Multipoleanalysisoftheelectrondensityandelectrostaticpotential in germanium by high-resolution electron diffraction A. Avilov a, * , G. Lepeshov a,b , U. Pietsch c , V. Tsirelson b a Institute of Crystallography, Moscow 117333, Russia b Mendeleev University of Chemical Technology, Moscow 125047, Russia c Institute of Physics, University of Potsdam, Potsdam 14415, Germany Abstract Accurateelectronstructurefactorsmeasuredbysigni®cantlyimprovedtransmissionelectrondiffractiontechniquewereused in a high-resolution quantitative study of the electron density and electrostatic potential in Ge polycrystalline sample. The parameters of the multipole model adapted for electron diffraction were found and topological features of the electron density and electrostatic potential were determined with this model. q 2001 Published by Elsevier Science Ltd. Keywords: C. Electron diffraction 1. Introduction The rapidly developing ®eld of physics and chemistry of solids is the experimental study of the electron density, electrostatic potential and the peculiarities of the inner- crystal electric ®eld in crystals by diffraction methods. Up to now, only single-crystal X-ray diffraction was used for thesepurposes[1,2].Atthesametime,theFourierimagesof the electrostatic potential ESP) of atoms, molecules and solids, the structure factors, can be measured by the high- energy electron diffraction exchange interaction between beam electrons and target electrons at these energies is negligible [3]). Electron diffraction is very sensitive to the details of distribution of valence electrons in crystals, parti- cularly at the small angles sinu /l , 0.4A Ê 21 ) [4,5]. It is also applicable to polycrystalline samples: that allows extending the available methods of the accurate X-ray diffraction [1] on the thin ®lms. These reasons make electron diffraction attractive for analysis of bonding in polycrystals. The electrostatic potentials for some simple molecules were obtained early using the gas-phase electron diffraction [6]. The electrostatic potential in solids was also studied by the electron diffraction [7±11]. However, the structure factors in the latter works were measured either for the low-angle re¯ections only or their accuracy was low. Recently, we have developed the new transmission elec- tron diffractometer for polycrystalline samples [12]. A considerable improvement in the accuracy of structure factors measurement for all re¯ections within the Ewald sphere was achieved with this device [13±15]. We also adapted the structural k -model allowing the analytical description of the crystal ESP as a superposition of the spherically deformed ions. Then we started the program of a high-resolution study of the ESP in different typed of crystals and performed the analysis of the electron density ED) and of the electrostatic potential features in binary crystalswiththerock-salt-typestructureusingthisapproach [16]. Solid Ge belongs to traditional objects for testing the new theoretical and experimental methods. Especially, bonding in this crystal has been investigated by X-ray diffraction, Hartree±Fock and density functional methods [2,17±26] and a reasonable agreement of the experimental and theo- retical structure factors and the electron densities was achieved.Electrondiffractionwasalsousedforthispurpose [9,27]. However, only a very limited number of low-angle structure factors was measured. At the same time, the quan- titative reconstruction of the ESP from the electron diffrac- tion data requires getting as complete as possible a set of structure factors with a statistical precision of about 1±2%. Our new electron diffractometer [12] is able to provide Journal of Physics and Chemistry of Solids 62 2001) 2135±2142 0022-3697/01/$ - see front matter q 2001 Published by Elsevier Science Ltd. PII:S0022-369701)00170-6 www.elsevier.com/locate/jpcs * Corresponding author. Fax: 17-95-135-1011. E-mail address: avilov@ns.crys.ras.ru A. Avilov).