Energy line positions

Figure 5 Density of states of Ni V clusters with N — 5, 6, and 7, calculated by the tight binding method sp (dashed lines) and d (continuous lines). Positive and negative values correspond to up and down J, spins, respectively. The Fermi level is at the energy zero. Adapted with permission from Ref. 45.

It varies strongly with DqlB and may be positive or negative, as shown in Figure 11.11. The dashed lines indicate the energy level positions of a number of different crystals. In a high-strength crystal field, DqlB 2.3 as in ruby (Al203 Cr3+) and alexandrite(41) AE > 0 (2350 cm-1 and -800 cm-1 for ruby(42) and alexandrite,(41) respectively), and the Cr3+ emission is dominated by the sharp R lines (2E 4A2 transition). However,... 

Figure 11.11. Simplified Tanabe-Sugono diagram. The dashed line represents the approximate energy level positions of the Cr3+ ion in various host crystals.

The effect of the nanoparticle volume fraction on the displacement of the contact line becomes pronounced only at higher volume fractions. For example, the displacement of the contact line is 10 times the nanoparticle diameter or approximately 0.2 im for a nanoparticle volume fraction of 0.25, while there is no appreciable change in the contact line position when the volume fraction is 0.2. This non-linear dependence of contact line position on nanoparticle volume fraction is consistent with the form of Eq. 10, where the film energy contribution due to structural disjoining pressure is subtracted from the surface energy contribution. The extent of displacement of the con-... 

Figure 13.2—Simplified schematic of an atom showing the origin, and the Siegbahn nomenclature, of some fluorescence radiation processes caused by impact of a photon having a high energy. The position of the spectral line is not significantly influenced by the chemical combination in which the atom is found. For example, the Kat line from sulphur is observed at 0.5348 nm for S + and at 0.5350 nm for S°, yielding a shift of 1 eV, which is comparable to the natural line width for X-rays.

Figure 27-6 Total magnetic energies and transition energies for the possible states of the protons of CI2CH—CH=0 at close to 60-MHz observing frequency. The energy levels on the left are without correction for the spin-spin interactions, those on the right include the corrections. The chemical shifts with respect to TMS at exactly 60 MHz are 350 Hz and 580 Hz. The resulting line positions are shown in Figure 27-7.

Now if we plot the energies of the transitions shown in Figures 27-5 and 27-6, we get the predicted line positions and intensities of Figure 27-7. Four lines in two equally spaced pairs appear for C12CH—CH=0, as expected from the naive rules for spin-spin splitting. 

Figure 38. Electron spectra for transfer ionization system He+-Ca at different collision energies. Nominal energy % and position e, are indicated.110 Solid line result of semiclassical calculations with model functions Vt and I ...

The lowest energy line belonging to the absorption spectrum of the 6T majority phase is a very weak shoulder at 18,350 cm-1. The line a0 of the dominant trap (17,325 cm is found very close to the position of the above-mentioned small peak at 17,450 cm-1 in the absorption spectrum. The important finding is that additional lines (which were not observed on glass) are found at 18,340 cm-1 (0-0) 17,645 cm-1 (v3) 16,885 cm-1 (v4, intense) and 15,400 cm-1 (2v4). The very small Stokes shift of <10 cm-1 points again to the high structural order in this phase. 

It is possible to model the vibronic bands in some detail. This has been done, for example, by Liu et al. (2004) forthe 6d-5f emission spectrum of Pa4+ in Cs2ZrCl6, which is analogous to the emission spectrum of Ce3+. However, most of the simulations discussed in this chapter approximate the vibronic band shape with Gaussian bands. The energy level calculations yield zero-phonon line positions, and Gaussian bands are superimposed on the zero-phonon fines in order to reproduce the observed spectra. Peaks of the Gaussian band are offset from the zero phonon fine by a constant. Peak offset and band widths, which are mostly host-dependent, may be determined from examination of the lowest 5d level of the Ce3+ spectrum, as they will not vary much for different ions in the same host. It is also common to make the standard... 

Since kinetic energy is positive, the action integral A = flTt dx is nondecreasing. This suggests using a global parameter r = s defined by the Riemannian line... 

We saw in Chapter 2 that when the chemical-shift difference in Hz, An, decreases and approaches the order of magnitude of 7 we get distortions of the peak heights ( leaning ) and of the line positions. We are now in a position to derive this effect precisely for the AB system (Ha, Hb). The Hu and H44 terms of the Hamiltonian are enormous compared to the central portion of the matrix for a 600-MHz spectrometer va + n is around 1200 MHz So we can assume that these diagonal terms are the correct energies and that the aa and PP states are true stationary states. We only need to look at the central 2x2 matrix in the Schrodinger equation ... 

In some physical situations, namely when the hyperfine energy is small compared to the nuclear Zeeman energy, then the hyperfine splitting is linear in (the projection magnitude of) matrix A for example, see Ref. 131. Then, provided that g is anisotropic (note Ref. 128 on this point), one can detect asymmetry of A directly from EPR line-position measurement see below. 

We can call the terms shown explicitly in Equation (A3) quadratic terms other (higher-order) terms can exist [Ref. 7, Section 6.7]. The above spin-Hamiltonian is used to obtain the energies (and hence also transition energies) of the spin system considered. Another, similar spin-Hamiltonian (with the excitation magnetic-field amplitude vector B, replacing B in Equation (A3)), yields the transition relative intensities. The line positions and intensities obtained are expressed in terms of the scalar ( tensors of zeroth rank) parameters g, gn, D, P,A,..., derived as projections from the matrices g, gn, D, P and A.12... 

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