Lorentzian energy averaging

Long-range order theory, 35 4-5 Looper s walk capping, 32 438-445 Lorentzian energy averaging, 34 217 Lorentzian function, energy-dependent, 34 243 Losod, 33 215, 224, 258 Low-coordinated transition-metal ions, 34 133 Low energy... 

Gaussian-like distribution of energy around the energy average. Other ensembles with non-Boltzmann distributions can enhance the sampling considerably for example, in the multi-canonical approach [97, 98], all the conformations are equiprobable in energy in Tsallis statistics [99], the distribution function includes Boltzmann, Lorentzian, and Levy distributions. 

It should be noted that, because the interaction energy is averaged in a different way with respect to the case of shortest zs discussed above, the equations look very different. In particular, the J(io) do not have the usual Lorentzian form (Fig. 4.5). 

The intermediate-energy P-matrix method averages the T-matrix elements over energy using a Lorentzian averaging function of width /. 

As shown in figure 4.9, the time evolution P f) shows three distinct time dependencies, each characterized by either 7, F, or e (1) The Lorentzian-like envelope, with width F for the complete absorption spectrum transforms into an exponential decay with rate constant F/ft, which is for the short-time decay of PXt)- (2) The set of resonances, separated on average by an energy e, transform into a set of oscillations (i.e., recurrences) whose periods are approximately elh. (3) The envelope of each individual resonance also transforms into an exponential-like decay, characterized by the rate 7/ft, which corresponds to leakage from the sparse i) — /) subspace into the quasi-continuum ). The recurrences described above in (2) are damped out by this slow decay. 

Fig. 3.10. Sketches of the (average) density of states per atom n(E) for three different values of the halfwidth r of the Lorentzian distribution of single site energies. The mobility edges separate regions of localized states (shaded) from those of extended states, and always lie within the interval [ - E, E ]. is Anderson s critical value of the randomness.

There are several exact results available in this model to serve as checks on approximate calculations of and L(E). In the limit of small x or 6, all averaged quantities may be expanded in powers of a small parameter. For arbitrary x and 5, a number of moments of the density of states can be calculated exactly (Velicky, 1968). When 5 > 2, the density of states is split into two separated sub-bands, centered about and e , each of width B. Thus in the limit 5 -> a site containing a B atom is forbidden to an electron with energy near and percolation theory (Frisch, 1963) may be used to determine the probability that such an electron is trapped or free to move across the crystal. When x is greater than x, the criticd value for the onset of percolation, there will be extended states in the A sub-band. Since x, is less than 1/2 for all three-dimensional lattices, we observe that at least one of the strongly split sub-bands will always contain extended states, in contrast to the complete localization observed for r>Fg in the Lorentzian model of the preceding section. 

To set up the lifetime experiment, fluorescence excitation spectra were recorded using pulses of 90 ns duration corresponding to a spectral bandwidth of 15 MHz at a repetition rate of 1 MHz. The bandwidth was limited by the pulses rise and fall times, the pulse shapes and the frequency jitter of the laser. Fig. 8(a) shows a typical spectrum with four individual molecular resonances A, B, C and D. On average, the lines are about 25 MHz wide as compared to a homogeneous linewidth of about 8 MHz measured by earlier experiments using cw radiation and lower excitation energies. The best fit of the absorption profile of molecule C was obtained using a Lorentzian profile with a FWHM of 27 MHz. 

It was shown in Ref. [15] that for optical transitions in glasses the TLS dynamics results in spectral diffusion, which shows up in the experiment as a time and temperature-dependent Lorentzian line broadening. The width of this Lorentzian line must be calculated by averaging over the distribution of energies and relaxation rates P E, R). It can be written as ... 

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