Energy loss function calculation

Figure 3.1. (a) Experimental transmission of two thermal SI02 layers 26 nm thick each produced on both sides of Si plate by heating in air at 800°C for 20 min s- (dashed line) and p-polarization (solid line) = 74°. (6) The LO (solid line) and TO (dashed line) energy loss functions calculated on basis of optical constants of thermal Si dioxide grown at 1150°C [52]. 

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 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... 

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. 

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

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). 

Figure 3.11. (a) Calculated ATR spectra of SiOa layer 2.5 nm thick on Ge for s-polarization (short-dashed line) and p-polarizarion (solid line) at = 15° p) dispersion of TO (short-dashed line) and LO (long-dashed line) energy loss functions and s and Re(e) (solid line) of Si02 (c) dispersion of refractive index n and extinction coefficient k of CVD Si02 after Rakov [52]. 

To estimate the possible error in n2 and k2 caused by an inaccuracy in the layer thickness d, AR/R was calculated as a function of d using Eq. (1.75). The results, presented in Fig. 3.72, show a linear dependence of AR/R on d at small layer thickness and an exponential dependence as d increases. The limit of the linear dependence varies, depending on the maximum value Amax of the LO energy loss function Im(l/ 2) and the angle of incidence (Fig. 3.73). Thus, for (pi = 75°, the linear dependence occurs for thicknesses that satisfy d/X 1 X 10- -2 X 10 for a wide range of values of A ax- Th analysis of the change in AR/R as a function of the thickness d allows one to assume that for ultrathin layers the relative error A A ax/Amax caused by inaccuracy in specifying d is approximately equal to Ad/d. This result means that the inaccuracy in d makes a major contribution to the accuracy of experimental values of the optical constants. 

Fig. 153. Energy loss function, Im[-l/e( tf)], of PANI-CSA calculated from the real and imaginary parts of the dielectric function. Reproduced by permission of the American Physical Society from K. Lee, A. J. Heeger, and Y. Cao, Phys. Rev. B 48, 14884 (1993). Copyright 1993, American Physical Society.

Calculation of many trajectories at different impact parameters for each given incident energy yields the energy-dependent deflection function and energy loss, which can then, through equation (1), be used to calculate the stopping cross section. 

Figure 2, Plots of the efficiencies tje, rjY, rfp, and -qc as a function of the wavelength Xg corresponding to the band gap E. The distributions have been calculated for AM 1.2 solar radiation (taken from distribution T/S of Ref. 6). Curves, E, Y, P, and C are plots of -qEy my VPy rjc as defined in Equations 3,8,12, and 16, respectively. Tfc has been calculated for 0.6, 0.8, and 1.0 eV energy loss, respectively, as...

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