Level-width function

Assume that a noninteracting nanosystem is coupled weakly to a thermal bath (in addition to the leads). The effect of the thermal bath is to break phase coherence of the electron inside the system during some time Tph, called decoherence or phase-breaking time. rph is an important time-scale in the theory, it should be compared with the so-called tunneling time - the characteristic time for the electron to go from the nanosystem to the lead, which can be estimated as an inverse level-width function / 1. So that the criteria of sequential tunneling is... 

Using the level-width function (below without spin polarization of the leads)... 

We should stress once more that this formula is valid for finite voltage. Therefore, the voltage dependence of the level-width functions is important. 

At strong coupling to the leads and the finite level width the master equation approach can no longer be used, and we apply alternatively the nonequilibrium Green function technique which have been recently developed to treat vibronic effects in a perturbative or self-consistent way in the cases of weak and intermediate electron-vibron interaction [113-130]. 

This result describes quantitatively the energy distribution of the decaying nij hole-state. The function is symmetric in E around En(j. For E = E (j it has a maximum, and its fwhm value is given by En(j which is called the natural or inherent level width because it originates from the decaying hole-state which is inherent to the atom. As an example, a compilation of level widths r in neon is given in Table 2.2. Because of the replacement made in the derivation of equ. (2.18b) for xn(, one has (in atomic units)... 

Figure 2.7 Theoretical level widths for K-shell ionization as a function of the atomic number. The total level width T is the sum of two contributions that come from radiative (fluorescence) decay, TR, and non-radiative (Auger) decay, TA. From [Kra79].

To obtain a qualitative picture of the complex structure of SO3 adsorbed on the Pt (111) surface, the density of states (DOS) was calculated from the DV-Xa molecular orbital method. The level width was broadened by a Gaussian function (1.0 eV FWHM) to mimic the solid state. The results of the studies are summarized in the following paragraphs. In each DOS, Fermi energy is set at 0. In configuration A, the adsorbed O atoms are classified as of two types. One is two O atoms bound to Pt surface atom (0(1)) and the other is one O atom unbound to Pt surface (0(2)). 

The results in the first two columns of Table III imply that H2(v=1,j 2)-Ar complexes will predissociate almost 3 times as rapidly as H2(v 0,j=2)-Ar. However, within a first-order treatment, rotational inelasticity depends on the same type of squared matrix element of V2(v,j v j r) as does the level width, except that the (isoenergetic) wavefunctions being coupled are both continuum functions lying above the rotational threshold. In tenns of Figure 1, they would be continuum eigenfunctions of V (R) and V2(R) at... 

One obvious conclusion from the above study is that calculated vibrational and rotational predissociation levels widths are extremely sensitive to the quality of the wave function used to represent the "initial" and "final" states, as well as to the coupling function itself. While this makes it difficult to use observations of this type to obtain new information about the potential energy surface, it also means that they should provide an extremely stringent constraint on the properties of a surface so determined. We have also seen that while they sometimes give useful qualitative information, none of the approxioiate methods considered herein are reliable enough chat they may be used in a quantitative analysis of experimental level widths. 

An immediate consequence of this approximation is the neglect of the resonance level shifts. The reason is that the shifts, which are given as the Hilbert transform of the level widths, vanish for a constant function. Past treatments of FIT have also neglected multichannel effects, arising when each level is coupled to a multiplicity of continua. 

Fig. 2. The ratio of the Doppler broadened peak cross section aj to the true peak cross section as a function of the ratio of the Doppler width A to the level width F.

In larger molecules the phase-coherence time of excited levels may be shorter than the population lifetime because of perturbations between closely spaced levels of different electronic states, which cause a dephasing of the excited-level wave functions. One example is the NO2 molecule, where the width of the Hanle signal turns out to be more than one order of magnitude larger than expected from independent measurements of population lifetime and Landd factors [851, 852]. This discrepancy is explained by a short intramolecular decay time (dephasing time), but a much larger radiative lifetime [853]. 

Re-initialization Procedure Maintaining (p as a distance function becomes important for providing a uniform thickness at the interface with a fixed width in time and to avoid steep gradients. It is also essential that the level set function is re-initialized to a distance function without changing its zero level set. An iterative procedure is used to maintain 0 as a distance function. Their re-initialization procedure is based on solving the following partial differential equation to steady-state solution at each time step ... 

Porter and Thomas (42) have given strong theoretical arguments that the statistical frequency functions for all the partial resonance level widths should be chi-squared distributions with v degrees of freedom as given by (75). In this case, the number of degrees of freedom is the number of channels open for decay of the compound nucleus by the process to be... 

Spectral lines in discrete absorption or emission spectra are never strictly monochromatic. Even with the very high resolution of interferometers, one observes a spectral distribution l v) of the absorbed or emitted intensity around the central frequency v — Ei — Ek)/h corresponding to a molecular transition with the energy difference AE = Ei — Ek between upper and lower levels. The function I(v) in the vicinity of vq is called the line profile (Fig. 3.1). The frequency interval 8v = v2 — I between the two frequencies v and V2 for which 7(i i) = /(v 2) = ( o)/2 is ihe full-width at half-maximum of the line (FWHM), often shortened to the linewidth or halfwidth of the spectral line. 

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