Electron density in rutile (TiO<sub>2</sub>) by the maximum entropy method

Sakata, Makoto; Uno, Tatsuya; Takata, Masaki; Mori, Rentaro

doi:10.1107/s0108768192001770

Cited by 32 publications

(16 citation statements)

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“…MEM analysis of the rutile data by this approach has been successfully completed. As expected, and in contrast to the electron-density distribution obtained by the MEM [Sakata, Uno, Takata & Mori (1992). Acta Cryst.…”

Section: Introductionmentioning

confidence: 51%

“…The theoretical treatment of the MEM that is used in the present work has been described in some detail elsewhere (Sakata, Uno, Takata & Mori, 1992). The following notation is defined: p(r), electron (or nuclear) density at position r; T(r), prior density for p(r); A, Lagrange undetermined multiplier; NI and N2, the number of reflections…”

Section: The Mem Analysis For Neutron Datamentioning

confidence: 99%

“…As explained in previous work (Sakata, Uno, Takata & Mori, 1992), (3) results from the maximization of entropy subject to the constraint…”

Section: Introductionmentioning

confidence: 99%

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Maximum-entropy-method analysis of neutron diffraction data

Sakata¹,

Uno²,

Takata³

et al. 1993

J Appl Cryst

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The maximum-entropy method (MEM) is a very powerful method for deriving accurate electron-density distributions from X-ray diffraction data. The success of the method depends on the fact that the electron density is always positive. In order to analyse neutron diffraction data by the MEM, it is necessary to overcome the difficulty of negative scattering lengths for some atoms, such as Ti and Mn. In this work, three approaches to the MEM analysis of neutron powder diffraction data are examined. The data, from rutile (TiO2), have been collected previously and analysed by the Rietveld method [Howard, Sabine & Dickson (1991). Acta Cryst. B47, [462][463][464][465][466][467][468]. The first approach is to add an artificial large constant to the scattering-length density to maintain that density positive, then to subtract the same constant at the completion of the MEM analysis. This approach, however, proves unsuccessful since unrealistic density distributions result. In the second approach, the observed structure factors are amended so that the sign of the contribution from the Ti atoms is reversed. This method produces plausible maps of scattering-length density but suffers the disadvantage that the observations must be corrected by a modeldependent calculated factor before the MEM analysis can proceed. The third approach is based not on scatteringlength densities but on nuclear densities, which are always positive. Two equations are obtained, one for atoms with nuclei of positive scattering length and the other for atoms with negative scattering length. From these two equations, the nuclear densities of Ti and O atoms can be calculated separately. This procedure, like its X-ray counterpart, requires no structural model. MEM analysis of the rutile data by this approach has been successfully completed. As expected, and in contrast to the electron-density distribution obtained by the MEM [Sakata, Uno, Takata & Mori (1992). Acta Cryst. B48, 591-598], the map shows both Ti and O nuclear densities localized in very small regions around the atomic centres. It is concluded that the MEM applied to neutron powder diffraction data is superior to conventional Fourier transformation and, in spite of the longer computation time, is very well worthwhile.

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Section: Introductionmentioning

confidence: 51%

Section: The Mem Analysis For Neutron Datamentioning

confidence: 99%

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Maximum-entropy-method analysis of neutron diffraction data

Sakata¹,

Uno²,

Takata³

et al. 1993

J Appl Cryst

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“…The first application of the MEM to a chargedensity study has been done for Si and demonstrated the great power of the MEM in reducing truncation effects (Sakata & Sato, 1990). Since then, charge-density studies using X-ray diffraction data have been carried out for several substances and reveal some new aspects of the nature of bonding (Sakata, Uno, Takata & Mori, 1992;.…”

Section: Introductionmentioning

confidence: 99%

The Influence of the Completeness of the Data Set on the Charge Density Obtained with the Maximum-Entropy Method. A Re-examination of the Electron-Density Distribution in Si

Takata¹,

Sakata²

1996

Acta Cryst Sect A

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The charge densities derived with the maximum-entropy method (MEM) may be influenced to some extent by the completeness of the data set. In order to examine the effects of the incompleteness, structure-factor data of Si measured by the PendellOsung method [Saka & Kato (1986). Acta Cryst. A42, 469-478] were re-analysed by the MEM. This data set is incomplete: it contains all space-group-allowed reflections with sin 0/2 = 0.86i -I, and in addition 844 and 880 with sin0/2 = 1.04,~, -1. Results of a MEM analysis of the complete subset of data are compared with those from the full but incomplete set published previously [Sakata & Sato (1990). Acta Cryst. A46, 263-270]. The smaller but complete set was found to give a smooth charge-density distribution that is consistent with previous theoretical work. It is found that the sharp peak maximum at the bond midpoint reported previously is exaggerated owing to the highest-order reflection 880. The completeness of the data set appears to be one of the key factors for obtaining reliable charge densities with MEM. The incompleteness of the data set may cause non-physical fine features of the MEM density distribution.

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“…Sakata, Uno, Takata & Mori (1992) have applied the ME method to rutile (TiO2) using accurate long-scan powder diffraction data. This structure had previously been studied by single-crystal deformation-density methods by Restori, Schwarzenbach & Schneider (1987).…”

Section: Powder Diffraction Datamentioning

confidence: 99%

Maximum Entropy and Bayesian Statistics in Crystallography: a Review of Practical Applications

Gilmore¹

1996

Acta Crystallogr A Found Crystallogr

156

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You should call it 'entropy' and for two reasons: first, the function is already in use in thermodynamics under that name; second, and more importantly, most people don't know what entropy really is, and if you use the word 'entropy' you will win every time! von Neumann's advice to Shannon as to what to call the function -)-~'i Pi log p~ AbstractA survey of practical applications of the maximum entropy method to problems in crystallography is presented along with the related field of Bayesian statistical inference. Included in this review are problems of the processing of intensity data, the crystallographic phase problem, accurate electrondensity studies and algorithms for the constrained maximization of entropy. The discussion of the phase problem reviews the solution of structures from powder and single-crystal diffraction data sets, as well as applications to electron and macromolecular crystallography. It is believed that the methodology has a secure future and a sound statistical basis, but it is still not a technique that is regularly employed by crystallographers and the reasons for this are also examined.

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Electron density in rutile (TiO₂) by the maximum entropy method

Cited by 32 publications

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Maximum-entropy-method analysis of neutron diffraction data

Maximum-entropy-method analysis of neutron diffraction data

The Influence of the Completeness of the Data Set on the Charge Density Obtained with the Maximum-Entropy Method. A Re-examination of the Electron-Density Distribution in Si

Maximum Entropy and Bayesian Statistics in Crystallography: a Review of Practical Applications

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Electron density in rutile (TiO2) by the maximum entropy method

Cited by 32 publications

References 0 publications

Maximum-entropy-method analysis of neutron diffraction data

Maximum-entropy-method analysis of neutron diffraction data

The Influence of the Completeness of the Data Set on the Charge Density Obtained with the Maximum-Entropy Method. A Re-examination of the Electron-Density Distribution in Si

Maximum Entropy and Bayesian Statistics in Crystallography: a Review of Practical Applications

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Electron density in rutile (TiO₂) by the maximum entropy method