Energy-loss functions

Figure 1. Electronic and nuclear energy loss function of (a) Au implanted in Si02 and (b) He implanted in Si02. The fraction of the electronic or nuclear 5 stopping power with respect to the total (S tot = S + for Au (c) and He (d). (Reprinted from Ref [1], 2005, with permission from Italian Physical Society.)...

Considering an initial electron energy much larger than kBT, Rips and Silbey show that the distribution of thermalization time is given by the first two moments of the energy loss function (e) per unit time,... 

Parameters of Different Substances and the Resonance Frequencies a>r Corresponding to the Maximum of the Energy-Loss Function... 

In the vicinity of the maximum the energy-loss function (3.18) is of Lorentz form. With y- 0 it transforms into a delta function. In order to see this, let us use the representation of a delta function for a nonnegative variable (see the Mathematical Appendix A in Ref. 99) ... 

As one can easily check, if the damping is small, the energy-loss function has only one maximum corresponding to e(a>) = 0 at... 

As an example, let us consider liquid water (Fig. 8). The highest oscillator strength, fx = 0.43, corresponds to the transition ha>x = 13.5 eV. The peak the energy-loss function Im [-l/e(w)] has around 21 eV is of plasmon nature, that is, corresponds to longitudinal collective oscillations... 

It is very different to calculate the energy-loss function Im[-1/ e(q, a>)] theoretically. At <7—>0 one can use the optical data. This way Lindhard90 has obtained a formula for the permittivity of the electron gas, while the authors of Ref. 100 have constructed a semiempirical formula for e(q, n>) of water. 

The energy loss function is related to the dynamical structural factor S(q, ft)),90 which describes the scattering of particles in a liquid, namely,... 

Tan GL, DeNoyer LK, French RH, Guittet Ml, Gautier-Soyer M (2004) Kramers-Kronig transform for the surface energy loss function. J Electron Spectrosc Relat Phenom 142(2) 97-103... 

The a bonds in the backbone of vinyl polymers should not be describable in terms of local states of small model molecules because of overlap of carbon atomic orbitals only 1.5X apart. This concept can be tested in polyethylene where the least bound C-C bond band widths have been calculated to be about 3 eV (.8). The energy loss function, Im(-l/e), for polyethylene is given in Figure 3 where and the real and imaginary parts of the... 

Figure 3. Polyethylene measured energy loss function, Im(—l/e) ( ). Results of a Kramers-Kronig analysis e,(--------------) (---) (i).

The choice of basis sets is a particularly important technical issue for this type of calculation. Specifically, the small, highly contracted basis sets such as used in Ref. [16] generally are insufficient to calculate the real parts of dielectric functions and, consequently, the energy loss functions. We started from Partridge s 16sllp set [17], contracted the seven tightest s-functions and the four tightest p-functions, removed the most diffuse s and the two most diffuse p-functions to avoid approximate linear dependencies, and added a full set of three d-functions with exponents equal to those of the... 

Local Field Effects (LFEs) are illustrated in Fig. 4. Im Eo o reproduces some values from Fig. 3. For comparison, Im[l/ATq ol includes an estimate of the local field obtained by calculating Kq qi as the inverse of a 9 X 9 dielectric matrix which contains G = 0 and the eight vectors of the closest shell in the bcc reciprocal lattice. The reduction of the values without LFE (lim eo,oL open symbols) compared to those with LFE (llm(l/ATqo)I, filled symbols) is of the order reported by Van Vechten and Martin [24] (without their dynamical correlations ). The different sign of the effect for frequencies above and below the peak has been noticed before [25]. The differences are even smaller for the energy loss function. Hence the energy loss reported in the next paragraph was calculated from so o(q, w) alone. 

Figures 8.10(a) and (b) show electron energy loss functions at different momentum transfers for Sr2Cu02Cl2, along [100] and [110], respectively. There is an optically allowed transition in both directions, whose position is at...

The MD simulation by Marchi [8] of the librational band (Fig. 18) is rather poor in this example. A better simulation is for the translational band (solid line in Fig. 19). The dashed curve represents here the energy loss function determined by the dielectric constant s and calculated in the cited work by Marchi. The shift between the solid and dashed curves represents a specific parameter of the fluid... 

Figure 19 Frequency dependence of the loss factor e" measured for ice in the translational band at T= 100 K (solid curves). Dashed curve represents the energy loss function. (Reproduced from Marchi [8]).

Figure 25 Loss frequency dependences e"(v) (solid curve) and energy loss function Eioss(v) (dashed curve) calculated for ice at —7°C.

Now let us estimate (see Fig 25) the so-called transverse optic-longitudinal optic splitting [8,50] characteristic for ice at v 230 cm-1. Namely, the loss curve e"(v) is shifted on the frequency scale with respect to the energy loss function ... 

In Fig. 34 we depict the energy loss function Eioss(v) defined by Eq. (71). This function is calculated for the temperature 100 K with the parameters of the model presented in Table X. The splitting AvE = v2 — v1 comprising about 13 cm-1, is close to the experimental value [8]. This splitting is substantially less than at the temperature 266 K (cf. Figs. 34 and 25). 

The term 3(— l/e q, co)) is referred to as the dielectric loss function. Structures in this function can be correlated to bulk plasmon excitations. In the vicinity of a surface the differential cross section for inelastic scattering has to be modified to describe the excitation of surface plasmons. The surface energy loss function is proportional to 3(—l/e(, cu) + 1). In general, the dielectric function is not known with respect to energy and momentum transfer. Theoretical approaches to determine the cross section therefore have to rely on model dielectric functions. Experimentally, cross sections are determined by either optical absorption experiments or analysis of reflection energy loss spectra [107,108] (see Section 4.3). 

STUDY OF THE ENERGY LOSS FUNCTION OF ELECTRONS BY VACUUM ULTRAVIOLET TECHNIQUE. 1. EXPERIMENTAL APPARATUS AND REFLECTANCE OF POLYSTYRENE. 

The energy loss functions for fi-SiC are shown in Fig. 1.1 with the other optical parameters. 

Hence, the maximum of the TO energy loss function s") for an elementary damped harmonic oscillator occurs at [Pg.22]

It can be seen from Eq. (1.82) that an IR spectrum of a thin film depends essentially on the TO and LO energy loss functions of the film substance (1.1.18°, 1.1.19°) for 5- and p-polarization, respectively, the reflectivity values are positive, and the reflection spectrum is like the absorption spectrum. Moreover, the absorption of i -polarized radiation, 1 — Rs, is greater at small angles of incidence, whereas the quantity — Rp exhibits a maximum at grazing angles of incidence. This is not the case for a dielectric substrate [Eqs. (1.80) and (1.81)] (see Sections 2.2 and 2.3 for more detail). 

Let us compare the thin-film approximation formulas for (1) the transmissivity (1.98) (2) the reflectivities for the external reflection from this film deposited onto a metallic substrate (1.82) (3) the internal reflection at (p > (pc (1.84) and (4) the external reflection from this film deposited on a transparent substrate (dielectric or semiconducting) (1.81) (Table 1.2). In all cases s-polarized radiation is absorbed at the frequencies of the maxima of Im( 2), vroi (1.1.18°), whereas the jo-polarized external reflection spectrum of a layer on a metallic substrate is influenced only by the LO energy loss function Im(l/ 2) (1.1.19°). The /7-polarized internal and external reflection spectra of a layer on a transparent substrate has maxima at vro as well as at vlq. Such a polarization-dependent behavior of an IR spectrum of a thin film is manifestation of the optical effect (Section 3.1). 

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