The probability distribution of X-ray intensities. II. Experimental investigation and the X-ray detection of centres of symmetry

Howells, E.; Phillips, D. C.; Rogers, Dan

doi:10.1107/s0365110x50000513

Cited by 312 publications

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“…Nous pouvons donc exclure le groupe spatial B2. Pour lever l'ambigu'/t~ entre les deux autres groupes possibles (B2/m et Bm), nous avons appliqu6 la m6thode de Howells, Phillips & Rogers (1950); la r6partition statistique des facteurs de structure normalis6s de la strate hkO est en effet sensible ~ la pr+sence d'un centre de sym6trie (International Tables for X-ray Crystallography, 1967). Les r~sultats obtenus avec les 186 r~flexions de cette strate n'6tant pas probants, nous avons adoptb provisoirement le groupe spatial B2/m.…”

Section: D~termination De La Structureunclassified

Etude structurale des oxysulfures de chrome(III) et de terres rares. II. Structure de l'oxysulfure CeCrOS2

Dugué¹,

Voyan²,

Villers³

1980

Acta Crystallogr Sect B

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CeCrOS 2 is monoclinic, space group B2/m, with unit-cell constants a = 11.518 (6), b = 7.951 (4), c = 3.703 (4) A, y = 90.07 (6) °, z = 4, O m : 5.4 (1), D x = 5-33 Mg m -3. The crystal structure has been determined from four-circle diffractometer data using Patterson and Fourier syntheses and refined by the least-squares method to a final R value of 0.025 for 778 observed structure factors. The Ce atom is nine-coordinated in tricapped trigonal prisms containing three O and six S atoms; the Cr atom is six-coordinated in two types of octahedra: [CrS 6] and [CRO254]. The structure shows a three-dimensional arrangement and can be regarded as being built up from layers of [CeO3S 6] polyhedra alternating with layers of [CrOSs I octahedra.

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Section: D~termination De La Structureunclassified

Etude structurale des oxysulfures de chrome(III) et de terres rares. II. Structure de l'oxysulfure CeCrOS2

Dugué¹,

Voyan²,

Villers³

1980

Acta Crystallogr Sect B

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“…)N(Z) (Hargreaves, 1955). When h is odd the geometrical structure factor becomes zero; the heavy atom then makes no contribution to the reflexions and the intensity distribution, determined by the lighter atoms, is (1)N(Z) (Wilson, 1949;Howells et al, 1950).…”

Section: (Iii) Heavy Atom At Special Position (0 O) or (0 ½) Onmentioning

confidence: 99%

“…Centrosymmetrical and non-centrosymmetrical structures give rise to different distributions, represented by (1)N(Z) and (1)N(Z), when the unit cell contains a sufficient number of atoms distributed effectively at random (Wilson, 1949;Howells, Phillips & Rogers, 1950). Hargreaves (1955) has shown that the crystal symmetry modifies the intensity distribution in a more complex fashion when the scattering power of a single heavy atom in a general position in the unit cell dominates that of the remaining lighter atoms.…”

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confidence: 99%

The probability distribution of X-ray intensities: atoms in special positions

Hargreaves¹

1956

Acta Cryst

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Useful information about the symmetry of a structure may be provided by the probability distribution of the intensities of its X-ray reflexions. Centrosymmetrical and non-centrosymmetrical structures give rise to different distributions, represented by (1)N(Z) and (1)N(Z), when the unit cell contains a sufficient number of atoms distributed effectively at random (Wilson, 1949;Howells, Phillips & Rogers, 1950). Hargreaves (1955) has shown that the crystal symmetry modifies the intensity distribution in a more complex fashion when the scattering power of a single heavy atom in a general position in the unit cell dominates that of the remaining lighter atoms. He has considered the intensity distributions of zones of reflexions which provide information about the symmetry of the nine oblique and rectangular plane groups and finds that they are of three different types. These In a recent paper Collin (1955) considers the effect on the intensity distributions of atoms in fixed positions, an atom in a fixed position being defined as one which has no variable position parameter. Collin shows that the (1)N(Z) and (i)N(Z) distributions are seriously modified when the atoms in fixed positions are heavy compared with the other atoms in the structure, and that the distribution curve for centrosymmetrical structures becomes (max.)N(Z) if the atom in the fixed position is very heavy. Collin states that it is difficult to relate his own work on atoms in fixed positions to that of Hargreaves, which is concerned with atoms in general positions, because in one case (Collin) the reflexions are divided into groups, each with its characteristic N(Z) function, whilst in the other (Hargreaves) all reflexions are considered together.The reason why it is possible to consider all' reflexions together (excluding, in general, those for which h or k = 0) when the heavy atoms are in general positions is that, for each of the nine plane groups examined, all types of h/c reflexions give rise to the same N(Z) distribution. For example, in the plane group ping the reflexions may be divided into two groups--those with h even and those with h odd; the geometrical structure factor is 4 cos 2zhx.eos 2=ky for the first group and --4 sin 2=hx.sin 2=ky for the second group: but for an atom in a general position both expressions lead to the same distribution, namely (CC)N(Z).It is implicit within the earlier paper by the author that reflexions of types h0 and Ok must, in general, be considered separately from those of type hk. Thus if hk reflexions have a geometrical structure factor of the form It is evident, therefore, that making h or k = 0 may modify the geometrical structure factor so as to produce a different type of distribution. Similar modifications of the geometrical structure factor, giving different N(Z) distributions, are obtained by making x or y = 0, t, ½ etc., and in the following it will be shown that the work of Hargreaves (1955) can easily be extended to cover single heavy atoms in special (including fixed) positions. It will be see...

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“…Stefanidis & Nord chose the noncentrosymmetric space group because the statistical N(z) test after Howells, Phillips & Rogers (1950) gave an unequivocally acentric result. Hoggins et al (1983), however, report that their data agree closely with a centrosymmetric distribution.…”

Section: (Q) Diiron(ii) Diphosphate Fe2p207mentioning

confidence: 99%

How to avoid unnecessarily low symmetry in crystal structure determinations

Baur¹,

Тиллманнс²

1986

Acta Crystallogr Sect B

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A check of recent volumes of Acta Crystallographica and Crystal Structure Communications shows that about 3% of all recently published crystal structures were described with too low symmetry. Three categories of error are recognized: (1) both Laue class and crystal system are wrong; (2) only the Laue class is wrong; (3) Laue class and crystal system are correct, but an inversion center is missing. Category (1) cases can be most easily diagnosed by calculating the reduced cell and its Niggli matrix. Category (3) cases are most easily recognized during full-matrix leastsquares refinement from singularities, high correlations between parameters and poor convergence. The omission of an inversion center will not result in a singular matrix when all atoms in the centrosymmetric cell are in special positions which become general positions in the noncentrosymmetric cell. Once the refinement is completed structures with a missing inversion center can be recognized by unusually high e.s.d.'s, especially for the highly correlated parameters, and by large distortions of observed bond * Present address: Mineralogisches Institut, Universitiit Wiirzburg, Am Hubland, D-8700 Wiirzburg, Federal Republic of Germany. distances and angles from accepted values. Such distortions often show up as splitting of the centrosymmetrically related atomic positions into positions whose average is close to the true centrosymmetric positions. A mechanical application of the R-ratio test should be avoided; it can easily lead to wrong conclusions. Proof of higher symmetry must be obtained from the diffraction data. The presence of more than one effective formula per asymmetric unit should always be reason to check for higher symmetry. Cases in all three categories can be checked for the occurrence of higher symmetry by searching for regularities in bond lengths and angles or in the positional coordinates of the atoms. The most systematic way for such searches is the simple, but powerful method of topological analysis of crystal structures. For six structures which previously had been described with too low symmetry higher-symmetry descriptions are provided.

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The probability distribution of X-ray intensities. II. Experimental investigation and the X-ray detection of centres of symmetry

Cited by 312 publications

References 0 publications

Etude structurale des oxysulfures de chrome(III) et de terres rares. II. Structure de l'oxysulfure CeCrOS2

Etude structurale des oxysulfures de chrome(III) et de terres rares. II. Structure de l'oxysulfure CeCrOS2

The probability distribution of X-ray intensities: atoms in special positions

How to avoid unnecessarily low symmetry in crystal structure determinations

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