponent calculated may be compared with that obtained by experiment. Critical exponents calculated by the RG will generally differ from that obtained by classical mean field e.g. SCF approaches - Mackenzie " in a recent review has pointed out that discrimination between the two lies with experiment. For example, Le Guillou and Zinn-Justin have calculated v in equation (7) to be 0.588 (c/. the SCF-fifth-power law value of 0.60). However, to discriminate between these values is beyond the capability of current experimental techniques. Moore has used the RG to explore the asymptotic limit, and recently demonstrated that when the ternary cluster integral vanishes, an expression for the osmotic pressure may be derived which holds for both poor and good solvents, in semi-dilute solutions. 

It is important to notice that with the aid of viscosity measurements one can discriminate between ideal and expanded chains. One finds an exponent fji = 0.5 corresponding to u = 0.5 for ideal chains, and larger values for expanded chains, up to the limit of /x = 0.8 corresponding to i/ = 0.6 expected... 

According to the micellar theory [6, 7] the rate of polymerization (in the stationary interval 2) is proportional to the 0.4 power of the concentration of initiator and the 0.6 power of the concentration of emulsifier [4]. It was later shown by Roe [23], Fitch [22,25, 67] and Hansen and Ugelstad [18] that this behavior is also consistent with homogeneous nucleation. Indeed, identical exponents can be predicted by virtually any mechanism which assumes that (1) coagulation does not occur, (2) nucleation ceases when the surface area of the latex particles is equal to the total surface area capable of being occupied by the emulsifier molecules, and (3) the rate of production of free radicals is uniform. Here, determination of exponents for [I] and [E] fails to discriminate between competing micellar and homogeneous nucleation theories. 

With the aid of such molar mass-dependent [r] measurements one can discriminate between ideal and expanded chains. One finds an exponent /it = 0.5 corresponding to i/ = 0.5 for ideal chains and larger values for expanded chains, up to the limit of p = 0.8 corresponding to i/ = 0.6 expected for an expanded chain with ultra-high molar mass. In addition, the prefactor K can be used for determining the chain stiffness as expressed by the effective length per monomer a. Since the constant in Eq. (8.167) is known, a can be derived. In particular, by carrying out measurements of rj in theta solvents, the characteristic ratio. Coo = c / h (Eq- (2-32)), can be determined. 

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