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Nuclear and electron densities of anatase (TiO2)
PHYSICA ELSEVIER
Physica B 213&214 (1995) 384 386
Nuclear and electron densities of anatase
(TiO2)
M. Sakata a'*, M. Takagi a, M. T a k a t a a, C.J. Howard b "Department ~?fApplied Physics, Nagoya Universi~', Chikusa-ku. Nagoya 464-01. Japan bAustralian Nuclear Science and Technology Organization, Lueas Heights, NSW 2234, Australia
Abstract The nuclear-density distribution of anatase (TiO2) is obtained from neutron powder-diffraction data by the maximumentropy method (MEM). In MEM analysis, no structural model is required. The nuclear-density distribution indicates the smearing of nuclei by thermal motions, which is not necessarily harmonic. In order to show the complementarity of neutron and X-ray diffraction experiments in structural studies, the electron density distribution of anatase is also derived by the MEM. In comparison with the nuclear densities, the MEM electron-density distribution is more broadly distributed, which must represent the more wave-like nature of electrons. In addition, strong covalent bonding between the Ti and O atoms is evident. The thermal smearing may be intuitively understood in terms of the arrangement of the covalent bonds, i.e., isotropic thermal motions for Ti atoms which are octahedrally bonded to six O atoms, and anisotropic motion for the O atom which is bonded to three Ti atoms in a planar triangular configuration. For the O atoms, thermal vibrations perpendicular to the plane of the bonds, which involve bond bending but no bond compression, have the larger amplitudes.
1. Introduction It is often said that neutron and X-ray scattering are complementary. This complementarity has several aspects. For structural studies of non-magnetic materials, the important facts are that neutrons interact with nuclei, whereas X-ray photons interact with electrons. In other words, the nuclear- and electron-density distributions can be determined from neutron and X-ray diffraction experiments, separately. In conventional least-squares analysis, such a fundamental difference between neutron and X-ray scattering may be overlooked, because basically the same parameters are refined in the analyses. For a simple structure with a centre of symmetry, there is no practical problem in obtaining accurate structure factors from a diffraction experiment. Once the structure factors are obtained by an experiment, it becomes * Corresponding author.
possible to derive a corresponding density distribution, which is consistent with the observed structure factors and least biased with respect to unobserved structure factors. The method employed is called the maximumentropy method (MEM) [1]. The role of the MEM is shown schematically in Fig. 1. This kind of analysis may be called 'imaging from diffraction data'. The purpose of this work is to demonstrate the complementarity of neutron and X-ray diffraction by using the MEM to obtain the nuclear- and electron-density distributions for anatase. This is also a good example of imaging a crystal structure from neutron and X-ray diffraction data.
2. Theoretical development The present MEM analysis is developed based on Collins's formalism [2]. The basic MEM equation was
0921-4526/95/$09.50 ,~ 1995 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 5 ) 0 0 1 6 4 - 6
385
M. Sakata et al./Physica B 213&214 (1995) 384 386
X-ray Data
Neutron Data
Maximum Entropy Method
(a)
'~
] Ti
Electron Density
Nuclear Density
Fig. 1. The concept of M E M analysis.
originally derived under certain approximations. It was pointed out later [3] that the same equation could be derived without any approximation. The M E M equation can deal with only positive densities, in stark contrast to the conventional Fourier method, which is a great advantage when extracting electron densities by the MEM. In the neutron case, it is necessary to establish how to handle the negative scattering lengths for some nuclei, such as Ti. However, the negative scattering length problem has been completely overcome by considering nuclear densities instead of the densities of scattering length [4]. There are to date only two materials for which both nuclear- and electron-density distributions have been obtained by the MEM. They are Be [5] and rutile (TiO2) [4]. Compared with the huge n u m b e r of examples of the conventional structure analysis, the practical usage of M E M analysis in structural studies is in its very early stages.
Fig. 2. The density distribution map of anatase (TiO2) obtained by the MEM analysis of powder-diffraction data. The figure shows the Ti plane. (a) is the nuclear-density distribution and (b) is the electron-density distribution. The contour lines in (a) are on a logarithmic scale with 2" (n = 1. . . . . 10) n/A 3 and in (b) are on a linear scale ranging from 0.4 to 4 e/A 3 with 0.4 intervals.
3. Results of the MEM analysis From the M E M analysis of both neutron and X-ray powder data of anatase (TiO2), the nuclear- and electrondensity distributions are obtained. The M E M density maps of two planes, shown in Figs. 2 and 3, represent part of the results. Both planes are parallel to (001). In Fig. 2, the M E M maps of both nuclear (Fig. 2(a)) and electron (Fig. 2(b)) density distributions, in the plane containing the Ti atom, are shown. In Fig. 3, maps of the planes containing the O atom are shown. The O atoms are located slightly off the Ti planes. In Fig. 2(a), only nuclear densities of Ti appear, while in Fig. 2(b) electron densities of both Ti and O are recognized. This difference comes from the fact that the nuclei are confined within limited volumes around the atomic sites, while electrons tend to spread more widely throughout the unit cell. In Fig. 2(b), four Ti O covalent bonds can be clearly recognized. Taking into account the bonding of Ti to O atoms above and below the plane shown, it can be concluded that the Ti atom is connected with six oxygen atoms by strong covalent bonds. In
consequence, as can be seen in Fig. 2(a), the thermal smearing of the Ti nucleus is quite isotropic. In Fig. 3, the situation is reversed. The Ti atoms are slightly off the O plane. In Fig. 3(a), nuclear densities of oxygen are shown. In Fig. 3(b), two covalent bonds from O to Ti atoms can be seen. The O is bonded to one other Ti atom directly above (or below) the plane of the figure, so that in all the O is covalently bonded to three Ti atoms. The Ti atoms form a triangle with the O atom at its centre, so a planar configuration results. Thermal smearing of the O nucleus is elliptical (Fig. 3(b)), the ellipsoid being elongated perpendicular to the three T i - O covalent bonds. As for electron densities of O atoms, it is not noticeable that core electrons are strongly affected by the elliptical thermal motions. In contrast with the nuclear distributions, the electron distributions are elongated toward nearest neighbour oxygens in the lower electron density region.
386
M. Sakata et al./Physica B 213&214 (1995) 384 386
(a)
~Y
o
4. Concluding remarks The nuclear- and electron-density distributions of anatase (TiO2) have been obtained by the M E M , the results depending purely on the experimental data. Conceptually the method may be understood by the term, "imaging of diffraction data'. The method provides a good way of showing one aspect of complementarity of neutron and X-ray diffraction.
References
(5 Fig. 3. The density distribution map of anatase [TiO 2) obtained by the MEM analysis of powder-diffraction data. The figure shows the O plane. [a) is the nuclear-density distribution and (b) is the electron-density distribution. The contour line scales are the same as in Figs. 2(a), (b), respectively.
[1] M. Sakata and M. Sato, Acta Crystallogr. A 46 (1990~ 263. [2] D.M. Collins, Nature (London) 298 (1982) 49. [3] M. Sakata, T. Uno, M. Takata and R. Moil, Acta Crystallogr. B 48 11992) 591. [4] M. Sakata, T. Uno, M. Takata and C.J. Howard, J. Appl. Crystallogr. 26 (1993) 159. [5] M. Takata, Y. Kubota and M. Sakata, Z. Naturforsch. A 48 (1993) 75.
Physica B 213&214 (1995) 384 386
Nuclear and electron densities of anatase
(TiO2)
M. Sakata a'*, M. Takagi a, M. T a k a t a a, C.J. Howard b "Department ~?fApplied Physics, Nagoya Universi~', Chikusa-ku. Nagoya 464-01. Japan bAustralian Nuclear Science and Technology Organization, Lueas Heights, NSW 2234, Australia
Abstract The nuclear-density distribution of anatase (TiO2) is obtained from neutron powder-diffraction data by the maximumentropy method (MEM). In MEM analysis, no structural model is required. The nuclear-density distribution indicates the smearing of nuclei by thermal motions, which is not necessarily harmonic. In order to show the complementarity of neutron and X-ray diffraction experiments in structural studies, the electron density distribution of anatase is also derived by the MEM. In comparison with the nuclear densities, the MEM electron-density distribution is more broadly distributed, which must represent the more wave-like nature of electrons. In addition, strong covalent bonding between the Ti and O atoms is evident. The thermal smearing may be intuitively understood in terms of the arrangement of the covalent bonds, i.e., isotropic thermal motions for Ti atoms which are octahedrally bonded to six O atoms, and anisotropic motion for the O atom which is bonded to three Ti atoms in a planar triangular configuration. For the O atoms, thermal vibrations perpendicular to the plane of the bonds, which involve bond bending but no bond compression, have the larger amplitudes.
1. Introduction It is often said that neutron and X-ray scattering are complementary. This complementarity has several aspects. For structural studies of non-magnetic materials, the important facts are that neutrons interact with nuclei, whereas X-ray photons interact with electrons. In other words, the nuclear- and electron-density distributions can be determined from neutron and X-ray diffraction experiments, separately. In conventional least-squares analysis, such a fundamental difference between neutron and X-ray scattering may be overlooked, because basically the same parameters are refined in the analyses. For a simple structure with a centre of symmetry, there is no practical problem in obtaining accurate structure factors from a diffraction experiment. Once the structure factors are obtained by an experiment, it becomes * Corresponding author.
possible to derive a corresponding density distribution, which is consistent with the observed structure factors and least biased with respect to unobserved structure factors. The method employed is called the maximumentropy method (MEM) [1]. The role of the MEM is shown schematically in Fig. 1. This kind of analysis may be called 'imaging from diffraction data'. The purpose of this work is to demonstrate the complementarity of neutron and X-ray diffraction by using the MEM to obtain the nuclear- and electron-density distributions for anatase. This is also a good example of imaging a crystal structure from neutron and X-ray diffraction data.
2. Theoretical development The present MEM analysis is developed based on Collins's formalism [2]. The basic MEM equation was
0921-4526/95/$09.50 ,~ 1995 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 5 ) 0 0 1 6 4 - 6
385
M. Sakata et al./Physica B 213&214 (1995) 384 386
X-ray Data
Neutron Data
Maximum Entropy Method
(a)
'~
] Ti
Electron Density
Nuclear Density
Fig. 1. The concept of M E M analysis.
originally derived under certain approximations. It was pointed out later [3] that the same equation could be derived without any approximation. The M E M equation can deal with only positive densities, in stark contrast to the conventional Fourier method, which is a great advantage when extracting electron densities by the MEM. In the neutron case, it is necessary to establish how to handle the negative scattering lengths for some nuclei, such as Ti. However, the negative scattering length problem has been completely overcome by considering nuclear densities instead of the densities of scattering length [4]. There are to date only two materials for which both nuclear- and electron-density distributions have been obtained by the MEM. They are Be [5] and rutile (TiO2) [4]. Compared with the huge n u m b e r of examples of the conventional structure analysis, the practical usage of M E M analysis in structural studies is in its very early stages.
Fig. 2. The density distribution map of anatase (TiO2) obtained by the MEM analysis of powder-diffraction data. The figure shows the Ti plane. (a) is the nuclear-density distribution and (b) is the electron-density distribution. The contour lines in (a) are on a logarithmic scale with 2" (n = 1. . . . . 10) n/A 3 and in (b) are on a linear scale ranging from 0.4 to 4 e/A 3 with 0.4 intervals.
3. Results of the MEM analysis From the M E M analysis of both neutron and X-ray powder data of anatase (TiO2), the nuclear- and electrondensity distributions are obtained. The M E M density maps of two planes, shown in Figs. 2 and 3, represent part of the results. Both planes are parallel to (001). In Fig. 2, the M E M maps of both nuclear (Fig. 2(a)) and electron (Fig. 2(b)) density distributions, in the plane containing the Ti atom, are shown. In Fig. 3, maps of the planes containing the O atom are shown. The O atoms are located slightly off the Ti planes. In Fig. 2(a), only nuclear densities of Ti appear, while in Fig. 2(b) electron densities of both Ti and O are recognized. This difference comes from the fact that the nuclei are confined within limited volumes around the atomic sites, while electrons tend to spread more widely throughout the unit cell. In Fig. 2(b), four Ti O covalent bonds can be clearly recognized. Taking into account the bonding of Ti to O atoms above and below the plane shown, it can be concluded that the Ti atom is connected with six oxygen atoms by strong covalent bonds. In
consequence, as can be seen in Fig. 2(a), the thermal smearing of the Ti nucleus is quite isotropic. In Fig. 3, the situation is reversed. The Ti atoms are slightly off the O plane. In Fig. 3(a), nuclear densities of oxygen are shown. In Fig. 3(b), two covalent bonds from O to Ti atoms can be seen. The O is bonded to one other Ti atom directly above (or below) the plane of the figure, so that in all the O is covalently bonded to three Ti atoms. The Ti atoms form a triangle with the O atom at its centre, so a planar configuration results. Thermal smearing of the O nucleus is elliptical (Fig. 3(b)), the ellipsoid being elongated perpendicular to the three T i - O covalent bonds. As for electron densities of O atoms, it is not noticeable that core electrons are strongly affected by the elliptical thermal motions. In contrast with the nuclear distributions, the electron distributions are elongated toward nearest neighbour oxygens in the lower electron density region.
386
M. Sakata et al./Physica B 213&214 (1995) 384 386
(a)
~Y
o
4. Concluding remarks The nuclear- and electron-density distributions of anatase (TiO2) have been obtained by the M E M , the results depending purely on the experimental data. Conceptually the method may be understood by the term, "imaging of diffraction data'. The method provides a good way of showing one aspect of complementarity of neutron and X-ray diffraction.
References
(5 Fig. 3. The density distribution map of anatase [TiO 2) obtained by the MEM analysis of powder-diffraction data. The figure shows the O plane. [a) is the nuclear-density distribution and (b) is the electron-density distribution. The contour line scales are the same as in Figs. 2(a), (b), respectively.
[1] M. Sakata and M. Sato, Acta Crystallogr. A 46 (1990~ 263. [2] D.M. Collins, Nature (London) 298 (1982) 49. [3] M. Sakata, T. Uno, M. Takata and R. Moil, Acta Crystallogr. B 48 11992) 591. [4] M. Sakata, T. Uno, M. Takata and C.J. Howard, J. Appl. Crystallogr. 26 (1993) 159. [5] M. Takata, Y. Kubota and M. Sakata, Z. Naturforsch. A 48 (1993) 75.