Electron-Density Distribution from X-ray Powder Data by Use of Profile Fits and the Maximum-Entropy Method

Transcription

1 526 J. Appl. Cryst. (1990). 23, Electron-Density Distribution from X-ray Powder Data by Use of Profile Fits and the Maximum-Entropy Method By M. SAKATA, R. MORI, S. KUMAZAWA AND M. TAKATA Department of Applied Physics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Japan AND H. TORAYA Ceramic Engineering Research Laboratory, Nagoya Institute of Technology, Asahigaoka, Tajimi, Japan 507 (Received 9 March 1990; accepted 6 July 1990) Abstract Following the profile decomposition of CeO2 X-ray powder data into individual structure factors, the maximum-entropy method (MEM) has been used to obtain an electron-density-distribution map. In the profile decomposition process, it is impossible to avoid the problems of overlapping peaks which have the same magnitude of reciprocal vectors, such as d*(511) and d*(333), for a cubic crystal, or very severely overlapping reflections. The formalism to treat such overlapping reflections in the MEM analysis is to introduce combined structure factors. The maximum value of the scattering vector, 47r(sin0)/A, which was used in the present analysis is small (about ) but the resulting electrondensity-distribution map is of a high quality and much superior to the conventional map. As a consequence, the ionic charge of Ce and O ions can be obtained with reasonable accuracy from the MEM density map. Furthermore, the map reveals the existence of electrons around the supposedly vacant site surrounded by eight O atoms, which is probably related to the high ionic conductivity of this substance. 1. Introduction Since the advent of the Rietveld method (Rietveld, 1969) for profile refinement of powder diffraction data, this technique has been widely used for crystal structure refinement and analysis of powder samples. In the Rietveld method, the profile parameters and structural parameters are refined simultaneously based on a given structural model. Accordingly, the intensity at each measured point can be calculated, regardless of the existence of the overlapping powder lines. Its greatest advantage is obviously related to its high capability of solving problems involving overlapping powder lines. This method is basically suited to the refinement of a structural model (Lehmann, Christensen, Fjellv~g, Feidenhans'l & Nielson, 1987) /90/ The Rietveld method was not originally aimed at obtaining the electron-density distribution. However, it is possible to extend the Rietveld method for that purpose. It is said that the difference Fourier map obtained by the Rietveld method is sometimes useful to check the structural model used in the Rietveld method (Cheetham & Taylor, 1977). The Rietveld method, however, may not be suitable to draw a precise electron-density-distribution map from powder diffraction data, because it introduces the structural model in the first place and all the measured intensities including the overlapped intensities are interpreted so as to be consistent with the structural model as far as possible. This may cause prejudice in the interpretation of the observed data. For example, if the structural model is not good or precise enough compared with the accuracy of the measured intensities, the individual intensities derived from the overlapped intensities by the Rietveld method could be strongly biased.. Another approach to tackle the problems of overlapping peaks in the powder pattern is based on profile fits or the profile decomposition method (Parrish, Huang & Ayers, 1976). In this method, the integrated intensities, which are proportional to the square of the structure factors, are variables in calculating the overlapped powder pattern. Previously only a part of the whole powder pattern is treated in this method. Pawley (1981) has developed it in the neutron diffraction case into the WPPD (whole powder pattern decomposition) method, in which the whole powder pattern can be treated as one profile to be decomposed including simultaneous latticeparameter refinement. Toraya (1986) has extended WPPD as a tool for analyzing X-ray powder data where the complexity of Kal and Ka2 double peaks exists. These methods can be applied without prior knowledge about the crystal structure and hence enable one to evaluate the integrated intensities without any prejudice about the structure. The maximum-entropy method (MEM) for crystal structure refinement has been known for some years 1990 International Union of Crystallography

2 M. SAKATA, R. MORI, S. KUMAZAWA, M. TAKATA AND H. TORAYA 527 to give an electron-density distribution which is consistent with the given information and least biased with respect to missing information (Wilkins, Varghese & Lehmann, 1983). Besides this general feature of MEM, Sakata & Sato (1990) have recently revealed that MEM has some advantages over other methods for drawing high-quality electron-densitydistribution maps from accurately determined structure-factor data. Their main point is that MEM enables one to extract detailed information on the electron-density distribution, such as deformation of electron-density distribution from free atoms, while the conventional direct Fourier method fails to extract such information from the same data. Modern powder diffractometers have improved not only the instrumental resolution, particularly since the advent of synchrotron radiation, but also the peak-to-background ratios. There would be, therefore, no difficulty in measuring the integrated intensities and hence structure factors to high accuracy by powder diffraction. Accurate structure factors should include precise information on electron-density distribution. It seems that the accuracy of measuring structure factors in modern powder diffractometry is very high and the structural model involving the use of free atoms or ions for a simple crystalline substance is not sufficient to explain the detail present in the observed structure factors. In spite of the popularity and success of the Rietveld method in the analysis of powder X-ray data, it appears that it has not been considered very seriously for drawing precise electron-density distributions from powder X-ray data. In this work, we present a new method of drawing an electron-density distribution map from X-ray powder data, which is composed of the combination of the powder pattern decomposition and the maximum-entropy methods. As an example of the present method, the results for CeO2 (fluorite-type structure) will be given and possibilities for the future development of the method will be discussed. Because the MEM has great potential for ab initio structure determination (e.g. Gull, Livesey & Sivia, 1987), it seems likely that it will be possible to extend the present method to the ab initio structure determination of completely unknown structures from powder diffraction data. There have been quite a few publications on this subject (e.g. McCusker, 1988; Attfield, 1988). The main concern of the present work, however, is not ab initio structure determination. We will, therefore, not discuss the phase problem involved in deriving the structure factors from the integrated intensities, which is the central problem of ab initio structure determination. In this work, we are concerned with the question of how precisely we can draw the electron-density distribution for known structures from X-ray powder data. In this context, the recent works of Will, Bellotto, Parrish & Hart (1988) and Uno, Ishigaki, Ozawa & Yukino (1989) are good counterpart analyses done by the conventional method. 2. Method The sequence of the present method is given in Fig. 1. In the figure, the method is divided into five steps. Step 1 is an X-ray powder diffraction experiment to obtain the whole powder pattern. It is highly desirable to avoid preferred-orientation effects, since it is very difficult to correct for this. Step 2 is profile fits or decomposition. The theory and practice of profile fitting and WPPD are thoroughly described elsewhere (Pawley, 1981; Toraya, 1986). Step 3 is necessary to convert the integrated intensities to the structure factors on an absolute scale. The best way to accomplish step 3 would be absolute measurement of the intensities. From the practical viewpoint, there is an alternative. All we need in step 3 is the value of the scale factor. It can be determined by a least-squares refinement for powder data such as the PO WLS computer program (Will, 1979; Will, Parrish & Huang, 1983). We adopted the latter method. For known structures, it is fairly easy to analyze the powder data by least-squares refinement and to obtain the scale factor to a reasonable accuracy through which the conversion to absolute Step 1 Step 2 Step 3 Step 4 Step 5 Procedure and [Object] Powder Diffraction Experiment I Powder Profile Decomposition (WPPD if necessary) Least Squares refinement by POWLS (or Absolute Measurement) Maximum Entropy Method] I Integration of electron density around a certain site [Whole Powder Pattern] [Integrated Intensity for each component peak] [Structure Factors in Absolute Scale] [Electron Density] [Ionic Charge] Fig. 1. Sequence of the present method to draw the electrondensity distribution. In the figure, both 'Procedure' and 'Object', which can be obtained by the 'Procedure', are shown.

3 528 ELECTRON-DENSITY DISTRIBUTION FROM X-RAY POWDER DATA scale can be done. The scale factor is normally very well refined and shows very little correlation with other parameters. The biggest advantage of using least-squares refinement in step 3is its convenience. Step 4 is the MEM analysis using X-ray powder data. In the case of single-crystal work, the theory of MEM is outlined by Sakata & Sato (1990) which is based on Collins's (1982) formalism. Sakata & Sato (1990) considered two kinds of constraints, that is, for the phase-known and phase-unknown structure factors. In this paper, the phase-unknown case will not be included for simplicity. Furthermore, in order to apply the formula to powder diffraction data, a slight modification is necessary to make it possible to treat the overlapped peaks. It is, of course, not possible to evaluate the individual intensities of an overlapped peak when the overlap is very severe compared with the instrumental resolution. An extreme case is the overlap of two peaks which have the same magnitude of reciprocal vector, such as the 511 and 333 reflections in a cubic crystal. Because of this, we cannot avoid the overlap problem completely as long as powder diffraction data are used, even if very-high-resolution powder X-ray data, such as that obtained with synchrotron radiation, is used. Without any information on crystal structure, such severely overlapped peaks should be treated as the combined intensities. The information-theoretical entropy is written as S = - ~p'(r) In [p'(r)/z'(r)], (1) iv where the probability p'(r) and prior probability z'(r) are related to the actual electron density by p'(r) = p(r)/~p(r) (2) r C(r) = z(r)/z1"(r) (3) g and p(r) is the electron density at a certain pixel located at r and ~'(r) is the prior density for p(r). In the single-crystal case, a constraint C is introduced for phase-known structure-factor data as C = N-l~-'~ [Foal(k)- Fobs(k)lz/trz(k), (4) k where N is the number of phase-known structure factors in the data set, Fobs(k) is the observed structure factors for the reflection k, tr(k) is the standard deviation of Fobs(k) and F~(k) is the calculated structure factor given as Foal(k) = V~p(r)exp(- 27rik.r), (5) r where V is the unit-cell volume. For the powder diffraction case, we modify the constraint by using the combined structure factor, G(j), for the severely overlapped reflections as where C = N -~ N IF=,(k)- Fobs(k)12/o'2(k) Gcal(j) = [~'m(k)lf~al(k)lz/~'m(k)] 1/2 (7) and m(k) is the multiplicity of the structure factor, k, in an intensity peak involving overlapping. The observed combined structure factor, Gobs(j), can be obtained by taking the square root of the observed intensity peak involving overlapping reflections after the ordinary correction for absorption, Lorentz factor etc. The summation in (7) is performed over the overlapping reflections in an intensity peak, j. When n reflections overlap in an intensity peak, j, of a powder pattern and cannot be separated by profile fitting, the summation is taken up to n. Therefore, n varies with j. In this way, the severely overlapped peaks can be treated as the combined structure factor, G(j). In (6), N is equal to N1 + N2 where N 1 is the number of structure factors evaluated individually and N2 is the number of combined structure factors. Using the Lagrange method of undetermined multipliers, we have Q(A) = - ~p'(r)lnlo'(r)/r'(r)] - (M2)C (8) F where a is the Lagrange multiplier. By setting dq(a)/dp(r) = 0 (9) and using some approximations shown by Collins (1982), we have p(r) = exp[lnz(r) + (A/N)E[o-2(k)] -1 where X (Fobs(k) -- F~l(k)} exp(- 27rik.r) + ~[o'2(j)g~l(j)~m(k)]-'{g~(j) - Gobs(j)} x ~m(k)f~a,(k)exp(- 2~oik.r)], (lo) A=AFo (11) and F0 is equal to the number of electrons, Z, in a unit cell. Using (8) in an iterative procedure, the electron-density-distribution map can be obtained from powder data. Once the electron-density-distribution map is obtained, various values may be calculated in step 5, such as the structure factors for unmeasured reflections. Step 5, therefore, is not a single procedure. In this work, we concentrated on calculating the values of the ionic charge of the constituent atoms.

4 M. SAKATA, R. MORI, S. KUMAZAWA, M. TAKATA AND H. TORAYA Experiment and data analysis In order to be free from preferred orientation as much as possible, a NIST intensity standard CeO2 powder was used as the specimen for testing the present method in a flat-sample configuration for X-ray powder diffraction. An electron micrograph of the specimen is shown in Fig. 2. The particles in the h CeO2 showed beautiful crystal facets and each par- 1 ticle examined was found to be a single crystal from 2 observation of the electron diffraction pattern. The 2 particles are not disc-like nor needle-like, for which 2 1 the preferred orientation is known to be strong. The 4 typical particle size in the powder sample is about 3 3 I~m and the particle size distribution seems to be 4 fairly uniform. In addition to the above electron- 4 microscope observation, the preferred orientation of 4 5 the sample was investigated by X-ray diffraction 6 using a wide-range o~ scan, which is sometimes called 3 an e scan (Yukino & Uno, 1986). The result of the to 2 scan for the 111 Bragg reflection is shown in Fig Above all, it is confirmed that the specimen shows 5 virtually no preferred orientation. 6 The powder X-ray diffraction experiment was 4 done using a Rigaku RAD-C diffractometer with Fig. 2. An electron micrograph of CeO2 powder specimen. ~ ,o i- 2OOOO E 0 i I0? ~o(degree) Fig. 3. The profile of the to scan of the CeO2 powder specimen. Table. 1. The observed integrated intensity, Iobs, the observed structure factor, Fobs, which is converted from Iobs using the scale factor and the calculated structure factor from the MEM density distribution, Fcat, for each peak obs k l (countss -~) Fobs F=~ (3) (9) (1) (8) (1) (7) (1) (6) (3) (7) (3) (9) (4) (6) (3) (5) (4) (6) (2) (7) (5) (4) (4) (5) (5) (5) I00"I (4) 86-7 (4) ~ 49"6 (3) 117 (3) 118"2 l IJ 0 0~ 32"8 (3) 93 (3) 94"1 4 2J Cu Ka radiation monochromatized by PG(200). The X-ray powder patterns were recorded in steps of 0-02 in 20, and the counting time at each point was 20 s. The scan range in 20 was from 15 to 145 and the maximum intensity recorded was counts at in 28. The profile fits were carried out by the computer program for profile fitting in a small 20 range, PROFIT (Toraya, 1986) at the first stage. We examined two different profile functions which were split Pearson VII and pseudo-voigt functions. It was found that the pseudo-voigt function always gave better agreement with the observed intensities and also that low-angle peaks showed strong asymmetry. We, therefore, included the asymmetry in the pseudo-voigt function as P(20) = g(20){ 1 - as(e0-2oj)e/tan0j}, (12) where g(20) is the profile function, a is the asymmetry parameter, s is 1 for 20 > 20j and - 1 for 20 < 20j and 20j is the Bragg peak position. At the final stage, we estimated integrated intensities by the computer program for WPPD, WPPF (Toraya, 1986) with the asymmetric pseudo-voigt function and they are listed in the last column of Table 1. In order to derive structure factors, the integrated intensities have to be converted to the structure factors. It would be desirable to accomplish this process experimentally; i.e. by the absolute measurement of intensities. It is, however, not very common to perform an absolute measurement in powder diffraction, since it needs very careful experimental methods not only for the intensity measurement but

5 530 ELECTRON-DENSITY DISTRIBUTION FROM X-RAY POWDER DATA also for the volume and weight measurements of the sample. If an absolute measurement was essential in the present method, the convenience of powder diffraction would have been lost. There is, however, an alternative to accomplish the process. That is, the determination of the scale factor by an ordinary least-squares analysis using, for example, POWLS (Will, 1979; Will, Parrish & Huang, 1983). The adoption of the least-squares refinement implies the phase problem can be solved in this process. In the present study, the scale factor is determined by least-squares refinement and all the integrated intensities were converted to structure factors. In the analysis of the least-squares refinement, cerium and oxygen were assumed to be free ions, which is, strictly speaking, not true. Although the scale factor showed little correlation with other parameters and refined very well, the scale factor determined by the least-squares refinement may be somewhat different from the true value, because the structural model adopted in the least-squares-refinement analysis was inadequate to some extent. This may cause the introduction of some kinds of bias to the data in this process. In order to understand how significant would be the bias possibly introduced by the process of the scalefactor determination, we will discuss in 4 the effects Table 2. Summary of the results of the least-squares analysis by PO WLS; only the value of the scale factor is used for further MEM analysis Number of data used in PO WLS analysis lsotropic temperature parameter of O, Bo(,~, 2) lsotropic temperature parameter of Ce, Bc~(A 2) Scale factor R factor Weighted R factor (98) (14) (5) of the mis-estimation of the scale factor onto the final electron-density-distribution map. In the least-squares refinement, anomalous dispersion, possible anharmonic thermal vibration and non-stoichiometry of oxygen were considered. At the present accuracy level with the limited number of data, the anharmonicity and non-stoichiometry were not found. It should be mentioned here again that all we need for the least-squares refinement is the value of the scale factor despite the fact that we considered the various effects mentioned above. We considered these effects because they may affect the value of the scale factor. The final results of the least-squares refinement is shown in Table 2. The structure factors obtained by the least-squares refinement are also included in Table 1. i (a) (b) (c) (d) Fig. 4. The electron-density-distribution map of CeO2 obtained by MEM analysis. (a) and (b) are the higher and lower electron-density regions for the (ll0) plane, respectively. The Ce and O ions are indicated. (c) and (a t) are the higher and lower density regions for the (400) plane, respectively, where only O ions are located. The contour lines (e A -3) are drawn from 5-0 to 50.0 with 5.0 intervals in (a), (c) and from 0.2 to 5.0 with 0.2 intervals in (b), (at).

6 M. SAKATA, R. MORI, S. KUMAZAWA, M. TAKATA AND H. TORAYA 531 From the least-squares refinement, we obtained the estimated values of 14 independent structure factors and 2 combined structure factors which are the two pairs 333 and 511 and 600 and 442[ 4. MEM density distribution The procedure to obtain MEM density distribution from the structure-factor data is very simple and straightforward. For MEM analysis, we always start from the uniform electron-density distribution which is the maximum-entropy state. For CeO2, the total number of electrons in a unit cell is 296 and the number of pixels used is The electron density of all pixels, therefore, is set to (p)= Se A -3 at the initial state. Next we initialize the value of the Lagrange multiplier, A, which was chosen as 0.01 in the present case. A little experience may be needed to choose the value of A. We found from experience that the convergence will never be attained if the initial A value is too large and that it will converge very slowly if a small ~ value is chosen. Since it is known that the MEM density distribution does not depend on the A value as long as the convergence is attained (Gull & Daniel, 1978; Sakata & Sato, 1990), the choice of A value will not cause a real problem. In this context, it can be said that the MEM density distribution is obtained in a unique way for a given data set. The MEM deduction is done in an iterative way using (10). Taking advantage of symmetry, we calculate the electron density only for the minimum asymmetric unit from (10) and obtain the electron density for the whole unit cell by symmetry operations. The space group of CeO2 is Fm3m. The minimum asymmetric unit, therefore, is 0 <_ x _< 1/2; 0 <_ y _< 1/4; z < 1/4; Ymin (X, 1/2 - x); z <_ y (International Tables for Crystallography, 1983). The iteration is continued until the condition C<_ 1 is satisfied. In the present case, iteration 285 times gave convergence. The density distribution of CeO2 obtained from the present analysis is given in Fig. 4. For comparison, the conventional direct Fourier map is also shown in Fig. 5. In the conventional method, only the independent structure-factor data are used, since it is not known how to use the combined structurefactor data in the conventional Fourier synthesis without assuming a structural model to divide them into individual structure factors. It is obvious that the MEM density-distribution map is much superior to the conventional direct Fourier map which is severely affected by termination effects as a consequence of the very limited amount of available structure-factor data. In the MEM density-distribution map of CeO2, there is no bonding electron between the two nearest Ce and O, Ce and Ce or O and O ions. This result is very reasonable for an ionic crystal and is in strong contrast with the previous MEM analysis of silicon by Sakata & Sato (1990), where the covalent electron is clearly shown between two Si atoms. It is fairly straightforward in principle to calculate the ionic charge of the constituent atoms from the MEM density-distribution map in principle. All we have to do is to sum up the number of electrons in a certain region. In the present study, we calculated the ionic charge in two different ways. One was to sum up the number of electrons enclosed by a sphere, and the other is to sum them up in the polyhedra centered at the Ce and O sites. In the sphere case, the oxygen sphere and the cerium sphere are tangential. In the polyhedron case, all the electrons are attributed to either Ce or O ions. The results are given in Table 3 in both cases. Considering that the ionic charges of the Ce and O ions must be between + 3 and + 4 for Ce and between - 1 and -2 for O, the value seems to be reasonable. To establish the ionic charge of CeO2 more precisely, we cannot avoid the...-~r,-~;',wt-'-c':/,,-; _.J~"" ~J'/~-,:-,'.,...,/i'"'" ',-;,77 )//,,,;-. ' ' ' ",._i---- ),.,.\ _...~/.'.\',. +,, x :=.,',, ~ ~,1 )t._.)/., t;~ --7;,_` \\~ :-kw (a) ;/'O---~J ~',.~ (b) U.i Fig. 5. The electron-density distribution maps of (a) (110) and (b) (400) planes obtained by the conventional direct Fourier synthesis using 14 structure factors in Table 1. The contour lines (e A-3) are drawn from to 50.0 with 5.0 intervals. The negative regions are shown by a dotted line.

7 532 ELECTRON-DENSITY DISTRIBUTION FROM X-RAY POWDER DATA problem of the electron which is not located around the Ce or O sites. In the present case, such electrons are visible in Fig. 4(b) within the vacant site surrounded by eight O atoms. This vacant site is denoted E in the figure. For convenience, we call such electrons 'excess electrons' in this paper. This causes some complexities in trying to determine the precise ionic charges in CeO2. These will be discussed in more detail in The influence of the mis-estimation of the scale factor In the present method, we are free from any prejudice concerning the assumption of a structural model except space group or the introduction of bias to the observed data through the process of the analysis except for the procedure of determining the scale factor by the least-squares refinement. In order to investigate the influence of the possible mis-estimation of the scale factor, we examined the MEM density-distribution map by changing the scale Table 3. The values of ionic charge for Ce and 0 ions /n CeO2 evaluated by two different means Sphere Polyhedron Ce O factor by _+ 5tr where tr is the standard deviation of the scale factor estimated in the least-squares refinement. The scale factor is very well refined in the least-squares refinement and it would be reasonable to suppose that _ 5tr would be the extreme case of mis-estimation. The MEM density-distribution maps are drawn using the structure factors calculated from these scale factors. They are shown in Fig. 6. For the higherdensity region, there is virtually no difference. Even for the lower-density region, the difference is very small. In order to make the difference as clear as possible, the electron density between Ce and O ions, where the difference looks the biggest in Fig. 6, are shown in Fig. 7. For the smaller scale factor, for which the values of all structure factors become larger, the minimum of density becomes very slightly smaller. The situation is reversed for the larger scale factor. It can, however, be said that the difference is not very significant. In consequence, the MEM density-distribution maps in Fig. 4 are not significantly influenced through the process of the determination of the scale factor by the least-squares refinement. (a) 6. Discussion In 4, it was mentioned that there exist some electrons around the vacant site E. Because of the existence of excess electrons, there is a difference between the ionic charge estimated by the sphere and the 104E J i i i J i ~ i ) i i i t i I // o~103.== "~!0 z i (b) Fig. 6. The MEM density-distribution map of the (110) plane obtained by using the structure factors calculated from the scale factor which differ by (a) + 5o" and (b) -5o'. Only the lowerdensity region is shown. The contour lines are the same as in Fig. 4(b). "~!0 I 10-1 i 1 L! [ I q I I I t I T l E 0 I [111) direction Fig. 7. The electron density between Ce and O ions estimated by the scale factor differing by + 5tr (C)) and -5tr (A). The solid line is for the correct scale factor.

8 M. SAKATA, R. MORI, S. KUMAZAWA, M. TAKATA AND H. TORAYA 533 polyhedron cases, as shown in Table 3. In a unit cell, there are four such vacant sites. Summing up, the number of excess electrons in each oxygen cube is evaluated as 0.8. The ionic charge of CeO2 depends on whether any nuclei exist at the vacant site E. It would be very unrealistic to believe there exists a Ce nucleus at the site E. We therefore will not discuss such a case. In the case where there is an O nucleus at the vacant site E, the excess electrons should be attributed to oxygen. It is possible that the O atom's ionic charge at the ordinary O-atom site and the E site is different. In such a case, we cannot estimate more precisely the ionic charge of oxygen than the values given in Table 3, until the occupancy of oxygen at the E site is known. If we could assume that the ionic charge of oxygen at the ordinary site and the E site is the same, it could be said that the ionic charge of CeO2 may be represented as Ce In this case, the occupancy of oxygen at the E site can be estimated as about 9% of the total oxygen. In the case that there is no nucleus near the E site, there is no justification to attribute the excess electron to oxygen. In this case, it would be more appropriate to express the ionic charge of CeO2 as Ce3Z+OlZ-2X 8-, where X represents the excess electron. Finally, it should be mentioned that the present estimation of the ionic charge is based on the maximum-entropy map with 32 x 32 x 32 pixels. The maximum-entropy map with pixels should give a better estimation. CeO2 is a material of high ionic conductivity, which is probably due to the excess electrons. In order to understand the relationship between excess electrons and the ionic conductivity, it will be very useful to find out whether any nucleus exists inside the oxygen cube or not. If there is a nucleus, the supposedly vacant site E is not really a vacant site. The MEM analysis for a neutron diffraction experiment can give us the answer to this question. In general, to perform the MEM analysis for both X-ray and neutron powder data will be very useful, since we can obtain the electron and nuclear distributions simultaneously. It should be emphasized that the distributions of electrons and nuclei are essentially different kinds of information. As for the distribution of excess electrons, two small electron-density maxima are shown in Fig.4(b) in the oxygen cube. Three dimensionally, the number of these maxima is 6. Considering that the disordered state is the higher-entropy state, it is not certain whether these maxima are real or not. These maxima may simply mean that the information about the density distribution inside the oxygen cube is not sufficient to exclude this possibility. In such a case, MEM would select a higher-entropy state as the best estimation. In the very recent work of Will (1988) using synchrotron radiation, it is claimed that there exists bonding electrons between the two Ce ions. Although the interpretation of Fig. 6 in his paper is insufficient, we believe that what he called chemical bonding is the same as the excess electron on the vacant site of the oxygen cube mentioned above. It is true that these excess electrons exist between two second-neighbor Ce ions but it would not be appropriate to call this chemical bonding of Ce ions. In the present work, we described how to treat powder diffraction data by MEM. For single-crystal data, MEM can be applied in a straightforward way using (4) as a constraint. We have to, however, be careful in handling by the MEM-analysis singlecrystal data which involve extinction. At the present stage of extinction theory, the extinction effect is corrected as depending on the calculated structure factor from a certain structural model. In other words, the extinction effect is corrected so as to be consistent with the structural model. In consequence, we cannot avoid the introduction of some bias to the structure-factor data through the extinction correction. For example, if a structural model with spherical density distribution of the constituent atoms is assumed for the extinction correction, the information on the asphericity which would be included in the structure factor in the extinction-free case will be lost by the extinction correction. In this context, powder diffraction data seems to be more suitable for treatment by MEM analysis. 7. Concluding remarks A new method for obtaining the electron-densitydistribution map from X-ray powder data has been proposed. It is composed of the combination of profile fits and the maximum-entropy method. The new method is much superior to the conventional method with respect to the resolution of the electrondensity-distribution map and it looks promising for drawing precise electron-density maps from X-ray powder data. All of the computations in this work were done at the Computer Center of Nagoya University which is gratefully acknowledged by the authors. References ATTFIELD, J. P. (1988). Acta Cryst. B44, CHEETHAM, A. K. & TAYLOR, J. C. (1977). J. Solid State Chem. 21, COLLINS, D. M. (1982). Nature (London), 298, GULL, S. F. & DANIELL, G. J. (1978). Nature (London), 272, GULL, S. F., LIVESEY, A. K. & SIVIA, D. S. (1987). Acta Cryst. A43,

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