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:

$$\begin{aligned} \varepsilon _M(\mathbf{q},\omega )\,=\,1+\frac{(1+i/\omega \tau )[\varepsilon ^0(\mathbf{q},\omega +i/\tau )-1]}{1+(i/\omega \tau )[\varepsilon ^0(\mathbf{q},\omega +i/\tau )-1]/[\varepsilon ^0(\mathbf{q},0)-1]}\,, \end{aligned}$$
(15.1)

where \(\mathbf{q}\) is the momentum, \(\omega \) the frequency, \(\tau \) the relaxation time, and \(\varepsilon ^0(\mathbf{q},\omega )\) the Lindhard dielectric constant [5]

$$\begin{aligned} \varepsilon ^0(\mathbf{q},\omega )\,=\,1+\frac{4\pi ^2q^2}{e^2}B(\mathbf{q},\omega )\,, \end{aligned}$$
(15.2)
$$\begin{aligned} B(\mathbf{q},\omega )\,=\,\int \frac{d\mathbf{p}}{4\pi ^3}\frac{f_{\mathbf{p}+\mathbf{q}/2}-f_{\mathbf{p}-\mathbf{q}/2}}{\omega -(\varepsilon _{\mathbf{p}+\mathbf{q}/2}-\varepsilon _{\mathbf{p}-\mathbf{q}/2})/\hbar }\,. \end{aligned}$$
(15.3)

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]:

$$\begin{aligned} \varepsilon ^0(q,\omega )\,=\,1+\frac{\chi ^2}{z^2}[f_1(u,z)\,+\,i\,f_2(u,z)]\,, \end{aligned}$$
(15.4)

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

$$\begin{aligned} f_1(u,z)\,=\,\frac{1}{2}\,+\,\frac{1}{8z}\,[g(z-u)\,+\,g(z+u)]\,, \end{aligned}$$
(15.5)
$$\begin{aligned} f_2(u,z)=\left\{ \begin{array}{lll} \frac{\pi }{2}u, &{} z+u<1 \\ \frac{\pi }{8z}[1-(z-u)^2], &{} |z-u|<1<z+u \\ 0, &{} |z-u|>1\,, \end{array} \right. \end{aligned}$$
(15.6)

where

$$\begin{aligned} g(x)\,=\,(1-x^2)\,\ln \left| \frac{1+x}{1-x}\right| \,. \end{aligned}$$
(15.7)

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:

$$\begin{aligned} {\text{ Im }}\left[ \frac{-1}{\varepsilon _M(\omega _i,\gamma _i\,;q,\omega )}\right] \,=\,\frac{\varepsilon _{M_2}}{\varepsilon ^2_{M_1}+\varepsilon ^2_{M_2}}\,, \end{aligned}$$
(15.8)

where

$$\begin{aligned} \varepsilon _M\,=\,\varepsilon _{M_1}+i\varepsilon _{M_2} \end{aligned}$$
(15.9)

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]:

$$\begin{aligned} {\text{ Im }}\left[ \frac{-1}{\varepsilon (q=0,\omega )}\right] \,=\,\sum _i A_i {\text{ Im }}\left[ \frac{-1}{\varepsilon _M(\omega _i,\gamma _i\,;q=0,\omega )}\right] \,. \end{aligned}$$
(15.10)

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]

$$\begin{aligned} {\text{ Im }}\left[ \frac{-1}{\varepsilon (q=0,\omega )}\right] \,=\,\sum _i A_i {\text{ Im }}\left[ \frac{-1}{\varepsilon _M(\omega _i,\gamma _i\,;q=0,\omega )}\right] \,= \nonumber \\ \,\sum _i A_i {\text{ Im }}\left[ \frac{-1}{\varepsilon _D(\omega _i,\gamma _i\,;q=0,\omega )}\right] \,, \end{aligned}$$
(15.11)

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]

$$\begin{aligned} {\text{ Im }}\left[ \frac{-1}{\varepsilon _D(\omega _i,\gamma _i\,;\mathbf{q}=0,\omega )}\right] \,=\,\frac{\gamma _i \omega }{(\omega _i^2-\omega ^2)^2+(\gamma _i \omega )^2}\,, \end{aligned}$$
(15.12)

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]

$$\begin{aligned} {\text{ Im }}\left[ \frac{-1}{\varepsilon (\mathbf{q},\omega )}\right] \,=\,\sum _i A_i {\text{ Im }}\left[ \frac{-1}{\varepsilon _M(\omega _i,\gamma _i\,;\mathbf{q},\omega )}\right] \,. \end{aligned}$$
(15.13)

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:

$$\begin{aligned} {\text{ Im }}\left[ \frac{-1}{\varepsilon (q=0,\omega )}\right] \,=\,\left\{ \begin{array}{lll} &{}&{} {\sum _{i}} A_i {\text{ Im }}\left[ \frac{-1}{\varepsilon _M(\omega _i,\gamma _i\,;q=0,\omega )}\right] \,\,\,\,\,\, \omega <\omega _{i,edge}\\ &{}&{} {\sum _{i,sh}} A_{i,sh} {\text{ Im }}\left[ \frac{-1}{\varepsilon _M(\omega _{i,sh},\gamma _{i,sh}\,;q=0,\omega )}\right] \,\,\,\,\,\, \omega \ge \omega _{i,edge}\\ &{}&{} \end{array} \right. \end{aligned}$$
(15.14)

where the first term represents the contribution of the outer electrons while the second one includes the electrons of the outermost atomic inner shells.

15.3 Summary

In this chapter, after a brief discussion about the Mermin theory [1], the Mermin energy loss function-generalized oscillator strength method (MELF-GOS method), in the framework of the dielectric formalism, was shortly described [2,3,4].