Differential cross section for energy loss

In Section 2, we provide a precis of stopping power theory and its connection to the END approach for which we outline the salient features in Section 3. In Section 3.2, we discuss the treatment of the END trajectories and their connection to the differential cross section and energy loss. In Section 4, we present some simple applications and results of our approach. In Section 5, we discussed future directions on the END approach to stopping cross section. Finally, Section 6 contains our conclusions. 

The theoretical basis for an effective charge z f[ E) associated with collisional energy loss has been investigated by several authors (see, e.g.. Refs. 69-72). Within the Born approximation, the doubly differential cross section for ejection of an electron with energy IF by a He ion can be written as... 

Differential cross sections for elastic scattering of electrons from THF have been determined together with vibrational and electronic energy loss spectra. Band assignments were verified by Cl calculations <2005MI411>. 

Quantum-mechanically, the process of energy loss by electrons can be considered as formally equivalent to the absorption of photons, so that we may express the differential cross-section for inelastic scattering (energy loss) under the dipole approximation as ... 

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). 

It should be noted that low-loss spectra are basically connected to optical properties of materials. This is because for small scattering angles the energy-differential cross-section dfj/dF, in other words the intensity of the EEL spectrum measured, is directly proportional to Im -l/ (E,q) [2.171]. Here e = ei + iez is the complex dielectric function, E the energy loss, and q the momentum vector. Owing to the comparison to optics (jqj = 0) the above quoted proportionality is fulfilled if the spectrum has been recorded with a reasonably small collection aperture. When Im -l/ is gathered its real part can be determined, by the Kramers-Kronig transformation, and subsequently such optical quantities as refraction index, absorption coefficient, and reflectivity. 

Another procedure for calculating the W value has been developed by La Verne and Mozumder (1992) and applied to electron and proton irradiation of gaseous water. Considering a small section Ax of an electron track, the energy loss of the primary electron is S(E) Ax, where S(E) is the stopping power at electron energy E. The average number of primary ionizations produced over Ax is No. Ax where o. is the total ionization cross section and N is the number density of molecules. Thus, the W value for primary ionization is 0)p = S(E)/No.(E). If the differential ionization cross section for the production... 

Heijmen TGA, Moszynski R, Wormer PES, Van der Avoird A, Buck U, Ettischer I, Krohne R (1997) Total differential cross sections and differential energy loss spectra for He—C2H2 from an ab initio potential. J Chem Phys 107 7260-7265... 

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