A direct investigation of thermal vibrations of beryllium in real space through the maximum-entropy method applied to single-crystal neutron diffraction data

Takata, Masaki; Sakata, Makoto; Kumazawa, Shigeru; Larsen, Finn; Iversen, Bo B.

doi:10.1107/s0108767393011523

Cited by 36 publications

(27 citation statements)

References 3 publications

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“…This has been explained as an effect of inferior quality of the high-order reflections because of, for instance, uncertainty in the corrections for thermal diffuse scattering (Takata, Sakata, Kumazawa, Larsen & Iversen, 1994). Another problem with the present procedure is the observation that the distribution of residual structure factors, F ° -F~, can be very nonuniform.…”

Section: Calculationsmentioning

confidence: 87%

Experimental evidence for the existence of non-nuclear maxima in the electron-density distribution of metallic beryllium. A comparative study of the maximum entropy method and the multipole refinement method

Iversen¹,

Larsen²,

Souhassou³

et al. 1995

Acta Crystallogr Sect B

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The electron-density distribution (EDD) of metallic beryllium has been derived from the structure factors of Larsen & Hansen [(1984). Acta Cryst. B40, 169-179] using the maximum entropy method (MEM). Subsequent topological analysis reveals non-nuclear maxima (NNM) in the EDD. Plots of the gradient field of the electron density illustrates this finding. A possible critical-point network for the hexagonal close-packed (h.c.p.) structure of beryllium is suggested. It is thus demonstrated that it is possible to obtain detailed topological information about the electron density in metallic beryllium without the use of a structural model. In order to test the findings of the MEM, the same set of structure factors were analysed using the multipole refinement method (MRM). Use of the MRM also reveals NNM. The results of the two different approaches to electron-density analysis are contrasted and discussed. Expressed within the framework of the theory of atoms in molecules, our results suggest that the h.c.p, structure of beryllium has no Be atoms directly bonded to other Be atoms. The structure is held together through a three-dimensional network of bonds between the NNM and Be atoms as well as between different NNM. The topological analysis thus reveals that the beryllium structure has important interactions connecting Be atoms of different basal plane layers. The breaking of these interactions when forming a surface may explain the abnormally large expansion of the inter-layer distance in the beryllium surface structure.

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Section: Calculationsmentioning

confidence: 87%

Experimental evidence for the existence of non-nuclear maxima in the electron-density distribution of metallic beryllium. A comparative study of the maximum entropy method and the multipole refinement method

Iversen¹,

Larsen²,

Souhassou³

et al. 1995

Acta Crystallogr Sect B

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“…Wang & Klein, 1981;Yin & Cohen, 1983) and experimental (Spackman, 1986) valence densities. In contrast, the MEM density of Be based on single-crystal data (Larsen & Hansen, 1984) is quite smooth and such fine features do not appear (Takata, Sakata, Kumazawa, Larsen & Iversen, 1994;Iversen, Larsen, Souhassou & Takata, 1995). In order to gain an insight into the cause of the difference in the quality of the MEM densities, the observed structure factors, Fob s, are listed in Table 1 together with the calculated ones, FME M, corresponding to Fig.…”

Section: The Mem Charge Density Of Si Obtained Previouslymentioning

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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“…On the other hand, MEM is tolerant to noise and provides only positive electron density, which enables us to determine the precise position of light elements such as hydrogen. In other MEM application examples, it has been used to extract the decay constant distribution from fluorimetry [8] and to analyze X-ray scattering [9,10], neutron diffraction [11,12] and scattering [13,14]. In such applications it provides very detailed structural information.…”

Section: Introductionmentioning

confidence: 99%

Application of Maximum Entropy Method to Semiconductor Engineering

Yonamoto

2013

Entropy

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Abstract:The maximum entropy method (MEM) is widely used in research fields such as linguistics, meteorology, physics, and chemistry. Recently, MEM application has become a subject of interest in the semiconductor engineering field, in which devices utilize very thin films composed of many materials. For thin film fabrication, it is essential to thoroughly understand atomic-scale structures, internal fixed charges, and bulk/interface traps, and many experimental techniques have been developed for evaluating these. However, the difficulty in interpreting the data they provide prevents the improvement of device fabrication processes. As a candidate for a very practical data analyzing technique, MEM is a promising approach to solve this problem. In this paper, we review the application of MEM to thin films used in semiconductor engineering. The method provides interesting and important information that cannot be obtained with conventional methods. This paper explains its theoretical background, important points for practical use, and application results.

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A direct investigation of thermal vibrations of beryllium in real space through the maximum-entropy method applied to single-crystal neutron diffraction data

Cited by 36 publications

References 3 publications

Experimental evidence for the existence of non-nuclear maxima in the electron-density distribution of metallic beryllium. A comparative study of the maximum entropy method and the multipole refinement method

Experimental evidence for the existence of non-nuclear maxima in the electron-density distribution of metallic beryllium. A comparative study of the maximum entropy method and the multipole refinement method

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

Application of Maximum Entropy Method to Semiconductor Engineering

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