Parabola method

THOMSON PARABOLA METHOD. The method of investigating the charge-to-mass ratio of positive ions in which the ions are acted upon by electric and magnetic fields applied in the same direction normal to the path of the ions. It can be shown that ions of a given charge-to-mass ratio but different velocities will be deflected so as to form a parabola. 

The parabola method makes it possible to measure the potential of cell walls. Usually, the cell is made of quartz, and the parabola method thus offers the possibility of determining the lEP of one material that has already been extensively studied. The potentials of macroscopic specimens of other materials can also be determined from the mobility profile [273-275] by replacement of the original cell wall of a commercial electrophoretic cell by a flat specimen of the material of interest. For example, in [276], the lEP of a basal plane of mica found from the mobility profile was different from the lEP of a mica dispersion. Only a few types of electrophoretic devices (most of which are no longer available on the market) can be used to determine potentials by means of electro-osmosis. 

We present the main results in Table 4.14 where the basic electronegativity and chemical hardness calculated by finite difference approximations in terms of IP and EA definitions are considered (Lackner Zweig, 1983). This is based on two positive arguments they are based on the Parr s DFT groimd state parabola method that is consistent with definition (4.234) of global electrophilicity electronegativity imder Mulliken... 

A modified version of the bent strip test is the Dow-Sdbel test (parabola method) for quick testing of brittle thermoplastics (typically polystyrene). Flat test specimens are clamped over a parabola-shaped stainless steel form. This causes a continuous change in peripheral liber strain in the clamped specimens. As a measure for stress cracking sensitivity, the peripheral fiber strain is recorded up to which stress cracking has proceeded after a defined time [268]. 

HyperChein has two synch ron ons transit meth ods im piemen ted. The linear synchronous transit method (LST) searches for a maximum along a linear path between reactants and products. It may happen that this method will end up with a structure having two or more negative eigenvalues. The quadratic synchronous transit method (QSTlisan improvement of LST approach and searches for a maximum along a parabola connecting reactants and products, and for a minimum in all directions perpendicular to the parabola. 

This method approximates/(x) by a parabola on each subintei val. This rule is generally more accurate than the trapezoidal rule. It is the most widely used integration formula. 

When the log /J/pH measurement of a peptide is performed by the shake-flask or the partition chromatography method (using hydrophilic buffers to control pH), usually the shape of the curve is that of a parabola (see Ref. 371 and Fig. 1 in Ref. 282), where the maximum log I) value corresponds to the pH at the isoelectric point (near pH 5-6). Surprisingly, when the potentiometric method is used to characterize the same peptide [275], the curve produced is a step function, as indicated by the thick line in Fig. 4.5 for dipeptide Trp-Phe. 

In the method presented in the previous section each vertical slice was defined by two successive points, xq, x x, x2 etc. If now the successive points are selected three-by-three, they can be connected by a parabola. The approximate integral over the first two slices can then be written as... 

For simplicity, we will assume that depth below the sea bottom varies linearly with time. The first step in the calculation consists in removing the long-term drift over the whole period by fitting the data with a parabola and determining a periodogram out of the residuals from this fit. Depth has also been scaled to 1 in order to minimize round-off errors. Applying the method shown above, the best-fit parabola is obtained for... 

Two successful automated procedures have been given. In each case one must first find the peak, and a binary search with doubled limits on each reversal is the fastest search method if the peak position is unknown. One is then on either the upper or the lower branch of either the substrate or the epilayer parabola (it is in fact a very shallow and distorted parabola). Fewster first showed that the branch conld be detected by driving down in nntil the intensity is about halved from that of the peak, then driving down in. If the intensity rises one is on the upper branch of the parabola, if it falls one is on the lower branch. Fewster then alternated steps in and steps in nntil the peak was fonnd. The latter is determined by the intensity falling instead of rising on beginning the step. 

One step in such a different direction can be referred to as a mixed method, and is used in the previously mentioned paper of Henry and Lang (1977), with an extension by Jaros and Brand (1976b). Thus, Henry and Lang use the usual adiabatic product function [Eq. (32)] up to a critical point Q = Q1 close to the intersection of the two parabolas, and assume that beyond this point the wave function no longer changes, i.e.,... 

Once the hardness of standards has been determined from the Mohs scale by more precise methods, it became apparent that the Mohs scale in the interval 1-9 (talc-corundum) represents a series of minerals with a uniformly increasing hardness whose magnitudes are spread roughly on a curve corresponding to a mathematical function approximating a parabola of third degree (y = 3) (Fig. 4.5.1). As a result, the introduction of a quantitative method of hardness measurement with diamond indenters has not reduced the importance of the Mohs scale in any way on the contrary, its position has been enhanced. It will continue to be the basic macrodiagnostic method. 

Let us also notice that slow variations of K with Z imply that the gauge condition K may be treated as a semi-empirical parameter in practical calculations to reproduce, with a chosen K, the accurate oscillator strength values for the whole isoelectronic sequence. Thus, dependence of transition quantities on K may serve as the criterion of the accuracy of wave functions used instead of the comparison of two forms of 1-transition operators. In particular, the relative quantities of the coefficients of the equation fEi = aK2 + bK +c (the smaller the a value, the more exact the result), the position of the minimum of the parabola Kf = 0 (the larger the K value for which / = 0, the more exact is the approximation used, in the ideal case / = 0 for K = +oo) may also help to estimate the accuracy of the method utilized. 

The integral in (4.33) may be evaluated by the steepest descent method, which leads to an optimum value of Q = Q. This amounts to minimization of the total action 5, + S2 over the positions of the bend point Q. In fact, in the sudden approximation one looks for the minimum of the barrier action taken on a certain class of paths, each consisting of two straightforward segments. If the actual extremal path is close to one of the paths from this class—and this is indeed the case for low enough ft—then the sudden approximation provides accurate results. In particular, the sudden approximation permits calculation of the rate constant to an accuracy of 10% at V /co0 = 3, ft = 0.1, C 0.05 [Hon-tscha et al., 1990], For the cubic parabola (n = 1 in (4.29)) at small coupling parameter the rate constant in the sudden approximation may be evaluated analytically by using the one-dimensional instanton result (3.68) for k D ... 

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