Tables for
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 4.3, pp. 411-412

Section Conclusions

C. Colliexa Conclusions

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Since the early work of Hillier & Baker (1944[link]), EELS spectroscopy has established itself as a prominent technique for investigating various aspects of the electronic structure of solids. As a fundamental application, it is now possible to construct a self-consistent set of data for a substance by combination of optical or energy-loss functions over a wide spectral range (Altarelli & Smith, 1974[link]; Shiles, Sazaki, Inokuti & Smith, 1980[link]: Hagemann, Gudat & Kunz, 1975[link]). Sum-rule tests provide useful guidance in selecting the best values from the available measurements. The Thomas–Reiche–Kuhn f-sum rule can be expressed in a number of equivalent forms, which all require the knowledge of a function [[\varepsilon_2,\kappa, {\rm Im}(-1/\varepsilon)]] describing dissipative processes over all frequencies: [\left. \eqalign{\int\limits^\infty_0\omega\varepsilon_2(\omega)\,{\rm d}\omega &= {\pi\over 2}\omega^2_p, \cr \int\limits^\infty_0\omega\kappa(\omega)\,{\rm d}\omega &= {\pi\over4}\omega^2_p, \cr \int\limits^\infty_0\omega\left(-\displaystyle{1\over\varepsilon(\omega)}\right){\rm d}\omega &={\pi\over 2}\omega^2_p.}\right\} \eqno (]One defines the effective number density [n_{\rm eff}] of electrons contributing to these various absorption processes at an energy [\hbar\omega] by the partial f sums: [\left. \eqalign{ n_{\rm eff}(\omega)|_{\varepsilon_2} &= {m_0\over 2\pi^2e^2} \int\limits^\omega_0\, \omega'\varepsilon_2(\omega')\,{\rm d}\omega', \cr n_{\rm eff}(\omega)|_\kappa &= {m_0\over \pi^2e^2}\int\limits^\omega_0\, \omega'\kappa(\omega')\,{\rm d}\omega', \cr n_{\rm eff}(\omega)|_{-1/\varepsilon} &={m_0 \over 2\pi^2e^2}\int\limits^\omega_0\, \omega'\left[-\displaystyle{1\over \varepsilon(\omega')}\right]{\rm d}\omega'.}\right\} \eqno (]As an example, the values of [n_{\rm eff}(\omega)] from the infrared to beyond the K-shell excitation energy for metallic aluminium are shown in Fig.[link] . In this case, the conduction and core-electron contributions are well separated. One sees that the excitation of conduction electrons is virtually completed above the plasmon resonance only, but the different behaviour of the integrands below this value is a consequence of the fact that they describe different properties of matter: [\varepsilon_2(\omega)] is a measure of the rate of energy dissipation from an electromagnetic wave, [\kappa(\omega)] describes the decrease in amplitude of the wave, and [{\rm Im}[-\varepsilon^{-1}(\omega)]] is related to the energy loss of a fast electron. The above curve shows some exchange of oscillator strength from core to valence electrons, arising from the Pauli principle, which forbids transitions to occupied states for the deeper electrons.


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Values of neff for metallic aluminium based on composite optical data [courtesy of Shiles et al. (1980[link])].

More practically, in the microanalytical domain, the combination of high performance attained by using EELS with parallel detection (i.e. energy resolution below 1 eV, spatial resolution below 1 nm, minimum concentration below 10−3 atom, time resolution below 10 ms) makes it a unique tool for studying local electronic properties in solid specimens.


Altarelli, M. & Smith, S. Y. (1974). Superconvergence and sum rules for the optical constants: physical meaning, comparison with experiment and generalization. Phys. Rev. B, 9, 1290–1298.
Hagemann, H. J., Gudat, W. & Kunz, C. (1975). Optical constants from the far infrared to the X-ray region: Mg, Al, Cu, Ag, Au, Bi, C, and Al2O3. J. Opt. Soc. Am. 65, 742–748.
Hillier, J. & Baker, R. F. (1944). Microanalysis by means of electrons. J. Appl. Phys. 15, 663–675.
Shiles, E., Sazaki, T., Inokuti, M. & Smith, D. Y. (1980). Self consistency and sum-rule tests in the Kramers Kronig analysis of optical data: applications to aluminium. Phys. Rev. B, 22, 1612–1628.

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