Equal-energy spectrum

When we introduce the color response characteristics of the source, combined with the detector of our instrument, we find that we must drastically modify the transmission characteristics of our filters in order to duplicate the CIE color matching functions for the equal-energy spectrum. However, this is not an impossible task and we find that an excellent match can be obtained to the transmission functions of 6.7.20. This is typical for commercially available instruments. Now, we have an instrument, called a Colorimeter, capable of measuring reflective color. 

CIE standard observer n. The observer data adopted by the Commission Internationale d Eclairage to represent the response of the average human eye, when light-adapted, to an equal-energy spectrum. Unless otherwise specified, the term applies to the data adopted in 1931 for a 2° field of vision. The data adopted in 1964, sometimes called the 1964 observer, were obtained for a 10°, annular field which excludes the 2° field of the 1931 observer functions. 

FIGURE 3 Tristimulus values of the equal-energy spectrum of the 1931 CIE system of colorimetry. [From Billmeyer, F. W, Jr, and Saltzmann, M. (1981). Principles of Color Technology, 2nd ed. Copyright 1981 John Wiley Sons, Inc. Reprinted by permission of John Wiley Sons, Inc.]... 

Calculation of chromaticity coordinates then proceeds as described above for sources. If an equal-energy spectrum is assumed, the source term S X) can be dropped fi om the definition of (j) X). When the chromaticity of a surface is specified without specification of the source, an equal-energy spectrum is usually implied. 

The required distribution of initial populations ntu can be obtained in the following manner (32). Let us consider a system with Ed mi = 20 kcal/ mole and Ed max = 45 kcal/mole. Assuming that kd = 1013 sec-1 and x = 1, we can calculate theoretical desorption rates dnai/dt for Ed = 20, 21, 22,..., 45 kcal/mole as a function of nBOi. With increasing temperature, 25 values of dnjdt are measured at temperatures corresponding to Ed of 20, 21, 22,. . ., 45 kcal/mole. Since the total desorption rate at any moment must be equal to the sum of the individual desorption processes, we obtain 25 linear equations. Their solution permits the computation of the initial populations of the surface sites in the energy spectrum considered, i.e. the function n,oi(Edi). From the form of this function, desorption processes can be determined which exhibit a substantial effect on the experimental desorption curve. 

Non-Kolbe electrolysis may lead to a large product spectrum, especially when there are equilibrating cations of about equal energy involved. However, in cases where the further reaction path leads to a particularly stabilized carbocation and either elimination or solvolysis can be favored, then non-Kolbe electrolysis can become an effi-yient synthetic method. This is demonstrated in the following chapters. 

The bonding in solids is similar to that in molecules except that the gap in the bonding energy spectrum is the minimum energy band gap. By analogy with molecules, the chemical hardness for covalent solids equals half the band gap. For metals there is no gap, but in the special case of the alkali metals, the electron affinity is very small, so the hardness is half the ionization energy. 

In contrast to the d.c. or microwave plasma apparatus, the sample environment produced by these directed beam sources has been reasonably well characterized. Studies of Kaufman source operation (Sharp et al., 1979) have established that H beams are typically composed of mixtures of H+ and H2+ ions and a roughly equal mixture of energetic neutrals. The ion energy spectrum of such a source is fairly sharply peaked at the maximum energy at low acceleration voltages (150-500 eV) but spreads out considerably if the source is operated at voltages above 1000V. 

In Fig. 2, the normalized model scalar energy spectrum is plotted for a fixed Reynolds number (ReL = 104) and a range of Schmidt numbers. In Fig. 3, it is shown for Sc = 1000 and a range of Reynolds numbers. The reader interested in the meaning of the different slopes observed in the scalar spectrum can consult Fox (2003). By definition, the ratio of the time scales is equal to the area under the normalized scalar energy spectrum as follows ... 

As discussed in Section 2.1, in high-Reynolds-number turbulent flows the scalar dissipation rate is equal to the rate of energy transfer through the inertial range of the turbulence energy spectrum. The usual modeling approach is thus to use a transport equation for the transfer rate instead of the detailed balance equation for the dissipation rate derived from (1.27). Nevertheless, in order to understand better the small-scale physical phenomena that determine e, we will derive its transport equation starting from (2.99). 

We shall first find a component P0(nl9..., nM) of P. Quantity P0 describes the a priori probability of the nm photons as they are incident upon the input slit. This is independent of the df. Hence, what is the probability P0 that nx slit photons will enter cell 1, and. .., and nM photons will enter cell Ml One possibility is that P0 is a constant, independent of nm. But that actually assumes the unknown spectrum to be equal-energy white (see Section VII), quite a restrictive assumption. We find next an expression for P0 that allows for a more general state of prior knowledge about the spectrum, in fact the most general. 

This corresponds to prior knowledge of an equal-energy white spectrum. Combining Eqs. (16) and (20) then yields... 

Principle (17) now consists of just the Bose-Einstein degeneracy factor, exactly Kikuchi and Soffer s form (1977). Also, as these authors showed (see also Section IX.B), in the case (11a) of sparsely occupied df it becomes Jaynes s maximum-entropy form (39) (Jaynes, 1968). Hence, both the Kikuchi-Soffer and Jaynes estimators are special cases of the ML approach, corresponding to the prior knowledge that the unknown spectrum is equal-energy white with the highest conviction. 

In summary, it makes more sense for maximum ignorance about a spectrum to mean an infinity of equally likely possibilities (33) than to imply the unique, equal-energy white spectrum (31). 

Here the prior probability law is of the general form (20). To review, this includes the case where the Qm are simply guessed at, based on the user s expectations, or where the image data im are used to represent <2m, or the case (24) of empirical data and high conviction, or the case (31) of an equal-energy white signal spectrum. 

Figure 2. Comparison of the NO+ spectra with equal energy scales of (a) ZEKE spectrum, showing ion states up to v = 26 (b) total ion spectrum and (c) photoelectron spectrum from Turner s book [2].

Figure 3-14. Experimental (solid line) and calculated (dashed line) spectra of [FeIU(PyPepS)2]- in water. Calculated spectrum obtained by collecting the vertical transitions of 25 solute-solvent configurations in 60 equal energy intervals, then convoluted with Gaussians with width of 0.3 eV. The dotted spectrum is collected in 200 intervals without convoluting...

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