Abstract
The Ritchie theory describes the relationship between the dielectric function and the electron energy loss in a solid. It allows to calculate the differential inverse inelastic mean free path, the elastic mean free path, and the stopping power.
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The Ritchie theory describes the relationship between the dielectric function and the electron energy loss in a solid. It allows to calculate the differential inverse inelastic mean free path, the elastic mean free path, and the stopping power. The original version of the Ritchie theory can be found in Ref. [1]. Also see Refs. [2,3,4,5,6] for further details.
13.1 Energy Loss and Dielectric Function
The response of the ensemble of conduction electrons to the electromagnetic disturbance due to electrons passing through a solid and losing energy in it, is described by a complex dielectric function \(\varepsilon (\varvec{k},\omega )\), where \(\varvec{k}\) is the wave vector and \(\omega \) is the frequency of the electromagnetic field. If, at time t, the electron position is \(\varvec{r}\) and its speed is \(\varvec{v}\), then, indicating with e the electron charge, the electron that passes through the solid can be represented by a charge distribution given by
The electric potential \(\varphi \) generated in the medium can be calculated asFootnote 1
In the Fourier space we have
In fact, on the one hand,
so that
and, on the other hand,
so that
Then, using Eqs. (13.2), (13.5), and (13.7), we obtain
which is equivalent to Eq. (13.3).
We are interested in calculating the energy loss \(-dE\) of an electron due to its interaction with the electric field \(\varvec{\mathcal{E}}\) generated by the electrons passing through the solid. Let us indicate with \(\mathcal{F}_z\) the z component of the electric force, so that
It should be noted that here and in the following equations, the electric force (and the electric field \(\varvec{\mathcal{E}}\,=\,\varvec{\mathcal{F}}/e\)) are considered at \(\varvec{r}\,=\,\varvec{v}\,t\). Since
the energy loss \(-dE\) per unit path length dz, \(-dE/dz\), is given by
Since
and \(\varphi (\varvec{k},\omega )\) is the Fourier transform of \(\varphi (\varvec{r},t)\) [see Eq. (13.4)], then
As a consequence
Taking into account (i) that the electric field has to be calculated at \(\varvec{r}\,=\,\varvec{v}\,t\) and (ii) of the presence in the integrand of the \(\delta (\varvec{k}\,\cdot \,\varvec{v}\,+\,\omega )\) distribution, we have
Since
we conclude thatFootnote 2
or
where
is the probability of an energy loss \(\omega \) per unit distance traveled by a non-relativistic electron of velocity \(\varvec{v}\) [1].
13.2 Homogeneous and Isotropic Solids
Let us assume now that the solid is homogeneous and isotropic, and \(\varepsilon \) is a scalar depending only on the magnitude of \(\varvec{k}\) and not on its direction
so that
where
and \(E\,=\,m\,v^2\,/\,2\). These limits of integration come from conservation of momentum (see Sect. 5.2.3).
Let us introduce the new variable \(\omega '\) defined as
so that
and, hence,
We thus can write that
Indicating with \(W=\hbar \omega \) the energy loss and with \(W_{\mathrm {max}}\) the maximum energy loss , the inverse electron inelastic mean free path , \(\lambda _{\mathrm {inel}}^{-1}\), can be calculated as
13.3 Summary
The Ritchie theory [1] was described. It allows to establish the relationship between electron energy loss and dielectric function, and to calculate the differential inverse inelastic mean free path, the inelastic means free path, and the stopping power.
Notes
- 1.
The vector potential is zero due to the chosen gauge.
- 2.
Note that, for any complex number z, \(\text{ Re }(i\,z)\,=\,-\,\text{ Im }(z)\).
References
R.H. Ritchie, Phys. Rev. 106, 874 (1957)
H. Raether, Excitation of Plasmons and Interband Transitions by Electrons (Springer, Berlin, 1982)
P. Sigmund, Particle Penetration and Radiation Effects (Springer, Berlin, 2006)
R.F. Egerton, Electron Energy-Loss Spectroscopy in the Electron Microscope, 3rd edn. (Springer, New York, 2011)
R.F. Egerton, Rep. Prog. Phys. 72, 016502 (2009)
S. Taioli, S. Simonucci, L. Calliari, M. Dapor, Phys. Rep. 493, 237 (2010)
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Dapor, M. (2017). Appendix D: The Ritchie Theory. 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_13
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DOI: https://doi.org/10.1007/978-3-319-47492-2_13
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