The valence electrons in alkali metals are typically considered as nature’s closest realisation of a free-electron gas, the so-called jellium, where the valence electrons move freely on a uniform positively charged background of ions. In jellium-like metals, e.g. aluminium and the alkali metals lithium, sodium, and potassium, it is assumed that the valence electrons interact only weakly with the ion cores. When the jellium is exposed to electromagnetic radiation with a sufficiently short wavelength, collective charge-density excitations called plasmons may be induced. Typically the required wavelength is shorter than that of visible light. As a consequence, jellium-like metals have a shiny luster because they cannot absorb visible light. When a photon has enough energy to excite a plasmon (e.g. 4 eV in potassium), it can be absorbed by the metal. These optical properties are described by the dielectric function (q,E) which depends on the momentum (q) and energy (E) transferred to the electrons. Optical light probes the response at q = 0, but studies as a function of both q and E are possible using, for example, inelastic X-ray scattering (IXS) spectroscopy [1].

Fig. 18: Experimental dynamical structure factor S(q,E) of potassium. With increasing q, a strong shoulder develops between 8 and 15 eV. This shoulder-like structure is isolated by subtracting a sloping background function (dashed lines) and shown in Figure 19a.

We investigated the dielectric response of potassium valence electrons at ID16 using inelastic X-ray scattering (IXS). The experiment measures the dynamic structure factor S(q,E) which is related to the dielectric function by being proportional to Im(-1/(q,E)). Figure 18 shows the measured spectra, which near to q=0 are dominated by a sharp peak at 4 eV, i.e. the plasmon, as expected for a jellium metal. This peak should broaden when q is increased since then the collective excitation starts decaying rapidly into electron-hole pairs. Indeed, this also happens in potassium, but to our surprise, a significant new structure appears at 8-15 eV energy transfers. This is highly untypical for the dielectric response of a simple metal [2]. To study the origin of this new unexpected excitation structure, a sloping background was subtracted from the spectra. The result is shown in Figure 19a. Clearly two separate peaks can be observed, labelled A and B. To understand these spectral features, we performed calculations within the frame of time-dependent density-functional theory, shown in Figure 19b. The theory shows a good agreement with experiment (with a small exception that the peak B splits into B1 and B2). The reason for the new structures can be understood in terms of the ground-state electron density of states (DOS), shown in Figure 19c. The DOS follows a jellium behaviour until the Fermi energy (EF), indicating that the ground state of potassium is indeed jellium-like. On the other hand, above EF deep gaps appear in the DOS at 6 and 9 eV, which we attribute to empty states of d symmetry. The comparison of the theoretical spectra and the DOS shows that the dip-and-peak structures of S(q,E) are caused by the corresponding structures of the DOS. The striking result of the study is that the dynamic response of potassium is strongly affected by the empty d bands and is far from being jellium-like. Why aren’t the excitations to the d bands visible in optical spectroscopies and only appear at increased q? The answer resides in the non-dipole character of the transitions: they reflect a change of electron’s state from s-type into d-type, forbidden by optical selection rules, but allowed in IXS when the measurements probe (q,E) at nonvanishing values of q. The remaining differences between experiment and theory, especially that of the experimentally observed merging of the peaks B1 and B2, may be due to further lifetime broadening or effects of electron-electron interactions.

Fig. 19: a) The structure obtained after subtraction of the background shown in Figure 18, b) the corresponding theoretical prediction, and c) the density of states of potassium valence electrons.

In conclusion, we have found surprisingly large deviations from a jellium behaviour in the dielectric response of potassium valence electrons when studied by IXS. Further experiments while modifying the band structure in situ, e.g. by high pressure, may become interesting in illuminating the surprisingly complicated character of this textbook example of a “free-electron” metal.



[1] W. Schülke, Electron dynamics by inelastic X-ray scattering, Oxford University Press, Oxford (2007).
[2] C. Sternemann, S. Huotari, G. Vankó, M. Volmer, G. Monaco, A. Gusarov, H. Lustfeld, K. Sturm and W. Schülke, Phys. Rev. Lett. 95, 157401 (2005).ory, Oak Ridge, Tennessee (USA)


Principal publication and authors

S. Huotari (a), C. Sternemann (b), M.C. Troparevsky (c), A.G. Eguiluz (c), M. Volmer (b), H. Sternemann (b), H. Müller (a), G. Monaco (a), and W. Schülke (b), Phys. Rev. B 80, 155107 (2009).
(a) ESRF
(b) Fakultät Physik/DELTA, Technische Universität Dortmund (Germany)
(c) Department of Physics & Astronomy, University of Tennessee; and Materials Science & Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee (USA)