Scattered square

Fig. 10 a UV-Vis DRS spectra of TS-1 (curve 1, full line), immediately after contact with H2O2/H2O solution (curve , dotted line), after time elapse of 24h (curve 3, dashed line) and after subsequent H2O dosage (curve 4, scattered squares), b as for a for the XANES spectra, c as for a for the -weighted, phase imcorrected, FT of the EXAFS spectra. Spectra 2-4 of b and c have been reordered at liquid nitrogen temperatime. Adapted from [49] with permission. Copyright (2004) by ACS... 

Fig. 11 High resolution XANES spectra collected at the GILDA BM8 heamline of the ESRF Grenoble (France) at liquid nitrogen temperatiu e on the TS-1 catalyst activated TS-1 catalyst (dotted line)-, after contact with anhydrous H2O2 from the gas phase (full line) after subsequent contact with water (scattered squares). Adapted from [50] with permission. Copyright (2004) by VCH...

Figure 27 Dependence of the overiap concentration c on chain s degree of polymerization Win salt-free polyelectrolyte solutions of NaPSS. Data obtained from X-ray scattering (squares) and from viscosity (circles). Data assembled by Boris, D. C. Colby, R. H.

The physical interpretation of the scattering matrix elements is best understood in tenns of its square modulus... 

Figure A3.9.2. Interaction potential for an atom or molecule physisorbed on a surface. A convenient model is obtained by squaring off the potential, which facilitates solution of the Sclirodinger equation for the scattering of a quantum particle.

The cross section for scattering into the differential solid angle dD centred in the direction (9,(l)), is proportional to the square of the scattering amplitude ... 

In this expression, factors that describe the incident and scattered projectile are separated from the square modulus of an integral that describes the role of the target in detemiining the differential cross section. The temi preceding the... 

A succinct picture of the nature of high-energy electron scattering is provided by the Bethe surface [4], a tlnee-dimensional plot of the generalized oscillator strength as a fiinction of the logaritlnn of the square of the... 

We have seen that the intensities of diffraction of x-rays or neutrons are proportional to the squared moduli of the Fourier transfomi of the scattering density of the diffracting object. This corresponds to the Fourier transfomi of a convolution, P(s), of the fomi... 

We have seen that the intensities of diffraction are proportional to the Fourier transfomi of the Patterson fimction, a self-convolution of the scattering matter and that, for a crystal, the Patterson fimction is periodic in tln-ee dimensions. Because the intensity is a positive, real number, the Patterson fimction is not dependent on phase and it can be computed directly from the data. The squared stmcture amplitude is... 

The intensity of light scattering, 7, for an isolated atom or molecule is proportional to the mean squared amplitude... 

The second method to calculate the scattered intensity or R the Rayleigh ratio) is to square the sum in I... 

One of the most important fiinctions in the application of light scattering is the ability to estimate the object dimensions. As we have discussed earlier for dilute solutions containing large molecules, equation (B 1.9.38) can be used to calculate tire radius of gyration , R, which is defined as the mean square distance from the centre of gravity [12]. The combined use of equation (B 1.9.3 8) equation (B 1.9.39) and equation (B 1.9.40) (tlie Zimm plot) will yield infonnation on R, A2 and molecular weight. 

If we consider the scattering from a general two-phase system (figure B 1.9.10) distinguished by indices 1 and 2) containing constant electron density in each phase, we can define an average electron density and a mean square density fluctuation as ... 

Figure C2.17.12. Exciton energy shift witli particle size. The lowest exciton energy is measured by optical absorjDtion for a number of different CdSe nanocrystal samples, and plotted against tire mean nanocrystal radius. The mean particle radii have been detennined using eitlier small-angle x-ray scattering (open circles) or TEM (squares). The solid curve is tire predicted exciton energy from tire Bms fonnula.

Interference of Waves. The coherent scattering property of x-rays is used in x-ray diffraction appHcations. Two waves traveling in the same direction with identical wavelengths, X, and equal ampHtudes (the intensity of a wave is equal to the square of its ampHtude) can interfere with each other so that the resultant wave can have anywhere from zero ampHtude to two times the ampHtude of one of the initial waves. This principle is illustrated in Figure 1. The resultant ampHtude is a function of the phase difference between the two initial waves. 

Once the form of the correlation is selected, the values of the constants in the equation must be determined so that the differences between calculated and observed values are within the range of assumed experimental error for the original data. However, when there is some scatter in a plot of the data, the best line that can be drawn representing the data must be determined. If it is assumed that all experimental errors (s) are in thejy values and the X values are known exacdy, the least-squares technique may be appHed. In this method the constants of the best line are those that minimise the sum of the squares of the residuals, ie, the difference, a, between the observed values,jy, and the calculated values, Y. In general, this sum of the squares of the residuals, R, is represented by... 

The data-reduction procedure just desciiDed provides parameters in the correlating equation for g that make the 8g residuals scatter about zero. This is usually accomphshed by finding the parameters that minimize the sum of squares of the residuals. Once these parameters are found, they can be used for the calculation of derived values of both the pressure P and the vapor composition y. Equation (4-282) is solved for yjP and written for species 1 and for species 2. Adding the two equations gives... 

If the experimental values P and w are closely reproduced by the correlating equation for g, then these residues, evaluated at the experimental values of X, scatter about zero. This is the result obtained when the data are thermodynamically consistent. When they are not, these residuals do not scatter about zero, and the correlation for g does not properly reproduce the experimental values P and y . Such a correlation is, in fact, unnecessarily divergent. An alternative is to process just the P-X data this is possible because the P-x -y data set includes more information than necessary. Assuming that the correlating equation is appropriate to the data, one merely searches for values of the parameters Ot, b, and so on, that yield pressures by Eq. (4-295) that are as close as possible to the measured values. The usual procedure is to minimize the sum of squares of the residuals 6P. Known as Barkers method Austral. ]. Chem., 6, pp. 207-210 [1953]), it provides the best possible fit of the experimental pressures. When the experimental data do not satisfy the Gibbs/Duhem equation, it cannot precisely represent the experimental y values however, it provides a better fit than does the procedure that minimizes the sum of the squares of the 6g residuals. 

A dynamic transition in the internal motions of proteins is seen with increasing temperamre [22]. The basic elements of this transition are reproduced by MD simulation [23]. As the temperature is increased, a transition from harmonic to anharmonic motion is seen, evidenced by a rapid increase in the atomic mean-square displacements. Comparison of simulation with quasielastic neutron scattering experiment has led to an interpretation of the dynamics involved in terms of rigid-body motions of the side chain atoms, in a way analogous to that shown above for the X-ray diffuse scattering [24]. 

Here is the position operator of atom j, or, if the correlation function is calculated classically as in an MD simulation, is a position vector N is the number of scatterers (i.e., H atoms) and the angular brackets denote an ensemble average. Note that in Eq. (3) we left out a factor equal to the square of the scattering length. This is convenient in the case of a single dominant scatterer because it gives 7(Q, 0) = 1 and 6 u,c(Q, CO) normalized to unity. 

Which measure of scatter is likely to be larger, the mean absolute error or the root-mean-square error ... 

Still be smooth, free of fouling material, and the edges sharp and square. It will be diffieult to achieve, but remember the better the instrumentation performs, the less the data scatter that will require rationalization later. 

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