Cross-over point

An alternative mechanism of excess energy release when electron relaxation occurs is through x-ray fluorescence. In fact, x-ray fluorescence favorably competes with Auger electron emission for atoms with large atomic numbers. Figure 16 shows a plot of the relative yields of these two processes as a function of atomic number for atoms with initial K level holes. The cross-over point between the two processes generally occurs at an atomic number of 30. Thus, aes has much greater sensitivity to low Z elements than x-ray fluorescence. 

Friction also changes the way k ) approaches its low-temperature limit and widens the intermediate region between the two asymptotes of k(P). At temperatures far below the cross-over point k T) behaves as... 

As another illustration, note that above the cross-over point the temperature dependence k T) displays an activation energy equal to... 

Next find the closed-loop gain cross-over frequency by deciding how much phase margin you desire in your system. A good value is 45 degrees. Ignoring any effect of the Q of the T-C filter, the gain cross-over point is found from... 

Figure 10-32. Typical temperature situations that contain cross-over points, preventing exchanger operation. (Adapted and used by permission Brown and Root, Inc.)...

The transfer of the electron takes place very rapidly compared to nuclear motion, and will only take place when the combination of internal and librational coordinates is such that the curves interact. Thus, the [Fe(H20)6] + species must first distort and/or experience a dipole moment field from the instantaneous positions of the water molecules such that it attains the cross-over point. At this point, the electron may tunnel from the [Fe(H20)6]2+ ion to the metal, leaving behind an [Fe(H20)6]3 + ion with a non-equilibrium geometry, This then relaxes by heat transfer to the solvent to the equilibrium point, q0. 

In addition to the 2 nm shift in the absorption maximum, the two cytochromes can be distinguished by the use of ethyl isocyanide interaction spectra (6, 7) and various inhibitors of the monooxygenase activity (Figure 2 and Table III). The relative magnitude of the ethyl isocyanide-cytochrome P-1+50 interaction spectral peaks at —1+30 and —1+55 nm is pH dependent (6j and if the absorbance differences are plotted as functions of pH, there is a cross-over point at a certain pH which is characteristic for a particular form of cytochrome P-1+50 pH 6.9 for cytochrome P-1+1+8 and pH 7-5-7.6 for PB induced or control cytochrome P-1+50 (6, 21). The cytochrome P-1+50 of apparently uninduced trout species (Salmo trutta lacustris) has been shown by us to have the pH cross-over point for ethyl isocyanide interaction spectrum at pH 7.8 (2l) and the absorption maximum of the reduced trout liver cytochrome P-1+50. 00 complex is 1+50 nm, nevertheless its catalytic and inhibitory properties (2l)(Table III) are similar to those of cytochrome P-1+1+8. 

The chemical shift 5, defined by Equation (22), was measured at 40.0 and 15.6 Mc./sec. and was found to be —3 2 relative to water for both SA and SG. The derivatives of the resonance absorptions were recorded in the measurements. If the total anisotropy of the chemical shift of protons in the solid is somewhat less than the line width, the cross-over point of the derivative will correspond to the average value of S as for liquids, and will be directly comparable with the shifts for protons in the liquid state. Comparison of the shift value with those of H3O+ (aqueous) (1 1), S = -1-11, OH (aqueous) (121), S = -1-10 dilute solutions of alcoholic-type protons... 

Fig. 26(b) is thus a superposition of the eight resulting envelopes of Fig. 27. From the humps and cross-over points of Fig. 26(b) and the value of go given above, one obtains...

Feed composition effects on cross-over points were investigated by cracking a heavy gas oil (with API gravity of 23.7) containing 71% slurry oil. The high (34%) aromatics content of this feed is believed responsible for the greater coke make and increased deactivation rate of the host catalyst. As a result, a cross-over point below 0.25 wt% vanadium is obtained, see Figure 6. A loss of carbon selectivity is observed also in the DFCC mixture. In fact. 

In several cases the melt viscosity of a series of lightly-branched polymers has been determined as a function of MW, and compared with that of linear polymers, and it has been found or may be deduced from the published data that there is a cross-over molecular weight, below which the branched polymer is less viscous, but above which it more viscous, than the linear polymer of equal MW. This behaviour is observed with some comb-shaped polystyrenes (35) and poly(vinyl acetate)s (59, 89), star polybutadienes (57, 58, 123), and randomly-branched polyethylenes (56,61). Jackson has found (141) that if the ratio ZJZC of the number of chain atoms at the cross-over point, Zx, to the number at the kink in the log 0 — logM curve, Zc, [as given in Ref. (52)], is plotted against nb, the number of branches, a reasonable straight line is obtained, as in Fig. 5.1. 

The value of the MW exponent for the low shear-rate melt viscosity, usually accepted as 3.4—3.5 for linear polymers, has been called in question for PE. Schreiber and Bagley (164) reported a value of 4.22 for linear PE Porter and Johnson (165) had earlier reported values up to 8 for LDPE. For branched polymers an effect of this kind could be due to the effect mentioned in Section 5, in which a cross-over point is found on a log-log plot of viscosity vs. MW when linear and slightly branched polymers are compared, since the curve for the branched polymers must be steeper than that for the linear ones near the cross-over point. 

Typical patterns of G and G" produced from frequency sweep experiments are classified into three categories (Figure H3.2.6). A dilute solution of macromolecules shows G < G" at most frequencies. With increasing concentration, interactions among molecules become pronounced in what is called an entangled or semi-dilute solution. An entangled solution has a distinguishing feature of a cross-over point where G = G" or tan(8) = 1. The material is fluid like at a frequency below the cross-over point, but solid like at a frequency above the cross-over point. The frequency at the crossover point is thus also referred to as the relaxa-... 

Reliable capital and operating cost comparisons between pervaporation and distillation are not available. Pervaporation is less capital and energy intensive than distillation or adsorption processes for small plants treating less than 5000 L/h of feed solution. However, because of the modular nature of the process, the costs of pervaporation are not as sensitive to economies of scale as are the costs of distillation and adsorption processes. Distillation costs, on the other hand, scale at a rate proportional to 0.6-0.7 times the power consumption. Thus, distillation remains the most economical process for large plants. The cross-over point at which distillation becomes preferable to pervaporation from an energy and economic point of view currently appears to be 5000 L/h processing capacity. Bergdorf has made an analysis of the comparative costs of pervaporation, distillation and other processes [43],... 

There are two basic approaches to the theory of atomic helium, depending on whether the nuclear charge Z is small or large. For low-Z atoms and ions, the principal challenge is the accurate calculation of nonrelativistic electron correlation effects. Relativistic corrections can then be included by perturbation theory. For high-if ions, relativistic effects become of dominant importance and must be taken into account to all orders via the one-electron Dirac equation. Corrections due to the electron-electron interaction can then be included by perturbation theory. The cross-over point between the two regimes is approximately Z = 27... 

In an asymmetric spectrum, the center of the spectrum may not coincide with the "cross-over" point referred to as the base line. This uncertainty is removed when a hyperfine pattern is observed. In the absence of hyperfine structure, Blois et al. (15) have shown that it is necessary to run the sample in a series of tubes of different diameter, and the true g-value is then obtained by extrapolating to zero tube diameter. 

The cross over point of the free energy curves of heptane and ethylene occur at a lower temperature than that of ethane, namely at about 800K. Pyrolysis of naphtha to ethylene can therefore be practiced at lower temperatures than that required for the pyrolysis of ethane. 

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