MEM analysis

Yamamoto, K., Takahashi, Y., Ohshima, K., Okamura, F.P. and Yukino, K. (1996) MEM analysis of electron-density distributions for silicon and diamond using short-wavelength X-rays (WKa, ), Acta Cryst., A52,606-613. 

Fig. 8.11. Cepstral and maximum entropy (MEM) analysis of time-resolved signals ...

In the MEM/Rietveld analysis, each of the observed structure factors of intrinsically overlapped reflections (for instance, 333 and 511 in a cubic system) can be deduced by the structure model based on a free atom model in the Rietveld refinement. In such a case, the obtained MEM charge density will be partially affected by the free atom model used. In order to reduce such a bias, the observed structure factors should be refined based on the deduced structure factors from the obtained MEM charge density. The detail of the process is described in the review article [9,22-24]. In addition, the phased values of structure factors based on the structure model used in Rietveld analysis are used in the MEM analysis. Thus, the phase refinement is also done for the noncentrosymmetric case as P2, of Sc C82 crystal by the iteration of MEM analysis. The detail of the process is also described elsewhere [25]. All of the charge densities shown in this article are obtained through these procedures. 

Acknowledgements The authors thank Dr. H. Tanaka for the computer program, ENIGMA, of the MEM analysis. For the construction of the experimental machine, the authors thank S. Hi-rano, Y. Wakui, K. Kumazawa, T. Sumi, and M. Kozuka of Nagoya University Machine Shop. This work has been supported by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan. A part of this work was financially supported by the Japan Atomic Energy Research Institute. 

Figure 3. Isosurfaces of electron densities calculated by a MEM analysis using from a preliminary structure model that does not contain H20 molecules. Balls indicate atoms used in the structure model. Equidensity level 1.2 A-3.

Figure 4. Structure model (left) and isosurfaces of electron densities determined for the sodium cobalt oxide hydrate by MEM analysis (right). Equidensity level 1.0 A 3.

The space group was assigned to P2i, which is monoclinic for Y Cs2. The experimental data were analyzed in an iterative way of a combination of Rietveld analysis (Rietveld, 1969) and the maximum entropy method (MEM) (Bricogne, 1988 Collins, 1982). The MEM can produce an election density distribution map from a set of X-ray structure factors without using any structural model. By the MEM analysis (Kumazawa et al., 1993 Sakata and Sato, 1990), the Rj becomes as low as 1.5% for Y Cs2-... 

The detailed endohedral structure of La Cg2 was revealed experimentally (Nishibori et al., 2000). The electron density distribution of La Cs2 based on the MEM analysis of the powder X-ray diffraction data is presented in Figure 10. The result shows that the La atom is encapsulated by the C2V isomer of Cg2 as in Sc Cg2 described above. As is seen from the figure, the La atom is not at rest in the cage but rather is in a floating motion along the nearest six-membered ring at room temperature. The result is different from the Sc Cg2 and Y C82 cases in which Sc and Y atoms are almost at a standstill in the cage even at room temperature. 

Time-resolved fluorescence studies of fluorene-fluorenone copolymers in dilute toluene solution, enable us to identify different time regimes in the photoluminescence (PL) decay. Figure 18 shows the results of a maximum entropy method (MEM) analysis of the PL decays of fluorene-fluorenone copolymers [58] collected at the fluorene emission wavelength [66]. The different time regimes of the PL decay are associated with different kinetic species which migrate to the defects, most typically these are the CTS defects... 

Fig. 18 Maximum entropy method (MEM) analysis of PF2/6 and PF/FLx copolymers. For PF2/6 only a narrow distribution is observed at 360 ps. For PF/FLx copolymers, together with the distribution 60 ps, two additional distributions are observed around 20 and lOOps...

Structure is nonmetallic with the calculated energy gap of 6.8-7.0eV. Because there is little contribution of Li orbitals to the occupied states, Li atoms are thought to be ionized as Li" cations. The occupied states split into two peaks the low-energy states are composed of B-2s and H-ls orbitals and the high-energy states consist of B-2p and H-ls orbitals. A boron atom constructs sp hybrids and forms covalent bonds with surrounding four H atoms. The charge from the extra electron needed to form these bonds is compensated by a Lf cation. This character is also confirmed experimentally by synchrotron X-ray diffraction measurement and maximum entropy method (MEM) analysis. 

A three-dimensional contour surface (0.15 fin A ) of nuclear distribution of lithium atoms is shown in Fig. 14.17. The probability density of lithium nuclei strictly distributes into the continuous curved one-dimensional chain along the [010] direction, which is consistent with the computational predictions by Morgan et al. [22] and Islam et al. [23]. Other atoms, Fe, P, and O remained to be localized at the initial positions even after MEM analysis. Given the two possible diffusion paths in Fig. 14.11, the microscopic reason of the diffusimi anisotropy can be the difference... 

Rietveld method, has been applied to describe chemical bonding, dynamic disorder, and anharmonic thermal vibration. Computer programs such as Rietan for the Rietveld analysis and PRIMA for the MEM analysis, which incorporate MEM and optimized for powder diffraction are available. 

More recently, this method has been modified to use the wave slopes derived from the water surface elevation measured by these gages in the MEM analysis rather than using directly the water surface elevations. This modification is triggered by the fact that the previous method was efficient only over a limited frequency range where the energy contained in the spectrum was substantial. Computational effort... 

The MEM analysis of time-resolved fluorescence decays of several copolymers is shown in Fig. 15.18a. For the homopolymer PF2/6, only a narrow distribution is observed around 360 ps. However, for copolymers PF/FLx, with different fractions of fluorenone residues, distributed randomly along the polymer chain, the distribution at 360 ps is accompanied by two additional peaks. These are observed around 20 and 100 ps as a result of quenching of polyfluorene emission, due to energy transfer from the fluorene to the fluorenone sinks. Figure 15.18b shows the fluorescence decay of the copolymer labelled with 25 % of fluorenone groups, analysed with a sum of three exponential functions. Note the good agreement between MEM and multiexponential analysis [91]. 

Recent papers gave many new insights in other locations for Cu ions inside zeolite framework structures. Andersen et al. [37] concluded that the majority of the Cu " ions are in a specific site in the 8R of SSZ-13, as well as in the d6R, based on Rietveld and MEM analysis applied to synchrotron pXRD data, on a dehydrated Cu-SSZ-13. They used a system with Si/Al ratio of 15.5 and a Cu/Al ratio of 0.45. By DFT calculations they assigned this specific site in 8R to [Cu +OH ]+ close to a Al atom, and Cu " in the d6r. This was backed up by the work of Godiksen et al. [38] where they used EPR to study their Cu-SSZ-13 system (Si/Al = 14). They explain the loss of EPR signals upon dehydration due to EPR silent species, [Cu +OH ]+. Borfecchia et al. [39] concluded that the majority of the dehydrated Cu species are found preferentially in 8R units of the SSZ-13 framework, as tri-coordinated [CuOH]+ species at 0 activation, and bi-coordinated Cu+ ions upon... 

The experimental diffraction data were analyzed by a combined technique involving Rietveld analysis, the maximum entropy method (MEM), and MEM-based pattern fitting (MPF) [10-15]. Rietveld analysis, which is used to refine the crystal structure from the powder diffraction data by a least squares method, was carried out using the RIETAN-2000 program [27], which yields structure factors and their errors after structural refinement. It is known that MEM can be used to obtain a nuclear density distribution map based on neutron structure factors and their errors [5, 6, 8, 10-15, 26-29] any type of complicated nuclear density distribution is allowed so long as it satisfies the symmetry requirements. MEM calculations were carried out using the PRIMA program [29]. To reduce the bias imposed by the simple structural model in the Rietveld refinement, an iterative procedure known as the REMEDY cycle [29] was applied after MEM analysis (Fig. 6.3). In this procedure, structure factors... 

MEM analysis to obtain nuclear dei lvdfeifibution [Fourier transform... 

Standard Rietveld indices reliability factors MEM analysis V unit-cell volume of the pseudo-perovskite cell. 

---