Energy broadening

Figure 25. Ion mass peaks at different two-photon energies. Broadenings of trimethylamine (TMA ) ion peaks as a function of the ionization energy. A hv2 > 3.875 eV B hv2 = 3.688 eV C hv2 = 3.607 eV. Excitation energy of paraxylene (PX) in the Si state = 3.90 eV. The broadenings in B and C correspond to time constants of 160 20 and 200 20 ns, respectively. Peaks corresponding to TMA H+ are also observable. Taken with permission from Int. J. Mass Spectrom. Ion Proc. 1994, 131, 233-264.

As discussed in Chapter 4.3, the interaction with the metal broadens the electronic energy levels of an adsorbate. If r is the average lifetime of an electron before it is transferred to the metal, the associated energy broadening is A = h/r. In a simple approximation the density of states of the adsorbate has a Lorenz shape [6] ... 

As the temperature increases, the distribution of collision energies broadens and shifts to higher energies (Figure 12.15), resulting in a rapid increase in the fraction of collisions that lead to products. At 308 K, for example, the calculated value of / for the reaction with Ea = 75 kj/mol is 2 x 10-13. Thus, a temperature increase of just 3%, from 298 K to 308 K, increases the value of / by a factor of 3. Collision theory therefore accounts nicely for the exponential dependence of reaction rates on reciprocal temperature. As T increases (1 /T decreases), / = e E RT increases exponentially. Collision theory also explains why reaction rates are so much lower than collision rates. (Collision rates also increase with increasing temperature, but only by a small amount—less than 2% on going from 298 K to 308 K.)... 

Spectral functions, described before, determine single-particle properties of the system, such as quasiparticle energy, broadening of the levels (life-time), and density of states. These functions can be modified in nonequilibrium state, but most important kinetic properties, such as distribution function, charge, and current, are determined by lesser Green function... 

Before these partial quantities are discussed further, an important comment has to be made unlike the partial transition rates, the partial level widths have no direct physical meaning, because even for a selected decay branch it is always the total level width which determines the natural energy broadening. The partial level width is only a measure of the partial transition rate. Both aspects can be inferred from the Lorentzian distribution attached to a selected decay branch, e.g., Auger decay, which is given by... 

Fig. 5. Zero pressure extrapolated cross sections for the competitive collision-induced dissociation processes of (H20)Na+(NH3) with xenon in the threshold region as a function of kinetic energy in the center-of-mass frame (lower axis) and laboratory frame (upper axis). Solid lines show the best fits to the data using the model of Eq. (7) convoluted over the neutral and ion kinetic energies and the internal energies of the reactants, using common scaling factors. Dashed lines show the model cross sections in the absence of experimental energy broadening for reactants with an internal energy of 0 K. Adapted from [45]...

We can interpret the universal result (4) as an expression of the time-energy uncertainty relation for an unstable level with lifetime At, relating the energy broadening (uncertainty) of e) to At, the interval between measurements (Fig. 1),... 

The following other form is also convenient By introducing the energy broadening of the group linked to the electrode [51, 52, 90]... 

The electronic coupling between the linking group and the electrode surface is represented by the energy broadening of the linker group levels, Ai, Ex... 

Equations (8-11) and (8-12) can be inserted into eqns.(8-l)-(8-3) and the electrochemical current obtained. The following other transmission coefficient form is, however, also convenient. By introducing the energy broadening of the group linked to the electrode, i.e. the group 1 , as ... 

FlG. 10.4. An illustration of the wavevector and energy broadenings of polariton branches. Reprinted with permission from Agranovich et al. (15). Copyright 2003, American Physical Society. 

Fig. 10.4 An illustration of the wavevector and energy broadenings of polariton branches 283...

Fig. 3 Adiabatic potential-energy curves for various values of the energy broadening A full line A = 0.01 eV long dashes A = 0.05 eV short dashes A = 0.1 eV. The energy of reorganization was taken as 6 ksT.

The Convolution of brick with energy broadening depends on the detector resolution, energy straggling and depth. The representation is shown in Figs. 2.18 and 2.19. 

Interestingly, an equivalent equation can also be obtained from the imaginary part of the dynamical self-energy as the electronic response to a weak, slow time-dependent perturbation. The lifetime associated with a given mode q is then defined as the inverse transition rates for the fundamental transition l- 0, Xg = l/r[ o, and can be related to the state energy broadening as =... 

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