The asymmetric lineshape

It is often convenient to write this in terms of the real detuning e — Eq thus 

In fig. 6.2, we show examples of the line profiles which occur as a result of autoionisation, as predicted by Fano s theory. 

The quantity q is called the shape index and is constant for a Fano profile. In chapter 8, situations in which q varies within an excitation channel will be discussed. However, even in such cases, it can be regarded as a constant over one line. 

The shape index determines the symmetry of the line. From equation (6.21), we note that, for a specific value the resonance energy 

The important special cases are (i) if q = 0 one has a symmetrical resonance with no absorption maximum and only a minimum (inverted 

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How then, can one recover some quantity that scales with the local charge on the metal atoms if their valence electrons are inherently delocalized Beyond the asymmetric lineshape of the metal 2p3/2 peak, there is also a distinct satellite structure seen in the spectra for CoP and elemental Co. From reflection electron energy loss spectroscopy (REELS), we have determined that this satellite structure originates from plasmon loss events (instead of a two-core-hole final state effect as previously thought [67,68]) in which exiting photoelectrons lose some of their energy to valence electrons of atoms near the surface of the solid [58]. The intensity of these satellite peaks (relative to the main peak) is weaker in CoP than in elemental Co. This implies that the Co atoms have fewer valence electrons in CoP than in elemental Co, that is, they are definitely cationic, notwithstanding the lack of a BE shift. For the other compounds in the MP (M = Cr, Mn, Fe) series, the satellite structure is probably too weak to be observed, but solid solutions Coi -xMxl> and CoAs i yPv do show this feature (vide infra) [60,61]. 

One-dimensional quadrupole echo NMR lineshape analysis of powder samples is particularly informative when fast, discrete jumps occur between sites of well-defined geometry as, for example, in a phenyl group undergoing two-site exchange. In this case, the characteristic Pake-pattern is transformed into an axially asymmetric lineshape with an apparent asymmetry parameter r] 9 0 (see Equation (6.2.3)) [1-8]. The asymmetric lineshapes, shown on the left in Fig. 6.2.2, can be derived by considering how the individual components of the principal EFG tensor become averaged by the discrete jumps. Within the molecular frame, and in units of as defined by Equation (6.2.2), the static axially symmetric tensor consists of the components = 1, = — 1/2, and V y = — 112. This traceless tensor satisfies the... 

Figures 14.1(a,b) show typical CP/MAS spectra of two types of PET yarns, an amorphous yarn wound at relatively low speed and a 36% crystalline yarn wound at relatively high speed, respectively [2]. The ethylene and carbonyl carbon peaks of the amorphous yarn are shifted about 1 ppm downfield with respect to the semicrystalline yarn, as opposed to the aromatic carbons which are shifted slightly upheld. Besides differences in chemical shift, the spectrum of the 36% crystalline yarn shows narrower lines with a better S/N ratio than the spectrum of the amorphous yarn. The broader lines in Fig. 14.1(a) are attributed to a broader orientation distribution of polymer molecules, which results in a larger distribution of isotropic chemical shifts. Additional differences between both spectra are observed in the lineshape the ethylene and carbonyl carbon peaks in Fig. 14.1(a) have a symmetric lineshape, whereas, these lines in Fig. 14.1(b) are asymmetric. The asymmetric lineshape is resolvable into two partially overlapping resonances a relatively broad low-field component and a relatively narrow high-field...

The photoionisation continuum of H is clean and featureless. Its intensity declines monotonically with increasing energy. Many-electron systems, in general, always exhibit structure embedded in the continuum. Such features are neither purely discrete nor purely continuous, but of mixed character, and are referred to as autoionising resonances. They were discovered experimentally by Beutler [254], and the asymmetric lineshape which they can give rise to follows a simple analytic formula derived by Fano [256]. For this reason, they are often referred to as Beutler-Fano resonances. A typical autoionising resonance is shown in fig. 6.1... 

The Compton scattering cannot be neglected, but it is independent of molecular structure. Then, fitting experimental data to formulas from gas phase theory, the concentration of excited molecules can be determined. Another problem is that the undulator X-ray spectrum is not strictly monochromatic, but has a slightly asymmetric lineshape extending toward lower energies. This problem may be handled in different ways, for example, by approximating its spectral distribution by its first spectral moment [12]. 

Hagen, W.R. 1981. Dislocation strain broadening as a source of anisotropic linewidth and asymmetrical lineshape in the electron paramagnetic resonance spectrum of metal-loproteins and related systems. Journal of Magnetic Resonance 44 447-469. 

If the experimental lineshapes do not exhibit sub-bands and are asymmetric, or if they involve sub-bands, but with intensity anomalies with respect to the Franck-Condon progression law, then, together with the dephasing mechanism, damping of the slow mode ought to be also considered as occurring in a sensitive competitive way. 

As in the parent binary phosphides MV, the metal 2p3 p XPS spectra for the mixed-metal phosphides M XM XV exhibit asymmetric lineshapes originating from final state effects [34] involving the metal-metal bonding network. Virtually no changes in BE are observed relative to the binary phosphides MV or the elemental metals... 

The amorphous lineshape becomes axially symmetric by -60° and then begins to collapse into a symmetric lineshape. The transformation of the amorphous lineshape from asymmetric to axially symmetric is imprecisely defined because the process which transforms it into a symmetric lineshape becomes dominant before the lineshape becomes completely axially symmetric. 

Figure 2.15. Spectra of the damping y(u>) and of the absorption c"(w) (with arbitrary absorption units) derived from our model (2.127)—(2.130), at temperatures ranging between 3 and 70 K. At low temperatures we may distinguish in the spectrum pure acoustical effects below the threshold, and combined effects (acoustical + optical) above the threshold. At high temperatures, both branches contribute to yield the broad and asymmetrical lineshape. The energy origin has been chosen at the unperturbed exciton band botton. So the absorption spectra show a red shift even at low temperatures, which should be considered when comparing the model with the experimental spectra of Figs. 2.12-13.

For a constant interaction continuum a pure Lorentzian function is obtained for the lineshape. When the continuum is allowed to vary exponentially the Lorentzian becomes asymmetric. For the coupling of the primary state (associated with an oscillator strength) with a manifold of discrete states and with the radiation continuum, a structured lineshape is obtained. The radiation field continuum may be replaced by any other interacting continuum, and in this sense the model approaches that discussed in the phenanthrene lineshape problem. However, the model lacks complete generality in that the manifold of discrete states is not fully random. 

Figure 12 (a) Schematic representation of how the projection of Bragg rods leads to the asymmetric shape shown in the Warren lineshape figure illustrated in panel (b). The origin of this asymmetry is discussed in the text. This diffraction pattern uses a Lorentzian profile for the structure factor of the Bragg rod. (Reprinted with permission Arnold, Chanaa, Clarke, Cook and Larese 2006, American Physical Society)... 

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