The MEM/Rietveld method with nano-applications – accurate charge-density studies of nano-structured materials by synchrotron-radiation powder diffraction

Structural studies of materials with nano-sized spaces, called nano-structured materials, have been carried out by high-resolution powder diffraction. Our developed analytical method, which is the combination of the maximum-entropy method (MEM) and Rietveld refinement, the so-called MEM/Rietveld method, has been successfully applied to the analysis of synchrotron-radiation (SR) powder diffraction data measured at SPring-8, a third-generation SR light source. In this article, structural studies of nano-porous coordination polymers and endohedral metallofullerenes are presented with the advanced technique of SR powder experiment. The structure of the adsorbed guest molecule in the coordination polymer and encapsulated atoms in the fullerene cage are clearly revealed by the MEM charge density. The methodology of MEM/Rietveld analysis is also presented.

1. Introduction: MEM application to nano-materials crystallography

Nano-structured materials, materials with nano-space, have attracted wide interest in the field of nano-materials science due to the huge potential for discovering novel physical properties. During the past two decades, several intriguing physical properties, such as ferromagnetism of K-doped zeolites (Nozue et al., 1992), superconductivities of Ba-doped clathrate (Kawaji et al., 1995) and electride (Miyakawa et al., 2007), and photodimerization in a coordination cage (Takaoka et al., 2006), have been found in nano-structured materials by doping of atoms and molecules into their nano-space.

The accurate structures of the nano-structured materials including doped atoms/molecules are always required for understanding the origin of characteristic properties. However, it is normally difficult to reveal structural information of doped guest atoms and molecules due to disorder and/or movement of guests. In addition, newly synthesized nano-structured materials are usually obtained in powder form. Thus, it is important to develop the powder diffraction instruments and data-analysis techniques for the structural analysis of novel nano-structured materials.

Since 1995, we have developed an elaborate charge-density refinement method, MEM/Rietveld method, which is based on information theory, the maximum-entropy method (MEM) using synchrotron-radiation powder data. The application of the MEM to crystallography extended the limits of the crystal-structure-analysis technique (Collins, 1982; Wilkins et al., 1983; Bricogne, 1988; Bricogne & Gilmore, 1990; Sakata & Sato, 1990; Bricogne, 1993; de Vries et al., 1994; Sivia & David, 1994; Papoular & Cox, 1995; Gilmore, 1996; Takata & Sakata, 1996; Marks & Landree, 1998; Burger & Prandl, 1999; Graafsmaa & de Vries, 1999). The MEM can yield a high-resolution electron-density distribution from a limited number of diffraction data. This ability of the MEM can be interpreted as `imaging of diffraction data' (Takata & Sakata, 1996) as if the MEM were the virtual imaging lens by means of a computer program as shown in Fig. 1. Consequently, the concept led to the subsequent advancement of the MEM application, which is the development of the MEM/Rietveld method.

Figure 1 MEM virtual imaging lens by means of a computer program.

The first successful application of the MEM/Rietveld method was the determination of the endohedral nature of a metallofullerene, Y@C₈₂ (Takata et al., 1995). The encapsulation of a Y atom in the nano-space of the fullerene cage was clearly visualized. The integration of the MEM charge-density imaging with Rietveld powder data fitting analysis (Rietveld, 1969) allowed us the systematic prediction of the Y-atom position and the rotational order of the fullerene cage in the iterative structure refinement. Since then, the method has been further applied to the structure determination of nano-structured materials such as a gas molecule adsorbed in a nano-porous coordination polymer (Kitaura et al., 2002) and electrides (Kim et al., 2007). The powder diffractometer installed at the SPring-8, BL02B2 beamline (Nishibori, Takata, Kato et al., 2001; Takata et al., 2002) has significantly contributed to the development and application of the MEM/Rietveld method. Characteristic structural features of other materials, orbital ordering (Takata, Nishibori, Kato et al., 1999; Kato, Takata et al., 2005; Kato, Ohishi et al., 2005) and/or charge-transfer ordering (Aoyagi et al., 2004; Kato et al., 2003) as well as bonding nature (Nishibori, Takata, Sakata, Tanaka et al., 2001) have also been successfully visualized by MEM/Rietveld analysis using a diffractometer. This review describes recent structural studies of nano-structured materials, coordination polymers and endohedral metallofullerenes by the MEM/Rietveld method using high-resolution powder data at SPring-8.

2. Synchrotron-radiation powder diffraction experiment

Following a great demand by materials scientists for synchrotron-radiation (SR) powder diffractometry, a large Debye–Scherrer camera was designed and installed at BL02B2 at SPring-8 (Nishibori, Takata, Kato et al., 2001; Takata et al., 2002). The schematic representation of the large Debye–Scherrer camera is shown in Fig. 2. To take advantage of the high-brilliance and high-energy beam, the camera method with transmission geometry was adopted. The 2θ dependence of the absorption effect by the specimen can be minimized even for materials containing heavy elements by selecting the proper X-ray energy and capillary diameter. The camera is equipped with a curved Imaging-Plate (IP) detector. The radius of the camera is 286.48 mm which determines the resolution to be 0.01° in 2θ due to the 50 × 50 µm pixel size of the IP. The size of the IP detector is 200 mm in width and 400 mm in length, which covers up to 80.0° in 2θ. Taking into account the utilization of short-wavelength incident X-rays, that is to say 0.35–1.0 Å, the maximum angle can cover a much wider range of reciprocal space (d > 0.3–0.8 Å) than in the case of Cu Kα. It is possible to record multi-powder patterns (maximum 30) on one IP detector using a mask with a vertical slit attached before the IP detector as shown in Fig. 2. This is particularly convenient for the investigation of structural changes induced by temperature (Kato et al., 2002), light (Kato et al., 2007) and gas adsorption (Kitaura et al., 2002). To change the temperature of powder specimens, three different devices are available depending on the temperature required. They are high-temperature (300–1000 K) and low-temperature (90–300 K) N₂ gas-flow apparatus, and the Displex-type He cryostat (15–300 K). For all the experiment modes, the sample capillary can be oscillated with an ω stage during measurement to reduce the intensity inhomogeneity of Debye–Scherrer rings caused by the coarse granularity effect of the powder specimen.

Figure 2 Schematic illustration of a large Debye–Scherrer camera and in situ gas adsorption experiment system at BL02B2 SPring-8. A photograph of the sample holder with a gas import tube is inserted.

A simple gas-import system for in situ gas-adsorption experiments is also available for the large Debye–Scherrer camera as schematically shown in Fig. 2. The first in situ SR powder diffraction study was carried out for hydrothermal conversion of zeolites by Norby (1997) with a similar system. The capillary containing the powder specimen was attached to the stainless-steel tube using an epoxy adhesive. It was mounted in the special sample holder for the gas introduction, which was constructed from Swagelok parts and a steel tube. A stainless-steel tube attached to the sample holder was connected to the gas-handling system with a number of valves and a pressure gauge. A helium leak detector was connected to the system for leak checking of the evacuation and vacuum.

As a typical example of an in situ powder diffraction measurement of gas adsorption, the oxygen-adsorption investigation on a type of copper coordination polymer is described (Kitaura et al., 2002). The sample used was [Cu₂(pzdc)₂(pyz)]_n (pzdc: pyrazine-2,3-dicarboxylate, pyz: pyrazine) called CPL-1 (coordination polymer 1 with a pillared-layer structure) with uniform nanochannels of 4 × 6 Å (Kondo et al., 1999). The crystal structure of CPL-1 is shown in Fig. 3. This material is well known to adsorb gas molecules inside the nanopores. However, there had been no structural evidence of the absorption site and the molecular structure inside the nanopores. Before the measurement of gas adsorption, the water molecules or organic solvent inside the nanopores must be removed. Using this system, the sample is easily degassed by heating the sample during evacuation. The sample of CPL-1 was degassed for approximately 10 min at 373 K to remove water molecules inside the nanopores. The oxygen was then introduced at 80 kPa at room temperature. The gas adsorption state was controlled by the temperature of the sample under a constant gas pressure. Because a powder sample is small, i.e. less than a few milligrams, the temperature is easier to control than the gas pressure. The temperature of the sample was gradually lowered from 300 to 90 K to initiate the gas adsorption. The measured powder diffraction patterns of CPL-1 with the adsorption of oxygen are shown in Fig. 4. Dramatic changes of the peak positions and relative intensities of reflections were observed between 150 and 130 K. These changes are thought to be caused by the adsorption of oxygen. To obtain data with much higher counting statistics for precise charge-density analysis, the exposure time is determined from the maximum limit of counting by an IP. In the present case, the data for charge-density analysis were measured with an exposure time of 65 min. After this measurement, the sample was heated to 300 K and the diffraction pattern changed to that of anhydrous CPL-1. The desorption of oxygen and the recovery of the empty structure without a collapse of the framework were confirmed.

Figure 3 Crystal structure of CPL-1 and schematic procedure of synthesis of pillared-layer structure.

Figure 4 In situ SR powder diffraction patterns of CPL-1 with the adsorption of oxygen at 80 kPa. The diffraction pattern of as-synthesized CPL-1 is also shown. The wavelength of the incident X-rays is 0.8 Å.

3. Structural analysis by the MEM/Rietveld method

In 1995, the MEM/Rietveld method (Takata et al., 1995) was developed for precise structure analysis using X-ray powder diffraction data. The method enables us to construct the fine structural model up to the charge-density level. This provides an efficient way for structure analysis with a self-consistent structure modeling by MEM charge-density imaging and Rietveld powder-pattern fitting based on the model derived from MEM charge density. At the start of the iteration, it is necessary to have a primitive structural model. The Rietveld method using this primitive structural model is called the pre-Rietveld analysis. It is well known that the MEM can provide useful information purely from observed structure-factor data beyond a presumed crystal structure model used in the pre-Rietveld analysis. This visualization ability of high-resolution charge density was useful to construct structure models of novel nano-structured materials.

The flow chart of the method is shown in Fig. 5 with an example of the structure-analysis procedure for a porous coordination polymer CPL-1 with the adsorption of O₂. In the first step of the analysis, the Rietveld refinement was carried out via a primitive structural model of only the CPL-1 framework determined by single-crystal X-ray diffraction (Kondo et al., 1999), without assuming any guest molecules inside the nanopores. In the refinement, a rigid-body model and restraints for intermolecular distances and the angles were appropriately applied so as to maintain intrinsic molecular forms. The result of this Rietveld fitting is shown in Fig. 6(a). The misfitting of the observed and calculated powder patterns was largely due to the lack of guest gas molecules in the primitive structural model. Reliability factors based on the whole powder profile, R_WP, and the Bragg integrated intensities, R_I, were 16.4 and 44.9%, respectively. Despite the unsatisfactory fit to the data, the integrated intensities of each reflection were evaluated from the observed diffraction patterns using the result of the Rietveld refinement. Actually, the diffraction intensity of each data point is partitioned between each contributing reflection using the proportion of the calculated intensities in accordance with the following equation:

where Y^obs(2θ_j) is the observed diffraction intensity at 2θ_j and has contributions from N reflections, is the calculated intensity from the structural model in the Rietveld analysis for the ith reflection at 2θ_j and is the contribution from the ith reflection at 2θ_j. This operation is carried out for all data points as shown in Fig. 7. As a result, all of the observed intensities were divided into each reflection without exception, and the integrated intensities of each reflection based on the observed data were estimated. The structure factors derived from these integrated intensities were used in the MEM analysis with the phases calculated from the structural model. MEM calculation was carried out using a computer program, ENIGMA (Tanaka et al., 2002). In the MEM calculation, the total number of electrons in the unit cell was given as a constraint. The number of electrons for adsorbed gas mol­ecules was included in this calculation. The number of adsorbed gas molecules was independently examined by the gas adsorption isotherm. The obtained three-dimensional MEM charge-density distribution is shown at the middle left in Fig. 5. The framework structure can be clearly seen. Despite the use of a structural model without assuming guest mol­ecules, two local peaks of charge density were recognized in each pore. It means that the diffraction intensities firmly contain the information of the adsorbed O₂ molecule.

Figure 5 Flow chart of the MEM/Rietveld method with structure modeling and imaging of charge density.

Figure 6 Fitting results of Rietveld refinement for CPL-1 with the adsorption of oxygen. (a) Initial refinement using a structural model of CPL-1 framework without guest molecules. (b) Final refinement using the modified structural model of CPL-1 with oxygen.

Figure 7 The procedure of evaluation of observed integrated intensities in MEM/Rietveld analysis. Difference plot (a) for the result of Rietveld refinement and (b) based on the evaluated intensities.

As the next step in the analysis, the modified structural model including O₂ molecules based on the local peak positions in the nanochannel was applied as a new model to the second Rietveld refinement. The results of the final Rietveld refinement are shown in Fig. 6(b), which indicates a satisfactory fitting as a published structure determination. Reliability factors R_WP and R_I were dramatically improved to 2.1 and 3.9%, respectively. The feature of the obtained MEM charge density was consistent with the structural model used in Rietveld refinement at the atomic level and thus it was judged to be the final result of this analysis. The determined MEM charge density is shown in Fig. 8 with the structural model. The details of the charge density will be explained in the next section.

Figure 8 (a) Crystal structure of CPL-1 with the adsorption of O2. O2 molecules are shown by red balls. Other atoms are shown by connecting lines. (b) MEM charge density of CPL-1 with the adsorption of O2 as an equi-density contour surface. The equi-density level is 1.0 e Å^−3. O2 molecules are indicated in red.

As shown here, the MEM/Rietveld method, which has the combined abilities of the structural model prediction of the MEM and the powder-pattern decomposition of the Rietveld method, was proved to be useful for the structure determination of porous coordination polymers undergoing gas adsorption. This method is also powerful for the structure determination of the rotational or positional disorder of molecules and organic ligands as well as metals and molecules in the fullerene cage for which their structural model is difficult to express and build.

4. Gas molecule in nanopores of a coordination polymer

Porous coordination polymers (Yaghi et al., 2003; Kitagawa et al., 2004) have attracted the attention of scientists for a number of reasons including the creation of nano-sized spaces and discovering novel phenomena as well as commercial interests such as their application in gas separation (Matsuda et al., 2005), gas storage (Noro et al., 2000; Férey et al., 2003; Rowsell et al., 2004) and heterogeneous catalysis (Seo et al., 2000; Ohmori & Fujita, 2004). To develop novel porous coordination polymers as functional materials, the precise structural information of guest molecules and host framework are indispensable. We have successfully revealed specific molecular arrangements in the nanopores and molecular interactions between the guest molecules and host framework by the MEM/Rietveld method and in situ SR powder diffraction experiments of gas adsorption.

4.1. One-dimensional array of the O₂ molecule

As shown in Fig. 8, we have succeeded in direct observation of the O₂ molecule adsorbed in CPL-1 (Kitaura et al., 2002). The result gave a conclusive answer to the long-standing question of where the gas molecules locate inside the nanopore. In Fig. 8, the dumbbell-shaped electron densities, which are presumably due to O₂ molecules, are clearly recognized in the middle of the nanochannels. There are 15.8 (1) electrons in total around the electron distribution, calculated on the basis of the MEM charge density, which virtually agrees with the number in the O₂ molecule. Therefore, we deduced that the dumbbell-shaped electron densities are O₂ molecules and one O₂ molecule was adsorbed per Cu atom without any charge transfer between O₂ molecules and/or O₂ molecules and pore wall. No electron density showing chemical bonding is seen between the O₂ molecule and the pore walls. The interatomic distances between the O₂ molecule and the atoms on the pore walls are from approximately 2.7 to 3.3 Å, which shows the van der Waals contact of these atoms. These observations support the idea that O₂ molecules are physisorbed in the nanochannels of CPL-1. The isotropic temperature factor B of the O₂ molecule was refined to be 4.1 Ų. This large value shows that the O₂ molecules have large thermal vibrations in the nanochannels. However, a clear charge-density distribution of adsorbed O₂ molecules shows that the molecules are strongly trapped at this position. The state of the oxygen in the nanochannels is thought to be closer to a solid state than a liquid state. As shown here, the distribution of the MEM charge density provides information on the molecular interaction, i.e. ionicity of atoms and the charge transfer between the molecules, because it is calculated for all of the electrons in the system.

The in situ SR powder experiment and the MEM/Rietveld analysis revealed that the restricted geometry achieved by one-dimensional nanochannels of CPL-1 leads to a specific molecular assembly, a one-dimensional ladder structure constructed by O₂ dimers, which is unlikely to be bulk fluid. This result will open up a new dimension of research on confined and ordered molecules within a restricted geometry. The O₂ molecule is well known to have paramagnetic property. The low-dimensional arrangement of molecular oxygen makes the investigation of the intermolecular potential and the quantum effect on magnetic properties attractive. Gas adsorption in the nanopores of a porous coordination polymer is a promising method by which to realize such an arrangement of molecular oxygen.

Fig. 9 shows the arrangement of the O₂ molecules in the nanochannel of CPL-1. Adsorbed O₂ molecules form a ladder structure. The intermolecular distance between the O₂–O₂ dimers is 3.28 Å, which is close to that in the α and β phases of solid O₂. On the other hand, the interdimer distance of 4.69 Å is relatively large. Theoretical studies have shown that the magnetic interaction between O₂ molecules depends strongly on the intermolecular distance and the orientation of the molecular axis (Bussery et al., 1994). These results indicate that the interdimer interaction is approximately a hundred times weaker than the intradimer interaction. Therefore, the magnetic properties of this system are assumed to be equivalent to those for a Heisenberg antiferromagnetic dimer with S = 1. The temperature dependence of susceptibility and the high-field magnetization process at low temperatures were also investigated in terms of the magnetic property emanating from the one-dimensional order of the molecular O₂ arrangement in the nanochannel. Consequently, we found that an additional antiferromagnetic nature was caused by the ordered O₂ molecular array.

Figure 9 Arrangement of O2 molecules in the nanochannel of CPL-1 viewed from the side of the nanochannel direction.

This work was the breakthrough,which gives a new concept of the novel functional material design by controlling mol­ecular alignment using ordered nanopore structure of coordination polymers and leads to studies on various gas adsorption to porous coordination polymers such as H₂ (Kubota et al., 2005), N₂, Ar, CH₄ (Kitaura et al., 2005) and C₂H₂ (Matsuda et al., 2005).

4.2. H₂ molecules align in nanochannels of CPL-1

It is a common assumption that the weak X-ray scattering amplitude of hydrogen makes it difficult to determine the H-atom position by X-ray structure analysis. However, the MEM/Rietveld analysis combined with high-brilliance SR powder diffraction enables us to reveal the position of H atoms as well as their chemical bonding (Noritake et al., 2002; Kubota et al., 2005). Adsorption of H₂ molecules in porous coordination polymers (Férey et al., 2003; Rowsell et al., 2004; Dybtsev et al., 2004) is one of the most promising candidates. To develop the rational synthetic strategy for a novel porous coordination polymer for high-performance hydrogen storage, the elucidation of the intermolecular interaction between H₂ molecules and pore walls is essential. Thus, Kubota et al. carried out the structure determination of adsorbed hydrogen in the nanochannel of CPL-1 by SR powder diffraction and revealed that the H₂ molecules also produce a one-dimensional array in the nanochannel analogous to the case of the O₂ molecule (Kubota et al., 2005).

Distributions of the MEM charge density of CPL-1 with H₂ molecules are shown in Fig. 10. The small peak maxima of elongated-shape electron distribution of the H₂ molecule were observed inside the nanochannels. The position of the H₂ molecule is close to that of the O atom which belongs to the carboxylate group. The H₂ molecule seems to be locked in the concave space formed by the O atom and the H atom of pyrazine of the CPL framework. Interatomic distances between the H atoms of the H₂ molecule and the nearest O atom are 2.7 (1) and 2.4 (1) Å, respectively. They are comparable with the distances calculated from the van der Waals radii. The O atom to which the H₂ molecule is close forms a coordination bond with the Cu atom and is slightly negatively charged. The Cu–OOC moiety is thought to be associated with the attractive hydrogen accommodation. It is interesting that Rosi et al. (2003) showed a similar result in a metal–organic framework material different from CPL-1. They claim that the site around Zn and O is one of the H₂-molecule binding sites, found by inelastic neutron scattering spectroscopy. This suggests that the metal and oxygen bonded to it may play an important role in hydrogen adsorption. Moreover, the pocket of the cavity formed by the carboxylate group and the pyrazine is suited to the size of adsorbed H₂ molecules. Both the suitability of the pocket to the size of adsorbed molecules and the Cu–OOC moieties are considered to be closely related to the H₂-molecule adsorption in this system. Consequently, the first direct observation of the H₂ molecules adsorbed in the nanochannels of CPL-1 gives us promising guidelines in hydrogen-gas storage-material design.

Figure 10 Close-up view of MEM charge-density distribution around an adsorbed H2 molecule as an equi-density contour surface. The equi-density level is 0.11 e Å^−3. The structural model of the CPL framework is superimposed on the charge-density map.

4.3. Chemical adsorption of acetylene

The features of acetylene (C₂H₂) adsorption were quite different from that of other gases (Matsuda et al., 2005). Fig. 11(a) shows the MEM charge density of a C₂H₂ molecule in CPL-1. The absorbed C₂H₂ molecule was clearly revealed as dumbbell-shaped electron densities with nipple-like projections. This interlayer dumbbell-shaped electron density of C₂H₂ molecules shows that one C₂H₂ molecule was adsorbed per two Cu atoms, which is different from the case of single molecules, O₂, N₂ and H₂, with one per Cu atom. Interestingly, C₂H₂ molecules align along the a axis with an inclination of 75.6° with an intermolecular distance of 4.78 Å. This long intermolecular distance means that C₂H₂ molecules are completely isolated from each other in the nanochannel of CPL-1. The section of electron density containing the mol­ecular axis of C₂H₂ in Fig. 11(b) reveals that there exist a few electrons between the H atom of C₂H₂ and the O atom of the carboxylate in the host framework. This indicates that O⋯H—C hydrogen bonding, which includes not only electrostatic interactions between positively charged H atoms and negatively charged O atoms but also some covalent-bond-like interaction, plays an important role in the stabilization of C₂H₂ in the nanochannel. It is the nano-sized specific space provided by CPL-1 that realizes the unusual stable molecular array of C₂H₂, which is unlikely to correspond to bulk fluid. This specific C₂H₂ array including hydrogen-bonding interactions was successfully characterized by the MEM/Rietveld method and SR powder diffraction.

Figure 11 MEM charge-density distribution of CPL-1 with the adsorption of acetylene. (a) Three-dimensional MEM map as an equal-density contour surface viewed from the nanochannel direction. The equi-density level is 1.0 e Å^−3. The central C2H2 molecule and O atoms are represented by green and red, respectively. Section MEM charge densities containing the molecular axis of C2H2 and the a axis of (b) fully adsorbed phase and (c) intermediate adsorbed phase. Contour lines are drawn from 0.00 to 1.00 e Å^−3 with intervals of 0.05 e Å^−3. Higher-density regions are omitted in the figure.

The flexible framework arising from coordination bonds and the degrees of freedom of organic molecules is one of the most interesting characteristic features of coordination polymers (Matsuda et al., 2004; Dybtsev et al., 2004). The in-depth understanding of the intermediate adsorption state provides us with a feasible design for a porous framework, which changes their structure into that well suited for desired guest molecules and results in an efficient accommodation system. The structure of the intermediate adsorption state was further studied and determined as shown in Fig. 12 (Kubota et al., 2006). The structural difference between fully and intermediate adsorption states was found to be the rotation of the C₂H₂ molecule in the nanochannel. The deformation of the CPL-1 framework structure was also found in association with molecular rotation. The obtained MEM charge density shows the clear disappearance of hydrogen bonding between the C₂H₂ molecule and pore wall indicating that the C₂H₂ mol­ecules in the intermediate state are more freely located with weak bonding inside the nanochannel (Fig. 11c). The results suggested a scenario of C₂H₂ adsorption in CPL-1 in which C₂H₂ molecules are adsorbed freely in the early stage and form an intermediate state and ratchet to lock in the final stage.

Figure 12 Acetylene adsorption isotherm and crystal structures of CPL-1 with adsorption of acetylene corresponding to the amount of adsorbed gas. These figures are viewed from the nanochannel direction. Acetylene molecules are shown by the space-filling model. Other molecules are shown by connecting lines.

Acetylene is one of the key molecules as a starting material for many chemical products and electric materials in the petrochemical and electronic industries. In order to obtain highly pure C₂H₂ for the preparation of these materials, the separation of C₂H₂ from the mixture gas containing CO₂ impurities without a large expenditure of energy is an important subject. In addition to this, acetylene is well known as a highly reactive molecule and, therefore, it cannot be compressed above 0.2 MPa; otherwise it explodes without oxygen, even at room temperature. Thus the result should open the door to create more feasible and safe materials for C₂H₂ separation.

Through this research we demonstrated that porous coordination polymers have a much more dynamic framework than is generally believed, and is characteristic of metal–organic species. The structural rearrangement of the host framework responding to guest molecules proceeds from the `open' phase to the `closed' phase. The structural findings by the MEM/Rietveld method would advance application of a unique class of materials, which cannot be obtained in rigid porous materials.

5. Metals in nano-space in a buckyball

Since the discovery of the buckyball molecule (Kroto et al., 1985), fullerene, many studies have attempted to encapsulate metal atoms inside the 1 nm-scale space inside the buckyball to create novel functional nanomaterials. However, X-ray structure analysis had not been carried out in the first 5 years since the research started. The difficulties of X-ray structure analysis for metallofullerenes come from the fact that intrinsic orientational disorder of the fullerene cage and encapsulated metal atoms is rather common. In such a case, it may be extremely difficult to construct an adequate structural model. The MEM/Rietveld method succeeded in overcoming such difficulties by the ability of the MEM in terms of a model-free reconstruction of the charge densities from measured X-ray diffraction data when it is applied to accurate synchrotron powder data. Recently, details of characteristic features for atoms inside the buckyball such as multiple disorders of metal atoms and encapsulation of the C₂ molecule are clearly visualized from high-resolution powder data measured at SPring-8.

One of the most intriguing observations is a perfect pentagonal dodecahedron charge density formed by two La atoms in an icosahedron C₈₀-I_h cage (Nishibori, Takata, Sakata, Taninaka et al., 2001). This novel shape of the charge density has been observed as a result of a highly selective trajectory of the two La atoms hopping along the hexagonal rings of the C₈₀-I_h polyhedral network. The three-dimensional equi-charge-density surface of charge density for the La atom in the La₂@C₈₀ molecule is shown in Fig. 13 with C₈₀ model. Figs. 13(a) and (b) represent the top views along the S₁₀ and C₂ axes of C₈₀-I_h, respectively. A pentagonal dodecahedron cage, which should be due to the charge density of the encaged La atoms, is clearly recognized inside the C₈₀ cage. The theor­etical study (Shimotani et al., 2004) succeeded in the reproduction of this highly selective trajectory for the internal motion of the encaged La atoms by examination of the energetically stable position of the La atoms inside the C₈₀ cage caused by charge transfer from fullerene cage to La atoms.

Figure 13 MEM charge densities of the encapsulated La2 pair as the equal density (1.7 e Å^−3) along the (a) S10 and (b) C2 axes. The model of the C80-Ih fullerene cage is represented by green sticks.

Recently similar pentagonal dodecahedron cages have also been found in C₈₂-based di-yttrium and di-scandium carbide endohedral metallofullerenes (Nishibori, Narioka et al., 2006, Nishibori, Ishihara et al., 2006). The MEM charge densities of encapsulated atoms Y₂C₂ for (Y₂C₂)@C₈₂(III) and Y₂ for Y₂@C₈₂(III) are shown in Figs. 14(a) and (b), respectively. Charge densities are inserted into the schematic structure of a C₈₂-C_3V(8) cage. The charge densities of two Y atoms for both Y₂C₂ and Y₂ are found to form pentagonal dodecahedra. In the case of (Y₂C₂)@C₈₂, the charge density colored blue is due to two C atoms encapsulated in the fullerene cage.

Figure 14 The equi-contour surface, 1.5 e Å^−3, of MEM charge density for (a) Y2C2, (b) Y2 and (c) La2. The charge densities are inserted into a schematic structure model of (a) and (b) C82-C3V(8) and (c) C80-Ih fullerene cage. The Y and La charge densities are colored red. The C2 charge densities are colored blue. (d) Schematic C-atom arrangement of the common part for C82-C3V(8) and C80-Ih cages.

The pentagonal dodecahedral shape of charge densities due to Y₂ in both (Y₂C₂)@C₈₂(III) and Y₂@C₈₂(III) is identical to that observed for La₂ in La₂@C₈₀. The MEM charge density of La₂ in La₂@C₈₂ with schematic structure model of C₈₀-I_h is also shown in Fig. 14(c). In Fig. 14(c), the C atoms shared by three six-membered rings are shown as red balls, forming a pent­agonal dodecahedron. The S₁₀ axis is perpendicular to the figure plane. This orientation of the pentagonal dodecahedron is identical to that of Y₂C₂ (Fig. 14a) and Y₂ (Fig. 14b). In Figs. 14(a) and (b), the C atoms shared by three six-membered rings on the C₈₂-C_3V(8) cage are shown as blue balls. It is important to note that the arrangement of 16 out of 22 blue balls in Figs. 14(a) and (b) can be superimposed on red balls in Fig. 14(c) except for the topmost part. The superimposed part of the carbon cage of C₈₂-C_3V(8) and C₈₀-I_h is extracted and shown in Fig. 14(d). It is found that 16 out of 20 Y-atom peaks are located under the C atoms shared by three six-membered rings. In La₂@C₈₀, the pentagonal dodecahedral charge density is caused by a time-averaged feature of hopping La atoms as revealed by MEM charge density and theoretical studies. The two Y atoms in both (Y₂C₂)@C₈₂(III) and Y₂@C₈₂(III) are also hopping inside the C₈₂-C_3V(8) cage. An arrangement of the 20 Y₂ sites maintains the threefold symmetry which explains the previous ¹³C NMR results (Inoue et al., 2004).

For two positively charged metal ions, the longer inter­atomic distance is favorable for reducing Coulomb repulsion. The formation of a pentagonal-dodecahedral charge density due to entrapped metal atoms is a rather common feature when two positively charged metal ions are trapped in a fullerene cage. In addition, encapsulated C₂ in (Y₂C₂)@C₈₂(III) are located at the center of the Y₂ pentagonal dodecahedron as if it is trapped in the Y₂ cage. The C₂ encapsulation to reduce the Coulomb repulsion between two negatively charged ions may frequently occur in other multi-metallofullerenes. The present results suggest the possibility of the existence of other carbon-encapsulated multi-metallofullerenes. Actually, the C₂ encapsulation of several multi-metallofullerenes such as Sc₃C₈₂ and Sc₂C₈₄ has been found after these studies.

6. Analysis resolution in the MEM/Rietveld method

It should be noted that the MEM/Rietveld method deduces the best possible averaged charge density of crystalline materials based simply on the maxent concept. Thus, the data reliability and resolution are strongly correlated to the result obtained. In the case of the endohedral metallofullerenes, critical examination had been required for the relationship between the observed data quality and the MEM charge density. The start of the discussion was the structure analysis of tri-Sc metallofullerene, Sc₃C₈₂. For the structure of Sc₃C₈₂, there had been two possibilities as (Sc₃C₂)@C₈₀ or Sc₃@C₈₂. Until then, we had reported a series of metal carbide (M₂C₂ species) encapsulated in a buckyball such as (Sc₂C₂)@C₈₄ (Wang et al., 2001) and (Y₂C₂)@C₈₂(III) (Inoue et al., 2004).

Our first report on the tri-Sc metallofullerene, Sc₃C₈₂, structure was the buckyball encapsulating an Sc₃ metal cluster. Sugai et al. (2001) have reported, however, the possibility of an endohedral metal carbide structure for (Sc₃C₂)@C₈₀ instead of Sc₃@C₈₂ by gas-phase ion mobility measurements. Actually, Iiduka et al. (2005) have recently shown that the fullerene cage of Sc₃C₈₂ is C₈₀-I_h instead of C₈₂ from ¹³C NMR results on the (Sc₃C₈₂)⁻ anion, indicating the intact (Sc₃C₂)@C₈₀ carbide structure. Single-crystal X-ray structures of chemically functionalized (Sc₃C₂)@C₈₀ molecules with an addition of adamantylidene–carbine (Ad) have also been reported. However, X-ray structures of the intact (Sc₃C₂)@C₈₀ fullerenes had not been known until we reported them. Finally, Nishibori has realized that the best possible structure model building depends on the data resolution in the MEM/Rietveld analysis just as much as in the case with conventional structure analysis (Nishibori, Terauchi et al., 2006). Here, the resolution effect in the MEM/Rietveld analysis of tri-scandium metallofullerene is described in some detail.

The key to the problem was a difference in quality of powder diffraction data measured at SPring-8 from that of the previous data (Takata, Nishibori, Sakata et al., 1999). In our previous report, the data resolution of the second-generation SR data was d > 2.1 Å. In the analysis, the fullerene cage for Sc₃C₈₂ was postulated as C₈₂ (i.e. Sc₃@C₈₂) based on the mass spectral data and the disorder of the Sc atoms was not considered. These led us to erroneously assign (Sc₃C₂)@C₈₀ as Sc₃@C₈₂ in the analysis. SPring-8 has great advantages over previous sources to improve the quality of data due to the availability of high flux and high energy resolution X-ray photons. The maximum intensity of the Bragg reflections for the present data was around 1500000 counts, which is approximately four times larger than for the previous data. The widths of Bragg peaks for the present data were from 0.04 to 0.08° in 2θ. These are approximately 2/3 of that of the previous data. The larger intensities and higher angular resolution enable us to perform accurate and reliable MEM/Rietveld analyses because even the weak Bragg intensities become detectable. The data set with d > 1.43 Å d-spacing range was used in the present analysis.

In the present MEM/Rietveld analysis, a spherical shell composed of 82 C atoms and a toluene molecule was used for the first Rietveld refinement to avoid model-biased effects. The model bias may lead to a wrong structure. An X-ray diffraction experiment only measures the amplitude of the structure factors. The initial phases of the structure factors are estimated from a structural model adopted at the beginning of the analysis. The structure obtained by the analysis can be biased by the model adopted at the beginning. In this study, the least-biased structural model for the fullerene cage, which must be a spherical shell, was used as an initial model.

In total, 837 observed structure factors were extracted after the refinement. The MEM analysis was carried out by the program ENIGMA (Tanaka et al., 2002). The MEM charge density at this stage is shown in Fig. 15(a). There are many peaks due to disorder of encapsulated Sc atoms in the MEM charge density of the fullerene cage. A relatively large peak seen at the center of the fullerene cage is assigned as C₂. The other peaks are assigned as disordered Sc atoms. Therefore, the fullerene cage is assigned as C₈₀.

Figure 15 The equi-contour (0.7 e Å^−3) mesh of MEM charge densities for the (Sc3C2)@C80 molecules based on a C82 spherical-shell model.

In the previous MEM/Rietveld study on Sc₃C₈₂, only 279 structure factors were used in the MEM analysis because of the weak Bragg intensities around d > 2.1 Å d-spacing range. No charge density at the center of the fullerene cage was found in that analysis. The limited data resolution failed to reveal C₂ encapsulation. To confirm this point, the MEM analysis was preformed using only 279 reflections with d > 2.1 Å d-spacing range with the SPring-8 data. The MEM charge density is shown in Fig. 15(b). The C₂ peak disappeared in the center of the fullerene cage, indicating that d > 2.1 Å resolution is not good enough to reveal a C₂ molecule in the C₈₀ carbon cage. In the previous analysis, we assigned three Sc atoms to the three large electron-density peaks in Fig. 15(b). Then we investigated possible directions for two isomers of C₈₂-C_3V cages. The structure with the lowest reliability factors of Rietveld refinement was a final solution. The resulting fit of the Rietveld refinement using the previous Sc₃@C₈₂ model for present d > 1.43 Å data is shown in Fig. 16(a). In the d > 2.6 Å (2θ < 22°) range, the fit is satisfactory. However, in the 2.6 > d > 1.43 Å range, there are large deviations indicating an incorrect structure. From these facts, it was concluded that the erroneous assignment of the previous analysis is mainly due to the lack of resolution in the analysis. The data accuracy and reliability in MEM electron-density analysis using powder data has also been discussed in the recent paper by Nishibori et al. (2007).

Figure 16 (a) The fitting results of Rietveld analysis using the Sc3@C82 structure model, which has been previously published (Takata, Nishibori, Sakata et al., 1999). A close up of the region between 19.0 and 41.0° is shown in the inset. (b) The fitting result of the final Rietveld analysis for (Sc3C2)@C80. A close up of the region between 19.0 and 41.0° is shown in the inset.

The result of the final Rietveld refinement for high-resolution d > 1.43 Å SPring-8 data is shown in Fig. 16(b). The reliability factors based on weighted profile, R_wp, and Bragg intensities, R_I, are as small as 2.60 and 4.36%, respectively. These values are lower than those of the previous study even in this high-resolution analysis. The lower reliability factors are satisfactory for the current precise structure determination.

The molecular structure of the major part of (Sc₃C₂)@C₈₀ determined by the MEM/Rietveld method is shown in Figs. 17(a) and (b). The C₂ molecule is shown as one yellow sphere, since the MEM charge density for C₂ appears as one peak. In Fig. 17(b), the three Sc atoms and C₂ are located on the belt of the six-membered ring. The present structure is similar to the theoretically predicted structures by recent density functional theory (DFT) calculations (Tan & Lu, 2006). The deviations of the Sc–Sc and Sc–C distances between the present and the theoretical structures are within 0.15 Å.

Figure 17 The molecular structure models of (Sc3C2)@C80 along the (a) S10 and (b) C2 axes determined by the MEM/Rietveld method from synchrotron X-ray powder diffraction data.

It is noted that the erroneous assignment of the previous analysis is due to the lack of resolution of the powder data. The available data resolution of powder data is dramatically increased by the third-generation SR sources such as SPring-8. High-resolution data with the MEM/Rietveld method can determine the structures of endohedral metallofullerenes including disorder and/or motion of metal atoms such as the present (Sc₃C₂)@C₈₀ study.

7. Advanced application to structure–property relationship investigation

In this review, some recent structural studies of novel nano-structured materials, CPL-1 and endohedral metallofullerene by the MEM/Rietveld method using high-resolution powder data measured at SPring-8 are described. Accurate structural information, such as bonding nature, positions of H atoms and multiple disorder of metal atoms, have been clearly revealed from powder diffraction data alone. One of our main purposes in carrying out a MEM charge-density study is to investigate structure–property relationships in novel materials. Recent progress of MEM charge-density studies has been made in terms of the investigation of structure–electronic-property relationships.

As a recent application to the structure–conductivity relationship study of a nano-material, the structure determination of the nano-cage compound 12CaO·7Al₂O₃ (C12A7) (Kim et al., 2007) is presented. This material is synthesized from lime and alumina, both typical insulators. However, it was converted into an inorganic electride C12A7:e⁻ by the reduction of O²⁻ ions encapsulated in the nano-cage (Matsuishi et al., 2003) (Fig. 18a). The electric and chemical properties of this material can be modified by the substitution of various encapsulated anions. In particular, the electrical transport nature from insulator to superconductor is recently reported (Miyakawa et al., 2007). We succeeded in determining the complicated structure of C12A7 by MEM/Rietveld analysis (Fig. 18). As shown in Fig. 18(b), in insulating C12A7:O²⁻, Ca atoms are displaced by the electrostatic interaction with a free O²⁻ ion in the cage and the cage is deformed. Free O²⁻ ions are randomly accommodated in two different cages out of twelve cages per unit cell to compensate for the charge of the framework. The splitting feature of the Ca-atom charge density indicates the mixture of regular cages and deformed cages. Consequently, the MEM charge density uncovered a new structural basis for the C12A7 conductivity (metal–insulator change), i.e. the cage structure homogeneity should be a key factor for the increase of the conductivity associated with substitution from O²⁻ ions to e⁻ doping (Fig. 18c).

Figure 18 (a) Nano-cage structure of C12A7. MEM charge densities on the (001) plane for (b) C12A7:O^2− and (c) C12A7:e^− electride. Contour lines are drawn from 0.0 to 6.0 e Å^−3 with intervals of 0.3 e Å^−3. Higher-density regions are omitted in the figure.

Another remarkable example of methodological progress is the recent achievement of electrostatic potential and electric field imaging based on MEM charge-density analysis (Tanaka et al., 2006). As an example of the successful application of the method, the visualized electrostatic potential and electric field of the typical ferroelectric material, tetragonal PbTiO₃, are shown as iso-surfaces of charge density in Figs. 19(a) and (b), respectively. In Fig. 19(a), most of the iso-surface around the Pb ion is colored green. Note that the upper area of the surface is red, while the bottom area is yellow. This means that the center of electronic charge is shifted in the up–down direction relative to the center of the nucleus charge, which is evidence for the polarization of the Pb ion at the atomic level. Furthermore, the distribution of electric field shown in Fig. 19(b) agrees well with that obtained from the ab initio calculation (Cohen, 1992). These facts mean that our method is efficient for the experimental analysis of the electrostatic potential in crystalline materials. The application of the method is growing in the research area of nano-materials science such as gas absorbed in CPL-1. The experimental electrostatic potential would give crucial information, for example, to investigate interplay between the guest molecule and host framework.

Figure 19 (a) Isosurface of charge density for PbTiO3 at 0.86 e Å^−3. It is colored by the value of electrostatic potential from −0.9 (red) to 0.9 e Å (blue). (b) Isosurface of charge density for PbTiO3 at 1 e Å^−3. It is colored by the amplitude of the z component of the electric field from −2.0 (red) to 2.0 e Å^−2.

Application of the MEM and the MEM/Rietveld method using SR data to nano-structured materials is coming into advanced use. For example, Sawa et al. (2005) have applied the MEM to the observation of a single H₂ molecule encapsulated in the fullerene cage of an aza-thia open-cage fullerene with SR single-crystal X-ray diffraction experiments. The clathrate-like cage structure by chemical bonding has been found in a binary CoSb₃ skutterudite (Ohno et al., 2007), a promising thermoelectric material, by the MEM/Rietveld method using multi-SR powder data taken at SPring-8 (Nishibori et al., 2007).

The method is now in a utilization phase as one of the proper techniques for thorough research into structure–property relationships of novel materials (for instance, Snyder et al., 2004) and will extend the innovative contribution of crystallography in the materials science research area, e.g. by further application to topological analysis (Cargnoni et al., 2004).