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Real-Space Dynamic Electron Correlation in Beryllium
Phys. Rev. Lett. 136, 256402 – Published 26 June, 2026
DOI: https://doi.org/10.1103/vzrc-5m39
Abstract
Electron correlation in solids has a major impact on material properties. However, it has been studied mainly by theory, with very limited direct experimental investigations. Here, we demonstrate that dynamic electron correlation function can be experimentally measured using inelastic x-ray scattering on polycrystalline beryllium. The data are expressed as the energy-resolved dynamic pair-distribution function. Our results confirm the size of the exchange-correlation hole as , consistent with theoretical expectations. However, at the plasmon energy of , the exchange-correlation hole is extended up to 4–5 Å, suggesting a unique influence of the dynamic plasmon state.
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References (54)
- D. Bohm and D. Pines, A collective description of electron interactions: III. Coulomb interactions in a degenerate electron gas, Phys. Rev. 92, 609 (1953).
- M. Gell-Mann and K. A. Brueckner, Correlation energy of an electron gas at high density, Phys. Rev. 106, 364 (1957).
- J. Hubbard, The description of collective motions in terms of many-body perturbation theory. II. The correlation energy of a free-electron gas, Proc. R. Soc. A 243, 336 (1958).
- L. Landau, Theory of Fermi liquids, Sov. Phys. JETP 3, 920 (1956).
- W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140, A1133 (1965).
- R. Dörner, V. Mergel, O. Jagutzki, L. Spielberger, J. Ullrich, R. Moshammer, and H. Schmidt-Böcking, Cold target recoil ion momentum spectroscopy: A ‘momentum microscope’ to view atomic collision dynamics, Phys. Rep. 330, 95 (2000).
- H. J. Wörner et al., Conical intersection dynamics in probed by homodyne high-harmonic spectroscopy, Science 334, 208 (2011).
- R. Haindl, A. Feist, T. Domröse, M. Möller, J. H. Gaida, S. V. Yalunin, and C. Ropers, Coulomb-correlated electron number states in a transmission electron microscope beam, Nat. Phys. 19, 1410 (2023).
- S. Sukiasyan, C. McDonald, C. Destefani, M. Yu. Ivanov, and T. Brabec, Multielectron correlation in high-harmonic generation: A 2D model analysis, Phys. Rev. Lett. 102, 223002 (2009).
- G. Fève, A. Mahé, J.-M. Berroir, T. Kontos, B. Plaçais, D. C. Glattli, A. Cavanna, B. Etienne, and Y. Jin, An on-demand coherent single-electron source, Science 316, 1169 (2007).
- M. A. Kastner, The single-electron transistor, Rev. Mod. Phys. 64, 849 (1992).
- E. Wigner and F. Seitz, On the constitution of metallic sodium, Phys. Rev. 43, 804 (1933).
- J. C. Slater, The electronic structure of metals, Rev. Mod. Phys. 6, 209 (1934).
- P. Fulde, Electron Correlations in Molecules and Solids, Third (Springer-Verlag, Heidelberg GmbH, Berlin, 2002).
- W. Schülke, U. Berg, and O. Brümmer, Evidence for the Coulomb interaction of Be valence electrons by compton scattering cross-section measurements, Phys. Status Solidi (b) 35, 227 (1969).
- G. Mazzone, F. Sacchetti, and V. Contini, Static structure factor and electron-electron correlations in Be. A comparison of experiment with electron-gas theory, Phys. Rev. B 28, 1772 (1983).
- V. Contini and F. Sacchetti, Electron-electron correlation in beryllium and related properties, J. Phys. F 11, L1 (1981).
- P. Eisenberger, W. C. Marra, and G. S. Brown, Measurement of the static structure factor for conduction electrons with use of synchrotron radiation, Phys. Rev. Lett. 45, 1439 (1980).
- W. Schülke, Electron Dynamics by Inelastic X-Ray Scattering (Oxford University Press, Oxford, 2007).
- See Supplemental Material at http://link.aps.org/supplemental/10.1103/vzrc-5m39 for experimental details, data reduction procedure, formal connection between differential scattering cross section and correlation function, which includes Refs. [21,22].
- G. L. Squires, Thermal Neutron Scattering (Cambridge University Press, Cambridge, England, 1978).
- P. M Platzman and N. Tzoar, X-ray scattering from an electron gas, Phys. Rev. 139, A410 (1965).
- T. Egami and W. Dmowski, Dynamic pair-density function method for neutron and x-ray inelastic scattering, Z. Kristallogr. 227, 233 (2012).
- K. A. Acosta, H. C. Walker, and A. M. Fry-Petit, Optimizing the dynamic pair distribution function method for inelastic neutron spectrometry, Nat. Rev. Phys. 5, 236 (2023).
- W. Dmowski, S. B. Vakhrushev, I.-K. Jeong, M. P. Hehlen, F. Trouw, and T. Egami, Local lattice dynamics and the origin of the relaxor ferroelectric behavior, Phys. Rev. Lett. 100, 137602 (2008).
- W. Dmowski, S. O. Diallo, K. Lokshin, G. Ehlers, G. Ferré, J. Boronat, and T. Egami, Observation of dynamic atom-atom correlation in liquid helium in real space, Nat. Commun. 8, 15294 (2017).
- W. Schülke, H. Nagasawa, S. Mourikis, and P. Lanzki, Dynamic structure of electrons in Li metal: Inelastic synchrotron x-ray scattering results and interpretation beyond the random-phase approximation, Phys. Rev. B 33, 6744 (1986).
- W. Schülke, H. Schulte-Schrepping, and J. R. Schmitz, Dynamic structure of electrons in Al metal studied by inelastic x-ray scattering, Phys. Rev. B 47, 12426 (1993).
- W. Schülke, H. Nagasawa, S. Mourikis, and A. Kaprolat, Dynamic structure of electrons in Be metal by inelastic x-ray scattering spectroscopy, Phys. Rev. B 40, 12215 (1989).
- P. M. Platzman, E. D. Isaacs, H. Williams, P. Zschack, and G. E. Ice, X-ray-scattering determination of the dynamic structure factor of Al metal, Phys. Rev. B 46, 12943 (1992).
- B. C. Larson, J. Z. Tischler, E. D. Isaacs, P. Zschack, A. Fleszar, and A. G. Eguiluz, Inelastic x-ray scattering as a probe of the many-body local-field factor in metals, Phys. Rev. Lett. 77, 1346 (1996).
- W. Schülke, Inelastic scattering by electronic excitations, in Handbook on Synchrotron Radiation (Elsevier, New York, 1991), Vol. 3, pp. 565–637.
- S. K. Sinha, Theory of inelastic x-ray scattering from condensed matter, J. Phys. Condens. Matter 13, 7511 (2001).
- D. Pines and P. Nozières, The Theory of Quantum Liquids (W. A. Benjamin Inc., New York, 1966).
- L. Van Hove, Correlations in space and time and born approximation scattering in systems of interacting particles, Phys. Rev. 95, 249 (1954).
- P. Abbamonte, K. D. Finkelstein, M. D. Collins, and S. M. Gruner, Imaging density disturbances in water with a 41.3-attosecond time resolution, Phys. Rev. Lett. 92, 237401 (2004).
- T. Egami and Y. Shinohara, Perspective: Correlated atomic dynamics in liquid seen in real space and time, J. Chem. Phys. 153, 180902 (2020).
- N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders College Publishing, Philadelphia, 1976).
- M. Y. Chou, P. K. Lam, and M. L. Cohen, Ab initio study of structural and electronic properties of beryllium, Phys. Rev. B 28, 4179 (1983).
- P. Eisenberger, P. M. Platzman, and K. C. Pandy, Investigation of x-ray plasmon scattering in single-crystal beryllium, Phys. Rev. Lett. 31, 311 (1973).
- A. Vradis and G. D. Priftis, Dynamic structure factor of beryllium by x-ray inelastic scattering experiments, Phys. Rev. B 32, 3556 (1985).
- P. M. Platzman and P. Eisenberger, Presence of an incipient wigner electron lattice in solid-state electron gases, Phys. Rev. Lett. 33, 152 (1974).
- T. Suzuki and A. Tanokura, X-ray plasmon scattering. I. Experiment on beryllium, J. Phys. Soc. Jpn. 29, 972 (1970).
- D. C. Creagh and J. H. Hubbel, X-ray absorption (or attenuation) coefficients, in International Tables for Crystallography (International Union of Crystallography, Dordrecht, 2006), Vol. C, pp. 220–229.
- D. Ketenoglu, G. Spiekermann, M. Harder, E. Oz, C. Koz, M. C. Yagci, E. Yilmaz, Z. Yin, C. J. Sahle, B. Detlefs, and H. Yavaş, X-ray Raman spectroscopy of lithium-ion battery electrolyte solutions in a flow cell, J. Synchrotron Radiat. 25, 537 (2018).
- K. Hirano, T. Ishikawa, and S. Kikuta, Development and application of x-ray phase retarders (invited), Rev. Sci. Instrum. 66, 1604 (1995).
- J. Als-Nielsen and D. McMorrow, Elements of Modern X-Ray Physics, 2nd ed. (Wiley, New York, 2011).
- M. Hasegawa and M. Watabe, Theory of plasmon damping in metals. I. General formulation and application to an electron gas, J. Phys. Soc. Jpn. 27, 1393 (1969).
- L. Landau, On the vibrations of the electronic plasma, J. Phys. USSR 10, 25 (1946).
- H. Nagasawa, S. Mourikis, and W. Schülke, X-ray Raman spectrum of Li, Be and graphite in a high-resolution inelastic synchrotron x-ray scattering experiment, J. Phys. Soc. Jpn. 58, 710 (1989).
- J. P. Perdew and Y. Wang, Pair-distribution function and its coupling-constant average for the spin-polarized electron gas, Phys. Rev. B 46, 12947 (1992).
- D. Ceperley, Ground state of the fermion one-component plasma: A Monte Carlo study in two and three dimensions, Phys. Rev. B 18, 3126 (1978).
- D. M. Ceperley and B. J. Alder, Ground state of the electron gas by a stochastic method, Phys. Rev. Lett. 45, 566 (1980).
- 10.13139/ORNLNCCS/3364341.