Solving the crystal structures of zeolites using electron diffraction data. II. Density-building functions

A density‐building function is used to solve the crystal structures of zeolites from electron diffraction data using both two‐ and three‐dimensional data sets. The observed data are normalized to give unitary structure factors |Uh|obs. An origin is defined using one to three reflections and a corres...

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Published in:Acta crystallographica. Section A, Foundations of crystallography Vol. 64; no. 2; pp. 295 - 302
Main Authors: Gilmore, Christopher J., Dong, Wei, Dorset, Douglas L.
Format: Journal Article
Language:English
Published: 5 Abbey Square, Chester, Cheshire CH1 2HU, England International Union of Crystallography 01.03.2008
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Abstract A density‐building function is used to solve the crystal structures of zeolites from electron diffraction data using both two‐ and three‐dimensional data sets. The observed data are normalized to give unitary structure factors |Uh|obs. An origin is defined using one to three reflections and a corresponding maximum‐entropy map, qME(x), is calculated in which the constraints are the amplitudes and phases of the origin‐defining reflections. Eight strong reflections are then given permuted phases and each phase combination is used to compute P(δq) = ∫Vδq(x)2/qME(x)dx, where δq(x) is the Fourier transform of |Uh|obsexp(i) − |Uh|MEexp(i), is the permuted phase for reflection h and is the phase angle for reflection h predicted from the Fourier transform of qME(x). The 64 phase sets with minimum values of P(δq) are subjected to entropy maximization and, following this procedure, those with the five highest log‐likelihood gains are examined. Sometimes auxiliary potential histogram information is also used. The method worked routinely with seven zeolite structures of varying complexity and data quality, but failed with an eighth structure.
AbstractList A density‐building function is used to solve the crystal structures of zeolites from electron diffraction data using both two‐ and three‐dimensional data sets. The observed data are normalized to give unitary structure factors |Uh|obs. An origin is defined using one to three reflections and a corresponding maximum‐entropy map, qME(x), is calculated in which the constraints are the amplitudes and phases of the origin‐defining reflections. Eight strong reflections are then given permuted phases and each phase combination is used to compute P(δq) = ∫Vδq(x)2/qME(x)dx, where δq(x) is the Fourier transform of |Uh|obsexp(i) − |Uh|MEexp(i), is the permuted phase for reflection h and is the phase angle for reflection h predicted from the Fourier transform of qME(x). The 64 phase sets with minimum values of P(δq) are subjected to entropy maximization and, following this procedure, those with the five highest log‐likelihood gains are examined. Sometimes auxiliary potential histogram information is also used. The method worked routinely with seven zeolite structures of varying complexity and data quality, but failed with an eighth structure.
A density-building function is used to solve the crystal structures of zeolites from electron diffraction data using both two- and three-dimensional data sets. The observed data are normalized to give unitary structure factors |U(h)|(obs). An origin is defined using one to three reflections and a corresponding maximum-entropy map, q(ME)(x), is calculated in which the constraints are the amplitudes and phases of the origin-defining reflections. Eight strong reflections are then given permuted phases and each phase combination is used to compute P(deltaq) = integral(V)deltaq(x)(2)/q(ME)(x)dx, where deltaq(x) is the Fourier transform of |U(h)|(obs)exp(i\phi_(perm)h - |U(h)|(ME)exp(i\phi_(ME)h), phi_(perm)h is the permuted phase for reflection h and phi_(ME)h is the phase angle for reflection h predicted from the Fourier transform of q(ME)(x). The 64 phase sets with minimum values of P(deltaq) are subjected to entropy maximization and, following this procedure, those with the five highest log-likelihood gains are examined. Sometimes auxiliary potential histogram information is also used. The method worked routinely with seven zeolite structures of varying complexity and data quality, but failed with an eighth structure.A density-building function is used to solve the crystal structures of zeolites from electron diffraction data using both two- and three-dimensional data sets. The observed data are normalized to give unitary structure factors |U(h)|(obs). An origin is defined using one to three reflections and a corresponding maximum-entropy map, q(ME)(x), is calculated in which the constraints are the amplitudes and phases of the origin-defining reflections. Eight strong reflections are then given permuted phases and each phase combination is used to compute P(deltaq) = integral(V)deltaq(x)(2)/q(ME)(x)dx, where deltaq(x) is the Fourier transform of |U(h)|(obs)exp(i\phi_(perm)h - |U(h)|(ME)exp(i\phi_(ME)h), phi_(perm)h is the permuted phase for reflection h and phi_(ME)h is the phase angle for reflection h predicted from the Fourier transform of q(ME)(x). The 64 phase sets with minimum values of P(deltaq) are subjected to entropy maximization and, following this procedure, those with the five highest log-likelihood gains are examined. Sometimes auxiliary potential histogram information is also used. The method worked routinely with seven zeolite structures of varying complexity and data quality, but failed with an eighth structure.
A density-building function is used to solve the crystal structures of zeolites from electron diffraction data using both two- and three-dimensional data sets. The observed data are normalized to give unitary structure factors |U(h)|(obs). An origin is defined using one to three reflections and a corresponding maximum-entropy map, q(ME)(x), is calculated in which the constraints are the amplitudes and phases of the origin-defining reflections. Eight strong reflections are then given permuted phases and each phase combination is used to compute P(deltaq) = integral(V)deltaq(x)(2)/q(ME)(x)dx, where deltaq(x) is the Fourier transform of |U(h)|(obs)exp(i\phi_(perm)h - |U(h)|(ME)exp(i\phi_(ME)h), phi_(perm)h is the permuted phase for reflection h and phi_(ME)h is the phase angle for reflection h predicted from the Fourier transform of q(ME)(x). The 64 phase sets with minimum values of P(deltaq) are subjected to entropy maximization and, following this procedure, those with the five highest log-likelihood gains are examined. Sometimes auxiliary potential histogram information is also used. The method worked routinely with seven zeolite structures of varying complexity and data quality, but failed with an eighth structure.
A density-building function is used to solve the crystal structures of zeolites from electron diffraction data using both two- and three-dimensional data sets. The observed data are normalized to give unitary structure factors tiles. An origin is defined using one to three reflections and a corresponding maximum-entropy map, qmE(x), is calculated in which the constraints are the amplitudes and phases of the origin-defining reflections. Eight strong reflections are then given permuted phases and each phase combination is used to compute P(Sq)= fv8q (x)2/qmE(x) dx, where Sq(x) is the Fourier transform of I Uhrbs exP(i(PPhenn) - I 1/.1' exp(iyon), caLerm is the permuted phase for reflection h and yor is the phase angle for reflection h predicted from the Fourier transform of qmE(x). The 64 phase sets with minimum values of P(Sq) are subjected to entropy maximization and, following this procedure, those with the five highest log-likelihood gains are examined. Sometimes auxiliary potential histogram information is also used. The method worked routinely with seven zeolite structures of varying complexity and data quality, but failed with an eighth structure.
A density-building function is used to solve the crystal structures of zeolites from electron diffraction data using both two- and three-dimensional data sets. The observed data are normalized to give unitary structure factors |U h |obs. An origin is defined using one to three reflections and a corresponding maximum-entropy map, q ME(x), is calculated in which the constraints are the amplitudes and phases of the origin-defining reflections. Eight strong reflections are then given permuted phases and each phase combination is used to compute P([delta]q) = [symbol omitted] V [delta]q(x)2/q ME(x)dx, where [delta]q(x) is the Fourier transform of |U h |obsexp(i) - |U h |MEexp(i), is the permuted phase for reflection h and is the phase angle for reflection h predicted from the Fourier transform of q ME(x). The 64 phase sets with minimum values of P([delta]q) are subjected to entropy maximization and, following this procedure, those with the five highest log-likelihood gains are examined. Sometimes auxiliary potential histogram information is also used. The method worked routinely with seven zeolite structures of varying complexity and data quality, but failed with an eighth structure. [PUBLICATION ABSTRACT]
Author Gilmore, Christopher J.
Dorset, Douglas L.
Dong, Wei
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Snippet A density‐building function is used to solve the crystal structures of zeolites from electron diffraction data using both two‐ and three‐dimensional data sets....
A density-building function is used to solve the crystal structures of zeolites from electron diffraction data using both two- and three-dimensional data sets....
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SubjectTerms Crystal structure
Crystallography
density building
Diffraction
electron crystallography
Entropy
entropy maximization
Fourier transforms
Zeolites
Title Solving the crystal structures of zeolites using electron diffraction data. II. Density-building functions
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