An aid to the structure analysis of incommensurate phases

McConnell, J. D. C.; Heine, Volker

doi:10.1107/s0108767384000921

Cited by 36 publications

(11 citation statements)

References 7 publications

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“…Let us have n modulation sine waves for some atom in the structure: We note that (16) is in full agreement with equation (1.1) of McConnell & Heine (1984), which states that incommensurate structures can always be expressed in the form of two different components, one modulated with cos q. r and the other with sin q. r.…”

Section: Combination Of Several Displacement Wavessupporting

confidence: 49%

Structure analysis of displacively modulated molecular crystals

Petřı́ček¹,

Coppens²,

Becker³

1985

Acta Cryst Sect A

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A structure-factor formalism for incommensurate modulated structures is derived. It allows for several simultaneous translational and rotational displacements of molecules or molecular segments, which are considered as being rigidly displaced. It incorporates treatment of several displacement waves in the crystal and makes full use of the four-dimensional symmetry description of de Wolff, Janssen & Janner [Acta Cryst.

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Section: Combination Of Several Displacement Wavessupporting

confidence: 49%

Structure analysis of displacively modulated molecular crystals

Petřı́ček¹,

Coppens²,

Becker³

1985

Acta Cryst Sect A

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“…If we suppose a modulation of the Cle(1) atom having the form: An interesting feature of this model is to exhibit the components C 1 and C 2 introduced in a two-component model of an incommensurate structure (McConnell & Heine, 1984) C 1 corresponds to octahedra rotated by an angle of 5 o (the rotational component of the modulation) and C 2 describes planes shifted by approximately 0.15 ,/k (the translational component of the modulation). We can easily deduce the symmetry of C 1 and C 2 by considering the elements of the superspace group leaving invariant hyperplanes defined by r= 0 and r= n/2 respectively .…”

Section: Dandorder Of the Octahedra Layersmentioning

confidence: 99%

Study of the modulated phase of (C₃H₇NH₃)₂CdCl₄ by single-crystal X-ray diffraction

Doudin¹,

Chapuis²

1988

Acta Crystallogr Sect B

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Bis(n-propylammonium) tetrachlorocadmate shows a modulated phase stable between two orthorhombic phases on the temperature scale. The superspace group could be determined unambiguously on the basis of the selection rules. In addition to Bragg main reflections, 1137 first-order and 107 second-order satellites with an intensity larger than 3a were measured. The refinement revealed a displacive modulation of harmonic type. The CdCI 6 octahedra show modulation amplitudes larger than 0.2 A. To a first approximation, the dynamics of the octahedra can be analyzed in terms of a rotation about the a axis and a translation along the c axis. The low-temperature phase can be considered as a lock-in of the modulated one with a change of q from 0.42b* to 1 b*. In this phase the rotation of the octahedra about a is frozen and a unique hydrogen-bonding scheme is adopted between the alkylammonium ion and the C1

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“…Another approach to the description of the structure and symmetry of an IC modulated phase has been given in McConnell & Heine (1984). We consider that this approach, which differs considerably from the present formulation and the standard superspace formalism, is not the most appropriate way to describe IC structures from a crystallographic viewpoint.…”

Section: Superspace Symmetrymentioning

confidence: 99%

“…Furthermore, the generalization of this approach to non-sinusoidal distortions becomes unpractical, because each new harmonic in the modulation requires two additional 'component structures' to be considered (in general with new symmetries). In addition, while superspace formulation includes those IC structures with more than one independent modulation wave vector (see Appendix), the approach in McConnell & Heine (1984) cannot be directly extended to this general case.…”

Section: Superspace Symmetrymentioning

confidence: 99%

On the structure and symmetry of incommensurate phases. A practical formulation

Perez-Mato¹,

Madariaga²,

Zúñiga³

et al. 1987

Acta Cryst Sect A

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PRECISE DIFFRACTION ANGLES BY FFTand the counter slit, t is the thickness of the specimen and 2A is the breadth of the spot on which the incident beam falls. In this case, where the vertical diversion of the beam is neglected, both t and A are variable factors in the experiment. If the value of /.1. = 143cm -1, estimated from the mass absorption coefficients of oxygen and silicon (Cullity, 1978) and the density of SiO2, and the goniometer radius of L= 175 mm are entered into (13), AO is found to be less than 9 x 10 -3° for 20 > 90 °. On the other hand, the analytical process of a personal computer is effective to six figures and its precision of calculation is better than 0.001 °. The difference between 20a and 20th is at most 0.02 ° because the goniometer has a precision of 20 =0.01 ° and is scanned with a step width of 0.01 ° in 20. All errors involved in the analysis can therefore be reduced through the mechanical accuracy of the goniometer. Consequently, it is definitely possible by adopting this analytical method to obtain a higher analytical accuracy when the accuracy of the goniometer is improved and the step width is reduced. AbstractThe structural description, symmetry and diffraction properties of incommensurate modulated phases are revised using a real-space framework. The superspace formalism usually employed is reformulated using a practical description where no multidimensional geometrical constructions are needed. The incommensurate structural distortion is described in terms of 'atomic modulation functions' where the internal space is only considered as a continuous label for the cells of the non-distorted structure. Hence, no atomic positions or thermal tensors in a multidimensional space are defined. By this means and with the introduction of the concept of 'atomic modulation factors' a general expression for the structure factor is proposed which constitutes a direct generalization of the standard expression for a commensurate structure. The concept of superspace symmetry is reduced in this approach to a simple relation between the defined atomic modulation functions, which can be 0108-7673/87/020216-11501.50 directly translated by means of the structure-factor expression into the symmetry and extinction rules of the diffraction diagram. The advantages of superspace formalism in the analysis of commensurate modulated phases are also discussed. The use of superspace groups for describing the symmetry of superstructures, contrary to some recent claims, does not formally reduce the number of structural parameters but may often allow some of them to be neglected.

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An aid to the structure analysis of incommensurate phases

Cited by 36 publications

References 7 publications

Structure analysis of displacively modulated molecular crystals

Structure analysis of displacively modulated molecular crystals

Study of the modulated phase of (C₃H₇NH₃)₂CdCl₄ by single-crystal X-ray diffraction

On the structure and symmetry of incommensurate phases. A practical formulation

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An aid to the structure analysis of incommensurate phases

Cited by 36 publications

References 7 publications

Structure analysis of displacively modulated molecular crystals

Structure analysis of displacively modulated molecular crystals

Study of the modulated phase of (C3H7NH3)2CdCl4 by single-crystal X-ray diffraction

On the structure and symmetry of incommensurate phases. A practical formulation

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Study of the modulated phase of (C₃H₇NH₃)₂CdCl₄ by single-crystal X-ray diffraction