Doublets

It is important to evaluate the surface distortion associated with the assymetric field at the surface, a difficult task often simplified by assuming that distortion is limited to the direction normal to the plane [64, 6S]. Benson and co-workers [6S] calculated displacements for the first five planes in the (100) face of sodium chloride and found the distortion correction to of about 100 ergs/cm or about half of itself The displacements show a tendency toward ion pair formation, suggesting that lateral displacements to produce ion doublets should be considered [66] however, other calculations yielded much smaller displacements [67]. 

Fig. XVin-6. Curve-fitted Mo XPS 3d spectra of a 5 wt% Mo/Ti02 catalyst (a) in the oxidic +6 valence state (b) after reduction at 304°C. Doublets A, B, and C refer to Mo oxidation states +6, +5, and +4, respectively [37]. (Reprinted with permission from American Chemical Society copyright 1974.)...

To nnderstand the internal molecnlar motions, we have placed great store in classical mechanics to obtain a picture of the dynamics of the molecnle and to predict associated patterns that can be observed in quantum spectra. Of course, the classical picture is at best an imprecise image, becanse the molecnlar dynamics are intrinsically quantum mechanical. Nonetheless, the classical metaphor mnst surely possess a large kernel of truth. The classical stnichire brought out by the bifiircation analysis has accounted for real patterns seen in wavefimctions and also for patterns observed in spectra, snch as the existence of local mode doublets, and the... 

Flowever, we have also seen that some of the properties of quantum spectra are mtrinsically non-classical, apart from the discreteness of qiiantnm states and energy levels implied by the very existence of quanta. An example is the splitting of the local mode doublets, which was ascribed to dynamical tiumelling, i.e. processes which classically are forbidden. We can ask if non-classical effects are ubiquitous in spectra and, if so, are there manifestations accessible to observation other than those we have encountered so far If there are such manifestations, it seems likely that they will constitute subtle peculiarities m spectral patterns, whose discennnent and interpretation will be an important challenge. 

The next two temis (Lorentzians) arise from the mechanical part of the density fluctuations, the pressure fluctuations at constant entropy. These are the adiabatic sound modes (l/y)exp[-FA t ]cos[co(A) t ] with (D(k) = ck, and lead to the two spectral lines (Lorentzians) which are shifted in frequency by -ck (Stokes line) and +ck (anti-Stokes line). These are known as the Brillouin-Mandehtarn, doublet. The half-width at... 

These are the same states as in Figure Bl.l 1.8, but attention is now drawn to the populations of the four spin states, each reduced by subtracting the 25% population that would exist at very low field, or alternatively at infinite temperature. The figures above each level are these relative differences, in convenient units. The intensity of any one transition, i.e. of the relevant peak in the doublet, is proportional to the difference of these differences, and is therefore proportionally relative to unity for any transition at Boltzmaim equilibrium, and 4 for any transition. 

B Spin popiiation ratios foloving the selective iituersion of the low frequency compel neri of the H doublet... 

Figure Bl.13.8. Schematic illustration of (a) an antiphase doublet, (b) an in-phase doublet and (c) a differentially broadened doublet. The splitting between the two lines is in each case equal to J, the indirect spin-spin coupling constant.

The cross-correlation effects between the DD and CSA interactions also influence the transverse relaxation and lead to the phenomenon known as differential line broadening in a doublet [40], cf Figure Bl.13.8. There is a recent experiment, designed for protein studies, that I wish to mention at tire end of this section. It has been proposed by Pervushin etal [4T], is called TROSY (transverse relaxation optimized spectroscopy) and... 

Photoelectron peaks are labelled according to the quantum numbers of the level from which the electron originates. An electron coming from an orbital with main quantum number n, orbital momentum / (0, 1, 2, 3,. .. indicated as s, p, d, f,. ..) and spin momentum s (+1/2 or -1/2) is indicated as For every orbital momentum / > 0 there are two values of the total momentum j = l+Ml and j = l-Ml, each state filled with 2j + 1 electrons. Flence, most XPS peaks come in doublets and the intensity ratio of the components is (/ + 1)//. When the doublet splitting is too small to be observed, tire subscript / + s is omitted. 

Figure Bl.25.2 shows the XPS spectra of two organoplatinum complexes which contain different amounts of chlorine. The spectrum shows the peaks of all elements expected from the compounds, the Pt 4f and 4d doublets (the 4f doublet is iimesolved due to the low energy resolution employed for broad energy range scans). Cl 2p and Cl 2s, N Is and C Is. Flowever, the C Is caimot be taken as characteristic for the complex only. All surfaces that have not been cleaned by sputtermg or oxidation in the XPS spectrometer contain carbon. The reason is that adsorbed hydrocarbons from the atmosphere give the optimum lowering of the surface free energy and hence, all surfaces are covered by hydrocarbon fragments [9].

More general situations have also been considered. For example. Mead [21] considers cases involving degeneracy between two Kramers doublets involving four electronic components a), a ), P), and P ). Equations (4) and (5), coupled with antisymmetry under lead to the following identities between the various matrix elements... 

This establishes our assertion that the former roots are overwhelmingly more numerous than those of the latter kind. Before embarking on a formal proof, let us illustrate the theorem with respect to a representative, though specific example. We consider the time development of a doublet subject to a Schrodinger equation whose Hamiltonian in a doublet representation is [13,29]... 

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