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Multipole analysis of the electron density and electrostatic potential in germanium by high-resolution electron diffraction
Journal of Physics and Chemistry of Solids 62 (2001) 2135±2142
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Multipole analysis of the electron density and electrostatic potential in germanium by high-resolution electron diffraction A. Avilov a,*, G. Lepeshov a,b, U. Pietsch c, V. Tsirelson b b
a Institute of Crystallography, Moscow 117333, Russia Mendeleev University of Chemical Technology, Moscow 125047, Russia c Institute of Physics, University of Potsdam, Potsdam 14415, Germany
Abstract Accurate electron structure factors measured by signi®cantly improved transmission electron diffraction technique were used 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 innercrystal electric ®eld in crystals by diffraction methods. Up to now, only single-crystal X-ray diffraction was used for these purposes [1,2]. At the same time, the Fourier images of the electrostatic potential (ESP) of atoms, molecules and solids, the structure factors, can be measured by the highenergy 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, partiÊ 21) [4,5]. It is cularly at the small angles (sinu /l , 0.4 A 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
* Corresponding author. Fax: 17-95-135-1011. E-mail address: [email protected] (A. Avilov).
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 electron 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 crystals with the rock-salt-type structure using this approach [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 theoretical structure factors and the electron densities was achieved. Electron diffraction was also used for this purpose [9,27]. However, only a very limited number of low-angle structure factors was measured. At the same time, the quantitative reconstruction of the ESP from the electron diffraction 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
0022-3697/01/$ - see front matter q 2001 Published by Elsevier Science Ltd. PII: S 0022-369 7(01)00170-6
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Fig. 1. The comparison of the electron diffraction and theoretical structure factors: DF/F (Fexp 2 Fmodel)/Fmodel, where Fexp and Fmodel are experimental and dynamic LAPW structure factors calculated in Ref. [24], correspondingly.
measuring the diffraction pattern for Ge polycrystalline ®lms with this precision. In this paper we will present the ®rst results of quantitative study of bonding in Ge crystal using the multipole model ®tting to the high-energy electron diffraction structure factors. The bonding features will be discussed in terms of the topological theory of atomic interactions [1,16,28]; some notions of this theory are given in Appendix. 2. Experimental Thin polycrystalline ®lms were prepared for the measurements by the vacuum evaporation of pure Ge on a fresh cleavages of NaCl, heated up to ,3008C. After annealing for 30 min at the same temperature, it was slowly (,1 h) reduced to 208C. Sample were investigated at the accelerating voltage 75 kV. Absence of the preferential orientation of micro-crystallites in a ®lm was checked by measurement of the intensity distributions at the different tilting of specimen with respect to the primary beam. The intensities were measured at room temperature in the accumulation regime. The primary data treatment showed that the intensity of a particular re¯ection is uniformly distributed over the diffraction circle. The local deviation from this uniformity was less 1%. The sample obtained gave a distinct diffraction pattern, which was indexed with the cubic unit cell parameter Ê. a 5.658(5) A Intensities of 51 re¯ection circles were measured up to Ê 21. The intensities of overlapping re¯ecsinu /l < 1.72 A tions were separated by the pro®le analysis [12] considering their intensity ratio expected from theory. The total number of the symmetry-independent re¯ections obtained after their separation was 91. The experimental intensities were then reduced to an absolute scale and used to determine the isotropic thermal parameters of atoms, B re®ning the structural model composed by a superposition of spherical atoms. The rela-
tivistic atomic scattering function of Ge was taken from Ref. [29]. To diminish an effect of atomic asphericity due to the chemical bond ignored in this model, the re®nement was performed with 59 high-angle re¯ections (sinu /l . Ê 21): the spherical approximation for atomic electron 0.90 A shells is valid at the high scattering angles [1]. The value of Ê 2 was obtained (discrepancy factors 2.32%). B 0.546(2) A This value is close to the X-ray diffraction value Ê 2 [24] and to early electron diffraction B 0.565A Ê 2 [9]. Debye temperature calculated from B 0.542(10) A our B is u D 295 K; that is very close to the u D 296 K obtained in Ref. [23] using the precise X-ray diffraction data. The experimental intensities were then analyzed for the presence of extinction. This analysis has shown that the intensities only four of the low-angle re¯ections (111, 400, 440, 620) were distorted by the primary extinction. They were corrected for extinction in the Blackman (twowave) approximation [11] using averaged crystallite size of Ê . Nevertheless, the 440 and 620 experitaver 205 ^ 7 A mental re¯ections still exhibited the lower intensities, as expected due to multiple scattering processes. The discrepancy factor R calculated over 89 independent re¯ections (excluding the 440 and 620 re¯ections) has diminished from 6.40 to 2.42% after extinction corrections. To estimate an accuracy of the results of our electron diffraction experiment, we compared the low-angle electron structure factors with those calculated in Ref. [24] by the linearized augmented-plane-wave (LAPW) method in the local density approximation (we converted the calculated values into electron diffraction structure factors using Ê 2). This comparison (Fig. 1) showed the reasonB 0.546 A able agreement both data sets with discrepancy factor R 2.07%. Then we reduced the set of the experimental structure factors to 0 K and recalculated them into X-ray structure factors using the Mott±Bethe formula [5]. At this stage it was recovered that some experimental structure factors,
A. Avilov et al. / Journal of Physics and Chemistry of Solids 62 (2001) 2135±2142
occasionally distributed over the re¯ection set, signi®cantly differed from corresponding values calculated in Ref. [24]. We assigned this discrepancy to errors arising during the background correction for the polycrystalline electron diffraction pattern with the overlapping re¯ections. After removing the re¯ections mentioned, the ®nal set of 43 independent re¯ections in the range of 0.15 , sinu /l , Ê 21 was obtained. The forbidden 222 re¯ection 1.72 A measured in Ref. [18] was also added to the structure factor set. 3. Multipole model re®nement The structural model used for the treatment of the experimental data was as follows. The electron density was approximated by the Hansen and Coppens [30] multipole model. Electron density of each pseudoatom is presented in this model as
ratom r rc r 1 Pn k 03 rn k 0 r 1
4 X l0
k 003 Rl k 00 r
1 X m0
Plm^ ylm^ r=r
The neutral atom wave functions from [31] were used to describe the core r c and valence r v spherical electron shells. The radial functions Rl r nl exp 2k 0 jr with nl 4, 4, 4, 4 (l # 4) and values of the orbital exponents j Ge 2.1 a.u. have been used for multipoles according to [24]. The symmetry of Ge atom position is 243 m, therefore, only octupole P 322 and hexadecupole P40 and P44 0.74045P40 terms are different from zero. The pseudoatomic electron occupancies Plm were re®ned by the least squares together the atomic valence-shell contraction/expansion parameter k 0 using MOLDOS96 program [32]. The k 0 parameter was found with all re¯ections, while the Plm parameters were re®ned using re¯ections Ê 21. This procedure was repeated a few with sinu /l # 1.0 A times and found that parameters of the model are stable. The ®nal results are listed in Table 1. The obtained multipole model parameters were used to calculate the static model electron density in Ge with the modi®ed version of the XPRO2000 program [33]. Cluster of Table 1 Results of the multipole model re®nement of Ge crystal
k0 P322 P40 R(%) Rw (%) GOF
Electron diffraction
Re®nement [26] with LAPW structure factors [24]
0.922 (47) 0.353 (221) 2 0.333 (302) 1.60 1.35 1.98
0.957 0.307 2 0.161 0.28 0.29 ±
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44 atoms have been used. The ED map in the (110) plane of the Ge unit cell is shown in the Fig. 2. We have also calculated the topological characteristics of the critical points (CP) in the electron density (Table 2) and the Laplacian of ED (Fig. 3). The electrostatic potential map in (110) plane of Ge (Fig. 4) has been calculated with the multipole parameters as well. 4. Results and discussion The study of the electron density features accompanying the chemical bond forming in crystals demands a high accuracy of the experimental determination of structure factors. The latter suffer from the errors of different nature: dynamical diffraction, inelastic background, overlapping of the re¯ections, model uncertainty, etc. In the case of the electron diffraction study of the polycrystalline Ge the signi®cant errors resulted from the re¯ection overlap in the diffraction pattern and corresponding problems with scaling and the background subtraction. That is why the independent criterion for the estimate of the accuracy of the experimental structure factors is need. For this purpose, in this work we used the very accurate theoretical structure factors of Ge obtained by LAPW method in Ref. [24]. Comparison of both kinds of data (Fig. 1) leads to the conclusion that the accuracy of our structure factors is about 2%. Note, that the experimental structure Ê 21 and, factor set consists of re¯ections up to sinu /l # 1.72 A therefore, provides the high-resolution maps of the electron density, electrostatic potential and the Laplacian of the electron density. Note also, that the model valence ED reconstructed with these data was positive everywhere, while shorter structure factor sets result sometimes in unphysical negative values in the interstitial regions [24]. Note, that the intensity of `forbidden' 222 re¯ection in diffraction pattern was signi®cant. However, we have preferred using the X-ray diffraction value of this re¯ection from Ref. [18]. A reason for that is the dependence of this re¯ection intensity from the accelerating voltage observed in early electron diffraction study [9]. This phenomenon is now under investigation by this laboratory. Analysis of the multipole model parameters listed in Table 1 shows that the probability of their statistically signi®cant determination is at least 70%, as it is following from the relation P40/s (P40). The values of the experimental multipole parameters differ from those obtained in Ref. [26] by re®nement of the same multipole model (see Tables 1 and 2) using theoretical structure factors. That led to a noticeable difference of the experimental and theoretical ED characteristics at the bond CP (Table 2). At the same time, the topological characteristics of the ED in other critical points are in very good agreement. The ®rst reason for the discrepancy mentioned is the use of the non-relativistic core and valence wave functions [31] in the multipole model re®nement: no tabulation of the relativistic orbital densities in suitable form is available. Indeed, the relativity affects
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Fig. 2. Distribution of electron density in plane (110) of Ge reconstructed with the multipole model from the electron diffraction structure factors. The bond, ring and cage critical points are shown by the dots, triangles and rectangles, correspondingly. Intervals are: (2, 4, 8) p 10 n Ê 23, 22 # n # 2 eA Ê 23. eA
Table 2 Topological characteristics of the electron density in Ge at the bond, cage and ring critical points [characteristics of the (3,2 1) critical point in Ê 23, l 1 l 2 20.65 eA Ê 25, l3 1:85 eA 25 , f2 rb 0:55 eA 25 ). First row presents the ED results, second row Ge procrystal: r b 0.357 eA presents the our calculations based on model parameters obtained in Ref. [26] using LAPW data [24] Critical point type and Wyckoff position
Ê 23) r (eA
Ê 25) l 1 (eA
Ê 25) l 2 (eA
Ê 25) l 3 (eA
Ê 25) f 2r (eA
Bond critical point, 16c
0.575(8) 0.504 0.027(5) 0.030 0.024(5) 0.022
21.87 21.43 20.02 20.02 0.05 0.05
21.87 21.43 0.013 0.014 0.05 0.05
2.04 1.68 0.013 0.014 0.05 0.05
21.70(5) 21.18 0.24(5) 0.26 0.15(5) 0.15
Ring critical point, 16d Cage critical point, 8b
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Fig. 3. The Laplacian on electron density in (110) plane of Ge reconstructed from the electron diffraction data. Intervals are: ^(2, 4, 8) p 10 n Ê 25, 22 # n # 2. Negative lines are solid. eA
mainly the core electron: keeping that in mind, we calculated the valence ED map (not given here). The Ê 23 was found at the middle of deviation of only 0.05 eA the interatomic distance as compare with Ref. [24], where the relativistic density functional orbitals calculated in the local density approximation were used. The second reason is that the electron population of the antisymmetric octupole term is, probably, slightly distorted by anharmonicity of atomic motion ignored in this work. It is worth noting that a multipole re®nement with 89 re¯ections, including ones removed from consideration, as we described above, resulted in the same values of model parameters. However, their e.s.d. enhanced noticeably. The re®ned value of k 0 0.922(47) indicates a 8% expansion of the spherical part of the Ge atomic valenceshell in a crystal. This result is in agreement with an earlier report that the valence shell expansion, which ranges from 4.5 to 17%, depends on the method and orbital type used
[24±26]. The same effect was also noted for diamond and silicon [26]. Topological characteristics of the bond CP (Table 2) indicate the shared interaction between atoms in Ge. Comparing the directional electron density curvatures, l i, at the bond critical point for crystal and procrystal (the set of noninteracting spherical atoms placed in the real atom positions) given in Table 2, we can conclude that the Ge crystal formation is accompanied by strong the electron density shift towards the Ge±Ge line. The electron density curvature along this line re¯ects a small shift of electrons towards the atomic basins. The electron concentration beyond the atomic cores takes place in compact region in the center of the Ge±Ge line (Fig. 3). Simultaneously, signi®cant electron density depletion is observed around the cage and ring critical points. Note that in contrast to silicon [28], no nonnuclear attractors was found on Ge±Ge lines (the same conclusion was reached in Ref. [26]).
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Fig. 4. The distribution of electrostatic potential in plane (110) of Ge overlaids with the critical points (see legend to Fig. 2). Intervals are Ê 21, 22 # n # 2. (2, 4, 8) p 10 n eA
Electrostatic potential in Ge is homeomorphic to the electron density (Figs. 2 and 4): it has the same set of critical points, which are placed in the same positions, correspondingly. The bond paths in the electron density and interatomic lines in the ESP also coincide. In general, the patterns of the critical points in the electron density and ESP in crystals are not the same [34,35]. However, in Ge, as well as in other element crystals, such as diamond and solid krypton and xenon [36], the location of the CPs is de®ned by a symmetry: namely that leads to the homeomorphism mentioned. The electric ®eld E(r) 2fq (r) and Coulomb force vanish at the CPs in the ESP. Because the ESP and electrostatic ®eld energy density w(r) are related by the expression w(r) (1/8p )[E(r)] 2 [37], the latter value is zero at the critical points as well [16]. Thus, the electric ®eld in Ge is well-structured, the interatomic lines in the ESP connect only nearest-bonded atoms. In conclusion, this study demonstrates that the electron
diffraction transmission technique in combination with a topological analysis of the electron density and electrostatic potential is now able to provide the reliable high-resolution quantitative information concerned with a bonding in the polycrystalline samples. Note that the experiments with the samples possessing the preferential orientation of micro-crystallites (texture or mosaic single-crystal type) will assure, in principle, dividing the overlapping re¯ections more correctly. This will allow getting more accurate experimental data at higher resolution. Acknowledgements The work was supported by the Deutsche Forschungsgemeinschaft (grant Pi217/13-2) and Russian Foundation for Basic Research (grant 98-03-32654). Authors thank Dr A. Stash for help in the electrostatic potential calculation and
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Dr Yu. Abramov for information concerned with the multipole model re®nement. [2]
Appendix A. Topological features of the electron density and electrostatic potential Topological theory [28] analyses a crystal in terms of the electron density r (r), its gradient vector ®eld fr (r), positions and characteristics of the critical points and the electron density curvature. The critical points are the points where fr (rc) 0. They are determined by their rank and signature: rank is the number of non-zero eigenvalues of the curvature (Hessian) matrix, l i, while signature is the sum of the algebraic signs of l i, correspondingly. The ED has four kinds of non-degenerate CPs of rank 3: maxima (3, 2 3), minima (3, 1 3) and two types of saddle points, (3, 1 1) and (3, 2 1), corresponding to nuclear positions, cages, rings and bonds. All types of the CPs are present in crystals and their positions are restricted by the space group symmetry [38]. The fr (r) ®eld exhibits the pairs of gradient lines originated at a (3, 2 1) CP and terminated at two neighboring nuclei. They form the atomic interaction lines along which the ED is maximal with respect to any lateral shift. In the equilibrium, these lines are called bond paths and the associated (3, 2 1) points are termed bond CPs. The l 1 , 0 and l 2 , 0 at the (3, 2 1) CP correspond to the directions normal to the bond path and measure the degree of the ED contraction towards this point. The l 3 . 0 measures the degree of ED contraction towards each of the neighboring nuclei. The existence of the (3, 2 1) critical point and the bond path between two interacting atoms is a necessary condition for the chemical bond to exist. The Laplacian of the ED, f 2r (r), characterizes the concentration and depletion of electrons in each point r of a crystal. The sign of the f 2r (rb) l 1 1 l 2 1 l 3 at the (3, 2 1) CP depends on the relation between the principal curvatures of the ED at rb and, therefore, re¯ects the character of the atomic interactions. If the electrons are locally concentrated around the bond CP (f 2r (rb) , 0), as it takes place in Ge, then electrons are shared by both nuclei (shared interactions). This is typical for the covalent bond. The inner-crystal electrostatic potential also exhibits maxima, saddle points and minima corresponding to the nuclear positions, internuclear lines, atomic rings and cages and, therefore, can be characterized by critical points as well [16]. The gradient lines of q (r) de®ne the classical electrostatic ®eld strength E(r) 2fq (r), the latter is tangential to the gradient line at r. The ESP de®nes also the value of the classic Coulomb force, dF 2dq7f (r), acting on element of the charge dq at r. References [1] V.G. Tsirelson, R.P. Ozerov, Electron Density and Bonding in
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Multipole analysis of the electron density and electrostatic potential in germanium by high-resolution electron diffraction A. Avilov a,*, G. Lepeshov a,b, U. Pietsch c, V. Tsirelson b b
a Institute of Crystallography, Moscow 117333, Russia Mendeleev University of Chemical Technology, Moscow 125047, Russia c Institute of Physics, University of Potsdam, Potsdam 14415, Germany
Abstract Accurate electron structure factors measured by signi®cantly improved transmission electron diffraction technique were used 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 innercrystal electric ®eld in crystals by diffraction methods. Up to now, only single-crystal X-ray diffraction was used for these purposes [1,2]. At the same time, the Fourier images of the electrostatic potential (ESP) of atoms, molecules and solids, the structure factors, can be measured by the highenergy 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, partiÊ 21) [4,5]. It is cularly at the small angles (sinu /l , 0.4 A 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
* Corresponding author. Fax: 17-95-135-1011. E-mail address: [email protected] (A. Avilov).
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 electron 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 crystals with the rock-salt-type structure using this approach [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 theoretical structure factors and the electron densities was achieved. Electron diffraction was also used for this purpose [9,27]. However, only a very limited number of low-angle structure factors was measured. At the same time, the quantitative reconstruction of the ESP from the electron diffraction 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
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Fig. 1. The comparison of the electron diffraction and theoretical structure factors: DF/F (Fexp 2 Fmodel)/Fmodel, where Fexp and Fmodel are experimental and dynamic LAPW structure factors calculated in Ref. [24], correspondingly.
measuring the diffraction pattern for Ge polycrystalline ®lms with this precision. In this paper we will present the ®rst results of quantitative study of bonding in Ge crystal using the multipole model ®tting to the high-energy electron diffraction structure factors. The bonding features will be discussed in terms of the topological theory of atomic interactions [1,16,28]; some notions of this theory are given in Appendix. 2. Experimental Thin polycrystalline ®lms were prepared for the measurements by the vacuum evaporation of pure Ge on a fresh cleavages of NaCl, heated up to ,3008C. After annealing for 30 min at the same temperature, it was slowly (,1 h) reduced to 208C. Sample were investigated at the accelerating voltage 75 kV. Absence of the preferential orientation of micro-crystallites in a ®lm was checked by measurement of the intensity distributions at the different tilting of specimen with respect to the primary beam. The intensities were measured at room temperature in the accumulation regime. The primary data treatment showed that the intensity of a particular re¯ection is uniformly distributed over the diffraction circle. The local deviation from this uniformity was less 1%. The sample obtained gave a distinct diffraction pattern, which was indexed with the cubic unit cell parameter Ê. a 5.658(5) A Intensities of 51 re¯ection circles were measured up to Ê 21. The intensities of overlapping re¯ecsinu /l < 1.72 A tions were separated by the pro®le analysis [12] considering their intensity ratio expected from theory. The total number of the symmetry-independent re¯ections obtained after their separation was 91. The experimental intensities were then reduced to an absolute scale and used to determine the isotropic thermal parameters of atoms, B re®ning the structural model composed by a superposition of spherical atoms. The rela-
tivistic atomic scattering function of Ge was taken from Ref. [29]. To diminish an effect of atomic asphericity due to the chemical bond ignored in this model, the re®nement was performed with 59 high-angle re¯ections (sinu /l . Ê 21): the spherical approximation for atomic electron 0.90 A shells is valid at the high scattering angles [1]. The value of Ê 2 was obtained (discrepancy factors 2.32%). B 0.546(2) A This value is close to the X-ray diffraction value Ê 2 [24] and to early electron diffraction B 0.565A Ê 2 [9]. Debye temperature calculated from B 0.542(10) A our B is u D 295 K; that is very close to the u D 296 K obtained in Ref. [23] using the precise X-ray diffraction data. The experimental intensities were then analyzed for the presence of extinction. This analysis has shown that the intensities only four of the low-angle re¯ections (111, 400, 440, 620) were distorted by the primary extinction. They were corrected for extinction in the Blackman (twowave) approximation [11] using averaged crystallite size of Ê . Nevertheless, the 440 and 620 experitaver 205 ^ 7 A mental re¯ections still exhibited the lower intensities, as expected due to multiple scattering processes. The discrepancy factor R calculated over 89 independent re¯ections (excluding the 440 and 620 re¯ections) has diminished from 6.40 to 2.42% after extinction corrections. To estimate an accuracy of the results of our electron diffraction experiment, we compared the low-angle electron structure factors with those calculated in Ref. [24] by the linearized augmented-plane-wave (LAPW) method in the local density approximation (we converted the calculated values into electron diffraction structure factors using Ê 2). This comparison (Fig. 1) showed the reasonB 0.546 A able agreement both data sets with discrepancy factor R 2.07%. Then we reduced the set of the experimental structure factors to 0 K and recalculated them into X-ray structure factors using the Mott±Bethe formula [5]. At this stage it was recovered that some experimental structure factors,
A. Avilov et al. / Journal of Physics and Chemistry of Solids 62 (2001) 2135±2142
occasionally distributed over the re¯ection set, signi®cantly differed from corresponding values calculated in Ref. [24]. We assigned this discrepancy to errors arising during the background correction for the polycrystalline electron diffraction pattern with the overlapping re¯ections. After removing the re¯ections mentioned, the ®nal set of 43 independent re¯ections in the range of 0.15 , sinu /l , Ê 21 was obtained. The forbidden 222 re¯ection 1.72 A measured in Ref. [18] was also added to the structure factor set. 3. Multipole model re®nement The structural model used for the treatment of the experimental data was as follows. The electron density was approximated by the Hansen and Coppens [30] multipole model. Electron density of each pseudoatom is presented in this model as
ratom r rc r 1 Pn k 03 rn k 0 r 1
4 X l0
k 003 Rl k 00 r
1 X m0
Plm^ ylm^ r=r
The neutral atom wave functions from [31] were used to describe the core r c and valence r v spherical electron shells. The radial functions Rl r nl exp 2k 0 jr with nl 4, 4, 4, 4 (l # 4) and values of the orbital exponents j Ge 2.1 a.u. have been used for multipoles according to [24]. The symmetry of Ge atom position is 243 m, therefore, only octupole P 322 and hexadecupole P40 and P44 0.74045P40 terms are different from zero. The pseudoatomic electron occupancies Plm were re®ned by the least squares together the atomic valence-shell contraction/expansion parameter k 0 using MOLDOS96 program [32]. The k 0 parameter was found with all re¯ections, while the Plm parameters were re®ned using re¯ections Ê 21. This procedure was repeated a few with sinu /l # 1.0 A times and found that parameters of the model are stable. The ®nal results are listed in Table 1. The obtained multipole model parameters were used to calculate the static model electron density in Ge with the modi®ed version of the XPRO2000 program [33]. Cluster of Table 1 Results of the multipole model re®nement of Ge crystal
k0 P322 P40 R(%) Rw (%) GOF
Electron diffraction
Re®nement [26] with LAPW structure factors [24]
0.922 (47) 0.353 (221) 2 0.333 (302) 1.60 1.35 1.98
0.957 0.307 2 0.161 0.28 0.29 ±
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44 atoms have been used. The ED map in the (110) plane of the Ge unit cell is shown in the Fig. 2. We have also calculated the topological characteristics of the critical points (CP) in the electron density (Table 2) and the Laplacian of ED (Fig. 3). The electrostatic potential map in (110) plane of Ge (Fig. 4) has been calculated with the multipole parameters as well. 4. Results and discussion The study of the electron density features accompanying the chemical bond forming in crystals demands a high accuracy of the experimental determination of structure factors. The latter suffer from the errors of different nature: dynamical diffraction, inelastic background, overlapping of the re¯ections, model uncertainty, etc. In the case of the electron diffraction study of the polycrystalline Ge the signi®cant errors resulted from the re¯ection overlap in the diffraction pattern and corresponding problems with scaling and the background subtraction. That is why the independent criterion for the estimate of the accuracy of the experimental structure factors is need. For this purpose, in this work we used the very accurate theoretical structure factors of Ge obtained by LAPW method in Ref. [24]. Comparison of both kinds of data (Fig. 1) leads to the conclusion that the accuracy of our structure factors is about 2%. Note, that the experimental structure Ê 21 and, factor set consists of re¯ections up to sinu /l # 1.72 A therefore, provides the high-resolution maps of the electron density, electrostatic potential and the Laplacian of the electron density. Note also, that the model valence ED reconstructed with these data was positive everywhere, while shorter structure factor sets result sometimes in unphysical negative values in the interstitial regions [24]. Note, that the intensity of `forbidden' 222 re¯ection in diffraction pattern was signi®cant. However, we have preferred using the X-ray diffraction value of this re¯ection from Ref. [18]. A reason for that is the dependence of this re¯ection intensity from the accelerating voltage observed in early electron diffraction study [9]. This phenomenon is now under investigation by this laboratory. Analysis of the multipole model parameters listed in Table 1 shows that the probability of their statistically signi®cant determination is at least 70%, as it is following from the relation P40/s (P40). The values of the experimental multipole parameters differ from those obtained in Ref. [26] by re®nement of the same multipole model (see Tables 1 and 2) using theoretical structure factors. That led to a noticeable difference of the experimental and theoretical ED characteristics at the bond CP (Table 2). At the same time, the topological characteristics of the ED in other critical points are in very good agreement. The ®rst reason for the discrepancy mentioned is the use of the non-relativistic core and valence wave functions [31] in the multipole model re®nement: no tabulation of the relativistic orbital densities in suitable form is available. Indeed, the relativity affects
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Fig. 2. Distribution of electron density in plane (110) of Ge reconstructed with the multipole model from the electron diffraction structure factors. The bond, ring and cage critical points are shown by the dots, triangles and rectangles, correspondingly. Intervals are: (2, 4, 8) p 10 n Ê 23, 22 # n # 2 eA Ê 23. eA
Table 2 Topological characteristics of the electron density in Ge at the bond, cage and ring critical points [characteristics of the (3,2 1) critical point in Ê 23, l 1 l 2 20.65 eA Ê 25, l3 1:85 eA 25 , f2 rb 0:55 eA 25 ). First row presents the ED results, second row Ge procrystal: r b 0.357 eA presents the our calculations based on model parameters obtained in Ref. [26] using LAPW data [24] Critical point type and Wyckoff position
Ê 23) r (eA
Ê 25) l 1 (eA
Ê 25) l 2 (eA
Ê 25) l 3 (eA
Ê 25) f 2r (eA
Bond critical point, 16c
0.575(8) 0.504 0.027(5) 0.030 0.024(5) 0.022
21.87 21.43 20.02 20.02 0.05 0.05
21.87 21.43 0.013 0.014 0.05 0.05
2.04 1.68 0.013 0.014 0.05 0.05
21.70(5) 21.18 0.24(5) 0.26 0.15(5) 0.15
Ring critical point, 16d Cage critical point, 8b
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Fig. 3. The Laplacian on electron density in (110) plane of Ge reconstructed from the electron diffraction data. Intervals are: ^(2, 4, 8) p 10 n Ê 25, 22 # n # 2. Negative lines are solid. eA
mainly the core electron: keeping that in mind, we calculated the valence ED map (not given here). The Ê 23 was found at the middle of deviation of only 0.05 eA the interatomic distance as compare with Ref. [24], where the relativistic density functional orbitals calculated in the local density approximation were used. The second reason is that the electron population of the antisymmetric octupole term is, probably, slightly distorted by anharmonicity of atomic motion ignored in this work. It is worth noting that a multipole re®nement with 89 re¯ections, including ones removed from consideration, as we described above, resulted in the same values of model parameters. However, their e.s.d. enhanced noticeably. The re®ned value of k 0 0.922(47) indicates a 8% expansion of the spherical part of the Ge atomic valenceshell in a crystal. This result is in agreement with an earlier report that the valence shell expansion, which ranges from 4.5 to 17%, depends on the method and orbital type used
[24±26]. The same effect was also noted for diamond and silicon [26]. Topological characteristics of the bond CP (Table 2) indicate the shared interaction between atoms in Ge. Comparing the directional electron density curvatures, l i, at the bond critical point for crystal and procrystal (the set of noninteracting spherical atoms placed in the real atom positions) given in Table 2, we can conclude that the Ge crystal formation is accompanied by strong the electron density shift towards the Ge±Ge line. The electron density curvature along this line re¯ects a small shift of electrons towards the atomic basins. The electron concentration beyond the atomic cores takes place in compact region in the center of the Ge±Ge line (Fig. 3). Simultaneously, signi®cant electron density depletion is observed around the cage and ring critical points. Note that in contrast to silicon [28], no nonnuclear attractors was found on Ge±Ge lines (the same conclusion was reached in Ref. [26]).
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Fig. 4. The distribution of electrostatic potential in plane (110) of Ge overlaids with the critical points (see legend to Fig. 2). Intervals are Ê 21, 22 # n # 2. (2, 4, 8) p 10 n eA
Electrostatic potential in Ge is homeomorphic to the electron density (Figs. 2 and 4): it has the same set of critical points, which are placed in the same positions, correspondingly. The bond paths in the electron density and interatomic lines in the ESP also coincide. In general, the patterns of the critical points in the electron density and ESP in crystals are not the same [34,35]. However, in Ge, as well as in other element crystals, such as diamond and solid krypton and xenon [36], the location of the CPs is de®ned by a symmetry: namely that leads to the homeomorphism mentioned. The electric ®eld E(r) 2fq (r) and Coulomb force vanish at the CPs in the ESP. Because the ESP and electrostatic ®eld energy density w(r) are related by the expression w(r) (1/8p )[E(r)] 2 [37], the latter value is zero at the critical points as well [16]. Thus, the electric ®eld in Ge is well-structured, the interatomic lines in the ESP connect only nearest-bonded atoms. In conclusion, this study demonstrates that the electron
diffraction transmission technique in combination with a topological analysis of the electron density and electrostatic potential is now able to provide the reliable high-resolution quantitative information concerned with a bonding in the polycrystalline samples. Note that the experiments with the samples possessing the preferential orientation of micro-crystallites (texture or mosaic single-crystal type) will assure, in principle, dividing the overlapping re¯ections more correctly. This will allow getting more accurate experimental data at higher resolution. Acknowledgements The work was supported by the Deutsche Forschungsgemeinschaft (grant Pi217/13-2) and Russian Foundation for Basic Research (grant 98-03-32654). Authors thank Dr A. Stash for help in the electrostatic potential calculation and
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Dr Yu. Abramov for information concerned with the multipole model re®nement. [2]
Appendix A. Topological features of the electron density and electrostatic potential Topological theory [28] analyses a crystal in terms of the electron density r (r), its gradient vector ®eld fr (r), positions and characteristics of the critical points and the electron density curvature. The critical points are the points where fr (rc) 0. They are determined by their rank and signature: rank is the number of non-zero eigenvalues of the curvature (Hessian) matrix, l i, while signature is the sum of the algebraic signs of l i, correspondingly. The ED has four kinds of non-degenerate CPs of rank 3: maxima (3, 2 3), minima (3, 1 3) and two types of saddle points, (3, 1 1) and (3, 2 1), corresponding to nuclear positions, cages, rings and bonds. All types of the CPs are present in crystals and their positions are restricted by the space group symmetry [38]. The fr (r) ®eld exhibits the pairs of gradient lines originated at a (3, 2 1) CP and terminated at two neighboring nuclei. They form the atomic interaction lines along which the ED is maximal with respect to any lateral shift. In the equilibrium, these lines are called bond paths and the associated (3, 2 1) points are termed bond CPs. The l 1 , 0 and l 2 , 0 at the (3, 2 1) CP correspond to the directions normal to the bond path and measure the degree of the ED contraction towards this point. The l 3 . 0 measures the degree of ED contraction towards each of the neighboring nuclei. The existence of the (3, 2 1) critical point and the bond path between two interacting atoms is a necessary condition for the chemical bond to exist. The Laplacian of the ED, f 2r (r), characterizes the concentration and depletion of electrons in each point r of a crystal. The sign of the f 2r (rb) l 1 1 l 2 1 l 3 at the (3, 2 1) CP depends on the relation between the principal curvatures of the ED at rb and, therefore, re¯ects the character of the atomic interactions. If the electrons are locally concentrated around the bond CP (f 2r (rb) , 0), as it takes place in Ge, then electrons are shared by both nuclei (shared interactions). This is typical for the covalent bond. The inner-crystal electrostatic potential also exhibits maxima, saddle points and minima corresponding to the nuclear positions, internuclear lines, atomic rings and cages and, therefore, can be characterized by critical points as well [16]. The gradient lines of q (r) de®ne the classical electrostatic ®eld strength E(r) 2fq (r), the latter is tangential to the gradient line at r. The ESP de®nes also the value of the classic Coulomb force, dF 2dq7f (r), acting on element of the charge dq at r. References [1] V.G. Tsirelson, R.P. Ozerov, Electron Density and Bonding in
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