Universal curve

Fig. 11. "Universal curve" of inelastic mean free path, X, as a function of electron kinetic energy. Solid line is universal curve, points are experimental data...

In a synchrotron, electrons are accelerated to near relativistic velocities and constrained magnetically into circular paths. When a charged particle is accelerated, it emits radiation, and when the near-relativistic electrons are forced into curved paths they emit photons over a continuous spectrum. The general shape of the spectrum is shown in Fig. 2.4. For a synchrotron with an energy of several gigaelectronvolts and a radius of some tens of meters, the energy of the emitted photons near the maximum is of the order of 1 keV (i.e., ideal for XPS). As can be seen from the universal curve, plenty of usable intensity exists down into the UV region. With suitable mono-... 

At higher pressures only Raman spectroscopy data are available. Because the rotational structure is smoothed, either quantum theory or classical theory may be used. At a mixture pressure above 10 atm the spectra of CO and N2 obtained in [230] were well described classically (Fig. 5.11). For the lowest densities (10-15 amagat) the band contours have a characteristic asymmetric shape. The asymmetry disappears at higher pressures when the contour is sufficiently narrowed. The decrease of width with 1/tj measured in [230] by NMR is closer to the strong collision model in the case of CO and to the weak collision model in the case of N2. This conclusion was confirmed in [215] by presenting the results in universal coordinates of Fig. 5.12. It is also seen that both systems are still far away from the fast modulation (perturbation theory) limit where the upper and lower borders established by alternative models merge into a universal curve independent of collision strength. 

Essentially, at a fixed monomer feed composition and temperature such a dependence is of universal character irrespective of the way of molecular weight regulation, which could be performed by changing the concentration either of the initiator or the transfer agent. Just on this universal curve the points fall which characterize the composition and molecular weight of the fractions isolated from copolymers synthesized under different contents of initiator and transfer agent. 

The dimensionless parameter % or xm is a function of Mach number and can be determined from a universal curve for the atomization of various liquid metals, i.e., a plot of % or xm vs- Mach number, as presented by Bradley in Ref. 329 for subsonic gas flow, and in... 

Derived from an analytical model for flat, infinitely thick liquid layer Effects of gas compressibility included Effects of gas/liquid ratio, liquid viscosity, and nozzle geometry not included X and Xm can be determined from the universal curves for metals in P29] for subsonic gas flow and in [330] for sonic/supersonic gas flow ... 

Fig. 6. Universal curve for the system-averaged on-top hole density in spin-unpolarized systems. The solid curve is for the uniform gas. The circles indicate values calculated within LSD, while the crosses indicate essentially exact results, and the plus signs indicate less accurate Cl results. The high-density (r, - 0) and low-density (r, ->oo) limits behave respectively like the weak-coupling U - 0) and strong-coup-ling (X- oo) limits...

Fig. 8. A universal curve (or band) of electron attenuation length vs. electron kinetic energy, as resulting from an inspection of many experimental results on heavy metals. The energy of different laboratory sources, representing the maximal kinetic energy of the electron, is drawn for comparison in the figure (from Ref. 5)...

Fig, 4.1. The universal curve for the electron mean free path as a function of electron kinetic energy. Dots show individual measurements... 

Analyzing the experimental data concerning the dependence of WZ. on vlv0Z, a number of authors have obtained the universal curves... 

A Universal Curve for Transfection Efficiency Versus Membrane Charge Density. 196... 

DOTAP/DOPC MVL2/DOPC MVL DOPC MVL5/DOPC TMVL5/DOPC5 Chol/DOTAP/DOPC Universal Curve fit)... 

Using the experimentally obtained values of aM calculated using (1), Fig. 5b plots TE of DOTAP/DOPC/Chol-DNA complexes (empty circles) as a function of membrane charge density, together with the universal curve and the TE data used for its derivation [21],... 

As we have elaborated in Sects. 2 and 4.3, the TE of lamellar DNA complexes of MVLs and UVLs shows universal behavior when plotted against the membrane charge density, implying that optimized complexes of MVLs and UVLs transfect equally well. Importantly, the fact that the universal curve is bell-shaped (where TE is plotted logarithmically) implies that optimization of the lipid composition is crucial for objectively comparing lipid performance. Interestingly, while complexes in the H c and HnC phase do not follow this universal curve, their TE no more than equals that of optimized lamellar complexes. 

FIGURE 7.9 B T) for a Lennard-Jones 6-12 potential. B is the ratio between B(T ) and the hard-sphere value 2n ALvogadroff3/3. T is the ratio of the thermal energy kB T to the well depth . B and T make Figure 7.10 a universal curve, valid for any gas. This curve agrees well with experiments on many molecules. 

Careful infrared study of CO chemisorbed by Pd/Si02 catalysts in an SMSI versus a non-SMSI state verified that after HTR, silicon species are distributed in the Pd surface layer. For the catalysts reduced at 300°C (LTR), the B/L intensity ratio (B = bridging CO L = linearly bound CO) is a monotonic function of Pd particle size (Fig. 17). On the other hand, the B/L ratios for Pd/Si02 catalysts that experienced HTR show considerable departure from this universal curve (Fig. 17) (208). Of course, a relatively higher proportion of linearly bound CO for Pd/Si02 catalysts in the SMSI state is believed to follow from the existence of silicon, rather uniformly interdispersed in the metal surface, resembling the case of CO adsorption on silica-supported Pd-Ag alloys (209). 

The optimization procedure outlined above is independent of the specific plate height equations chosen to represent chromatography. It depends only on the validity of the scaling relationships used to define reduced plate height and reduced velocity, and the emergence of universal curves from the scaling relationships. 

In the experiments the small particles (2.8 and 6.8 nm) appear more active than the largest ones (13 nm) especially in the medium range of temperature. First we have tried to take into account for the reverse spillover of CO by using Eq. (13). The reaction probability has been fitted from the universal curve given from experiments on various Pd extended surfaces [124] ... 

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