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Dielectric formalism is the most used method for investigating the interaction of swift electrons with solid targets. In this chapter the Mermin energy loss function-generalized oscillator strength method (MELF-GOS method) is briefly described within the framework of dielectric formalism [1,2,3,4].
15.1 The Mermin Theory
The Mermin dielectric function [1] is given by:
where \(\mathbf{q}\) is the momentum, \(\omega \) the frequency, \(\tau \) the relaxation time, and \(\varepsilon ^0(\mathbf{q},\omega )\) the Lindhard dielectric constant [5]
In these equations e is the electron charge, \(f_{\mathbf{p}}\) is the Fermi-Dirac distribution, and \(\varepsilon _{\mathbf{p}}\) the free electron energy.
Note that the Lindhard dielectric function [5] can be numerically calculated by using Eqs. (15.2) and (15.3). The integration can also be carried out in closed form. The result of the integration is the following [2, 3, 6]:
where \(u=\omega /(q v_F)\), \(z=q/(2q_F)\), and \(\chi ^2=e^2/(\pi \,\hbar \,v_F)\) is a measure of the electron density [6]. In this equation, \(v_F\) is the Fermi velocity of the valence electrons of the target and \(q_F=mv_F/\hbar \). The functions \(f_1(u,z)\) and \(f_2(u,z)\) are given by
where
15.2 The Mermin Energy Loss Function-Generalized Oscillator Strength Method (MELF-GOS)
Let us now consider a superposition of free and bound oscillators. For any oscillator, the energy loss function is given by the opposite of the imaginary part of the inverse of the Mermin dielectric function:
where
and \(\omega _i\) and \(\gamma _i\) are, respectively, the frequency and the damping constant associated to each specific oscillator. A linear combination of Mermin-type energy loss functions, one per oscillator, allows to calculate the energy loss function (ELF) for \(q\,=\,0\), for any specific material [2,3,4]:
In this equation, \(A_i\), \(\omega _i\), and \(\gamma _i\) are determined by looking for the best fit of the available experimental optical ELF. Actually, as both Mermin and Drude-Lorentz oscillators converge on the same values in the optical limit (i.e. for \(q\,=\,0\)) [7]
where the Drude-Lorentz functions \({\text{ Im }}\left[ \frac{-1}{\varepsilon _D(\omega _i,\gamma _i\,;\mathbf{q}=0,\omega )}\right] \) are given by [8]
the best fit can also be obtained using a linear combination of Drude-Lorentz functions, instead of Mermin functions. Once the values of the best fit parameters have been established (see, for example, Refs. [4, 9, 10]), the extension beyond the optical domain (\(\mathbf{q}\ne 0\)) can be obtained by [2,3,4]
Planes et al. [2], Abril et al. [3], and de Vera et al. [4] construct the ELF in the optical limit including the contribution of the electrons from the outermost atomic inner shells as follows:
where the first term represents the contribution of the outer electrons while the second one includes the electrons of the outermost atomic inner shells.
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Dapor, M. (2017). Appendix F: The Mermin Theory and the Generalized Oscillator Strength Method . In: Transport of Energetic Electrons in Solids. Springer Tracts in Modern Physics, vol 999. Springer, Cham. https://doi.org/10.1007/978-3-319-47492-2_15
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DOI: https://doi.org/10.1007/978-3-319-47492-2_15
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